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<h2>HL Paper 2</h2><div class="specification">
<p>The table gives data for Jupiter and three of its moons, including the radius <em>r</em> of each object.</p>
<p style="text-align: center;"><img 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BEfwUb+EzMxgJGaF3/65SPjwZvmcWOM/n0/whzfGWebsM83f4tK3j+NQArE0xBOayC5nZaapxxLXp+CL5ysXwfsrxrh4EUeRLmUP/svQnAtjIj82px8dg0Ekttr8LTXy+R1L2Jt41P8/8Epxf8fLJa9y41kWWpff6ZdN7r7RpY124f7FP9/cE3x/weLZe9yI1mW2hT/RanNml2U2AFhRbmU4j+AjLOXjMg/249c73gEktvjMZ3BYvI6Awtr+BDlUor/GvxWo4jIr07Kh5dAILm9BE27nUxed0OWEwIEolxK8Q8AW6k7In+lGO8aS3K7JvPJ65q8jogqyqUU/xFsnLxmRP7JbuRyHRBIbjuAOoHJ5HUCEhZxIcqlFP9FCK6FEZFfm5PProFAcnsNnvZ6mbzuRSzHRwhEufSF+DMoP4lB5kDmQOZA5kDmwDo5UDoYfCH+pQHZd30EKOJsayKQ3CavayKQUR2FQLRHpPgfhfDEdiLyJ3Y5XWtEILltBOpiw5LXixE2sbtRLqX4T0zaUa5F5B9lP+2MQyC5HYd9z5WT157o3st2lEsp/jfIg4j8G4S+fIjJ7ZoUJ69r8joiqiiXUvxHsHHymhH5J7uRy3VAILntAOoEJpPXCUhYxIUol1L8FyG4FkZEfm1OPrsGAsntNXja62XyuhexHB8hEOVSin+E2EL9EfkLhXjbUJLbNalPXtfkdURUUS6l+I9g4+Q1I/JPdmOJ5X777dfPfwvj+++/e40JfO2HMWe15PZ4pD98+OULPt+8+eb4RTYsJq8bAOXjZgSiXErxb4bwugMj8q8b0TjPv/76q5c///zj5dOnTy+IAv99cLWPHz++vH37rb6eck1u+8L8/v3PL2ce5hRN8iok8vosAlEupfg/i+wF5kfkX8D1qV1E6K3480vA2UKR3PZLEQ55I976iSh57cfr3SxHuXQp8WejJRDeuo5qnOx5m1Pj7c1u6Oq/8jUiv3dMYAm2f//91+ZSiCZj8ZUN9wwRRbz5ide3H3/84fPPvvhUGoMw2Lf8UUIxglv/szg+6AN2LS3CvmXunjHv3v30UvLJx8A430Yc5uTDCF61dl7XQiDKpduLv6eZTYkDwUotIr9njAg/Is7aW+Iv4bfjem+82Mc3L+xwTw7IF4kE4q7GM8bYQyjzvC2N73kdwW0pHmLnoATvWy3Cfmve3udwAj5e/Dnc46sOmPDpD3lwS5/leO/6z4yfhddnYsi5cyAQ5VKKv+Mnxd8B8sBXhEA4kngS0sgUYuDfvNiYe/zkKrsSdS/YrFnrU2xeFIjXHhCiWI/ujwr76HVq9vQLj8fNz9nC3o9/9Dv5Rk4h/vDpxZ9c47ltvo8Dgh9jx/e+n4HX3jGm/XMQiHLp0uKvtzQLoe1jEyBwCltvoVztz/qM54RPYwNnPB/10c+mZvvtJqe3CDYYxmB/toZfZzY2TcQRbFh7S/zBGh5s01wvshoD3pYH9cN1qd8+xx/s4psfiy82P5jHWoqJOfajt0f6Wt565cdRV9Yd3cCH2tlqcFPD3s+Hm5IAg3mpX/OZJ17xC/9sY67PN8b7msffUW0GXkfFnusei0CUS7cQf7uhU+SAIUGy4g/kbBZ2Y+CePm3sEnttLhIpbRSyeyx9z1mLyH/O6vZsYbOFCZux8JNVNnjLk/p1RbzhRTzQjw1vR+P9tST+cMyaXvyVA97GDN9HcavY4baEmZ6XriXsS+Pog08r9OQFvGOjpTHWiz8HdFvj2FGuttptWfuZMaN5fcb3nDsXAlEu3UL8faFT/BIJntkTP5uFHc8zvd2JUgRHb/jaNEb85Ct/tq4R+Vvznn0ubLbEX3jqgMV48MXvrbkIA/NLB4ia/yUBioSMfCAvZmyjuBUW1JFqQX1b1xL2tTnil+se4cdmSfypaVvjjFOupvjXmMhnV0Qg2iNuIf4Utm36GZe+mvhLDADPf3Rg0KaxJVJ2/bPvI/J7+7EHGzZ2YSxB57sOBDVfGbdXnEsClG/+NZTLz0pCWh75T28J+3+elu/ICXjeK84l8eewkuJfxjl710OAuim1FP/Km7/Ev/ZWv0fgSgSc0ReR33vtZ7DhMKADVs1PHRSOePNnHdb0v/T4n55r/pz9bBS3xEldsH6tPkp47BX/o9/8S7kC5y35VoqnR99IXnvEkzbHIRDl0tTi798qvCBwevc/OVLYeguUeDPPNnvyr735s0nhg59vbT0jcNZOz/uI/J5rYrsVG8RVnMmnLcGFG+ZYbpjDp6VFAkRuWJvY0gGjxe7ZY0ZxS5zg9IhgRtiXsPN5cMS/+XubigWeZ2kjeZ0Fg/TjGASiXJpa/Pl5ns2YzYJGcdoCZSMgMESGpu8SEok/G5R+lpfYy6a+vxr4/38jtALCPfO1Bj8N44P+j4haBU72R1wj8nv70oqNxgljrvBee6MEfy/SxANfpX4fayRAzGVt5YtyquaLt33m91HcEiMcqNb2xBxh723Aha13PYcT1Z/6oiv++bH6xQI7NLguHfoim2f0j+T1jPhyjfMQiHJpavFnk6D4cZ4P9xJtQYd42+ds/tqQJP4ScNnQxo4NL/7a7BmrtdiEtAZX7KlJuKxNPZvlis8jWoSN8LSYgTuHLHyFPx0EevkdCZDPOfzB31nbKG7BoySswkm1V8Iuwl5zj7xGPtp9Awz9AeFIHx6xNZLXR/zNOfMiEOXS1OL/LJzagHoLybN+9p4fkd973bTfH4Hktj/GI1ZIXkegvuaaUS6l+K/J9xdRReR/MSi/XBKB5PaStG06nbxuQpQDGhGIcinFvxHAKw+LyL9yTOn7/yGQ3K6ZCcnrmryOiCrKpaXFfwTQM64ZkT+jr+nTPgSS2314XWV08noVpub3M8qlFP/5uXvaw4j8pw2ngeEIJLfDKejiQPLaBdZbGo1yKcX/BukQkX+D0JcPMbldk+LkdU1eR0QV5VKK/wg2Tl4zIv9kN3K5Dggktx1AncBk8joBCYu4EOVSiv8iBNfCiMivzcln10Agub0GT3u9TF73IpbjIwSiXErxjxBbqD8if6EQbxtKcrsm9cnrmryOiCrKpRT/EWycvGZE/slu5HIdEEhuO4A6gcnkdQISFnEhyqUvxJ9B+UkMMgcyBzIHMgcyB9bJgdI55gvxLw3IvusjQBFnWxOB5DZ5XROBjOooBKI9IsX/KIQnthORP7HL6VojAsltI1AXG5a8Xoywid2NcinFf2LSjnItIv8o+2lnHALJ7Tjse66cvPZE9162o1xK8b9BHkTk3yD05UNMbtekOHldk9cRUUW5lOI/go2T14zIP9mNXK4DAsltB1AnMJm8TkDCIi5EuZTivwjBtTAi8mtz8tk1EEhur8HTXi+T172I5fgIgSiXUvwjxBbqj8hfKMTbhpLcrkl98romryOiinIpxX8EGyevGZF/shtLLPfbb79+/lsY33//3WtMpb6zgk1u+yD94cMvrzx//fVXL3///VefRSpWk9cKOJM8evfup9ccefPmmyE50gpDlEsp/q0IXnhcRP6FQxrmOmLw559/vHz69OmFov/99/+8lPrOcjC5PR5pxP7t229fDXMI+PHHH45fZMNi8roB0ODH5Aj1TyNH3r//ebBH8fJRLqX4x5gt8yQif5kABwWCQCD+tpX67POj75PboxH9v82cjR1sdbA7fpW6xeS1js/opx8/fvws/vzyl+I/mhG3PoVLEdkPb2x3a2dvJAgiIsi6cMDJeKtRQOKLjZfvvRs+lnzjTU85E/nP2z/zbSv12ec97s/mthQDP3+2vB2zYfJPJcKWeb1b5Bu8yw+u1heekYPsFfZXgN6+Wvsz8Gr9efae2iAmf1j2doW3uCnVp5/z7PdHcoQ1bS7j96wtyqXl3/xJNoI/I4muRn4Pf9kwEUyKnWZ/Ho/Wk/DbAqKweh4AVLg+LzjBI+ryhefkj+IhBm1Q9iBZ6oviPbI/Kuwj16jZAi98aBF/f9iCg55vTJFv7AnkqPIL7uwhD851GLBveDUcjn42mtcj49EeQEw18QdrywM+wKG4ONIn2Xo0R8gd5bxqXzZnu0a5tLz4QwzB+01+NoJ6+hOR32NNcKaAbaNI2OijxjNf4BSX/k0tmvdIv+xK1H1esGatj2eImBX+Ut8jvj0y50xurX/UlcQbzLQR2jH2XnXIVQ3cfK7o2TPXLd/INZ+Pto+DnnKPez/2Gd9a547itdW/PePADzyJqSb+5INwl30OBMzjemR7NkfwVXsWtvyvgEf6+qytKJduK/468QGMP20+C/Zs8yPye/hZ2tC3xB/84cM2/WJjRdY+xyZr+UZBlvo1jucUK3bBxY/FF79ByX/NYZ4+zNe9rnqj1Jo9r6w5ohG3sGPj6yH+2C8JL/iW+oXDlm/M9fnGHHsQIU/Alj7y5ew2itej49Rhm0MUMfnasuuRQ15EVXPRvFE5gl/4qhyxvwzamGa4j3LpluJPknHCVFGToABEIq3YIvJ7xEpRsGGqGPgO1jVBZDPWKVo+iRNxpH5dVXyWM2x4Oxrvr8z3nOstw280CIXflLy9Ud/P5DaKsUX8mcs4y1eJd78GfFqhJy+wA38treQb+ejFH87BstVuy9rPjJmB12f8Z65+xmcvoI6JydeWXYPciN78bd7YOdzfNUc8DtH3KJduJ/7a4L0YkUA+8SIwr9Yfkd8rDgqVNfVhA641Fb1+2mOjgAvmR+IvewgD81uERHO4lsQ/2qBS/C1y/74vCey/R728HgiVE1w5JLaIrfjlukf48aHkG+um+JcYOrYP7IVzVFt2RT+G3KCuyRW4r7U75kgND/sM/ErtduKvE74XFZIHkFo2oxKQM/dF5PfwGRwRbgk5a1DAWz8LC398laBzb+1E/jKOjWZPK4m/Dob+7STFv45sSWD9DL+x8xxcWw/cEoG99VnyjTUlSvJT+8Je+5p/9PXMmj3ad+zxMmVrssR/aV140MGfq34B9C9rpbl3y5ESBqW+KJdS/P8fLYnPLMVfIvHRvoj8R+3V5lHwFL5tj2ys8MEb2lbTQYGrX7c2tyT+jGdNv9FgF/sztjO5jeIvCawfWxJ6ceAPW34uuSCeWWtPjZZ8K+UKnLfkm/et1/cZeH0mNrAkhtLHH7xq6+jQoH9GjMbeMUciLHx/lEu3E3+93ZU2+Na3EA/u7N8j8nv4zWbri3tL/P1bAn5tCS4CwFoUvRpz+LQ0CY+dzzxywPdJeFrsnj3mTG6j2EoC68eSE4yzTRzUxN/nAXWLndYDQMk3bxOfJB7Wv5H3M/B6ZPwS8RrXOoBZbtVX88XzeZccqWFin0W5tLz4S3jshk6ysMmTkDSSBYDsGAve1e8j8nvEBYZsuMKWNRDP2s/+4kgbA1f4qZ32sVfiC25L/T5WCY8fy/dSbtR88bbP/H4mt1FcJYH1Y8kH/6uKsPZj9Z3n5I5v1Gstn+z4km9wCW7YoeFb6dBn7Zx9PwOvR8YMxsSkGi/ZpibJEb088KK2dfC+c46UMCz1Rbm0tPiXfnrSqZKkARQ+sxV+icBn+iLyn7FZmwu2FnsE2TZhbw8IbMSaw4Zd2ySsrUfvI/Gnnw1HucEVf2dtZ3NbwqEksNrsLXZwylhhC84t/zcdpTVb+0q+MReBkR9cWw8Tres+O24GXp+Nwc5XPti6Vp/NEfqUI+wH9pm1d+T9VXOkFYMol5YW/1ZwVh8Xkb963HeIL7ldk+XkdU1eR0QV5VKK/wg2Tl4zIv9kN3K5Dggktx1AncBk8joBCYu4EOVSiv8iBNfCiMivzcln10Agub0GT3u9TF73IpbjIwSiXErxjxBbqD8if6EQbxtKcrsm9cnrmryOiCrKpRT/EWycvGZE/slu5HIdEEhuO4A6gcnkdQISFnEhyqUU/0UIroURkV+bk8+ugUByew2e9nqZvO5FLMdHCES5lOIfIbZQf0T+QiHeNpTkdk3qk9c1eR0RVZRLKf4j2Dh5zYj8k93I5TogkNx2AHUCk8nrBCQs4kKUSyn+ixBcCyMiv6Fm7ksAABSGSURBVDYnn10DgeT2Gjzt9TJ53YtYjo8QiHIpxT9CbKH+iPyFQrxtKMntmtQnr2vyOiKqKJe+EH8G5ScxyBzIHMgcyBzIHFgnB0qHji/EvzQg+66PAEWcbU0EktvkdU0EMqqjEIj2iBT/oxCe2E5E/sQup2uNCCS3jUBdbFjyejHCJnY3yqUU/4lJO8q1iPyj7KedcQgkt+Ow77ly8toT3XvZjnIpxf8GeRCRf4PQlw8xuV2T4uR1TV5HRBXlUor/CDZOXjMi/2Q3crkOCCS3HUCdwGTyOgEJi7gQ5VKK/yIE18KIyK/NyWfXQCC5vQZPe71MXvciluMjBKJcSvGPEFuoPyJ/oRBvG0pyuyb1yeuavI6IKsqlFP8RbJy8ZkT+yW4ssdxvv/36+W9hfP/9d68x/fnnH5/7Pnz45dQ4k9s+cMMj2H799Vcvf//9V59FKlaT1wo4kzyi/uHpzZtvXj5+/DiJV/92I8qlFP9/Y7VcT0T+coGeEBBigNh/+vTpteh///0/L2/ffvvCoYANAKx5dlZLbo9HGrGHUxqHgB9//OH4RTYsJq8bAA1+TN3bw/+IHGmFIMqlU8VfgOGMPoA24mTdCtyz49hE3r//+VkzT82PyH/KaE5+FQhyWo085nCQ4i9ErnlF8Hmbo2502Ds7kqzZsxHftx458u7dT6+TOPSTJ7O2KJdOE3/9jMZVjU0S8cc53qZWbHcWfxVFy+FOP6GRC/r0Pk3Djc1H5Z9yEj8o6tIY8lVvh8zTPwfobUC2el/xcUSDU8sZWLTwzIYpfpnf+6DEeqU80n4kX7SRgyXPEH98IybL81lYj+L1kfjI/Vbxa6mtK+QIh37llXLpEezOmBPl0iniTwHhAElSahTX2ZtmyY8efcR2xzd/hF9vTy2iwNiSyPbgBJsSLr8mXFkhU2Hbw6kEoSRc2PU2e8WA3aiwe65J3PClzY+12LARgBIm8oXxdg5Y9RRWuAQfuya+sHHjq/Yj+LSHPPjTYUB5rBjOuo7g9ZHYwBJfW8S/pbaukiNgpT2EHKYeZm1RLp0i/hTSniKn4EgCnNYHoLWxKOEoUpKOMdinX6SU1lOBY4c5+CWB4sp8bRg8Z03b2CREuJ7jq23Wb+7xA5tqLTY09qgrvp7Z4EVxszYx1xoYMk781cY++4wNH67xkTW52qZnUZ9iUy4yjrxSjHDubVpbR9+fzS3+gyHrWgyEQxQ7+DDHHqK4p8/X0LMYqcaoO/iEE9uoe1/btg+/mEfj3o+1tnrdj+B1TyxwD65gw7VF/Ldq60o5wl5lc4T8mbVFuXSK+AOSFcAtkBAOW3AATQDaWPSdMdqAuBcZeq4NmfW0YbHRMAd7jNfGw5r0iUTm6rDAfObY7/TZNflOEWBHNvmOTcXeYgM7Rzd8OLOJlxIPJT8eEQGwVT5Ym/BX6tcYnsOtcsCPhWP8to21FBNY2g95Jf/pZ9yZ7Wxua7H5+rBjwVn1aftr98wp4QnmpX7ZYp54pR7hzzbmqibVz3j8VyNPwJY+u4/oee/rTLyWYgUT4QqWFrvSePoYE9UWz+HgSjlCHsGT3fOj2Ef2R7l0ivhDui82Egen7IcNOWrYkDBLVNh01STuskESaTxjIEobRmnjJ/HwRfOZgw35HSUmfrE2gs987m2zsW/ZsPOOvMevEU08bW2e4AJOKib8tdyVfIcnio65aszZmqexpRwQh36DIgdYa8Y2iluPhbi2NWnHCEP4wmc+8K2Dsh1r7+FTdUs/9QUXtk7teH/PWImUntm6Vp/8b7Wreb2us/DaEh/cUr+11lJbmSM1BB9/FuXSKeJfKjYbijYEW3gUOcmgxCIAFXGpUH2f5mGTD8kpoeA79pijxjOfwGwc2KGVDivY4MNcvf15odtjQ74cfcXHEU2ceEy8L2zw5IgVAvAW3368/Y4wgD/XVuFnvnJAOUEffvq8oJ8cgMcZ2yhuPRbgU8MfLqkvOwZc4X2riV+urAN3rY3xPo/wQ3UtO8rVPbY1t8d1Fl5bYgNLv3f6eS21lTniUTvme5RLp4g/BV/bPClqHFThsZGrQEksEofvKuJSofo+nTSxzYf5ss+V9ZijpjH6zhWftUmwtn0DseO4bxX/mg1v86jvEflH2Y/siJMt8S/NF572QFAaRx/x1fKrNE85AO9qyhmbFzwjB/bal83e11Hc2rjI6S182AO80IuD6NcCvwaxqobts9o9fmnf0LjSy4hyda992Tz6OgOvrTG1iH9LbWWOtCK+b1yUS6eIf3TqUwhswCpsjbUbAgW5V/yxTdFT/GxOVnS16dhNfkv8txJcPtZ+9t+yITyOvkbkH72Ot6cN9RHxVx5szYVXuONq3yq9L/67csCKP2PIM88hdm3+eFsjv4/iVjGDCzUGnrUmjvwY8PYc+DGay7VlLTu/JP6lXIFzfJmljeZ1Dw6t+9pWbYlnv3bmiEdk3/col04Rf1yFWJwgUdQ4DSLQ9Gvj1gmR8TQ2FYqVMTrBS1TshlPr80mnjX+P+OMXdqwIKCYJlE6ujKXx3cbcYuN14sH/gw8jmjgRPpEPvImJf41hLnhHDQ7Z2JUnjMOGt1ObDy52PmPxxffBue+L7J7dP4pbOAX/VjHmMO/f/FXr9qDv8YNPW3OIdOua2GKs9g3Z9jbph1+7jsaOuo7i9ZF4W8V/q7YyRx5Bf3tOlEuniT8usqFL7HGID9+tCDNOp3CNoTAZp81DorIl/thiDiJixz4i/thiw2ODkF9sLN53CkHP8Zkx9sDTYoO1jmz4M6KJpy3xh1940qEJX8HO4ub953lJkNnYS/1+vnLAj5Uv8plcBL+aQHnbZ34fwa0OseT2nsZ4izf81mwwtiTIcAL/LQ37fixcght2aHBdEqYW+73GjOD10VjgsXZQl92W2socEVrHXaNcOlX8jwun3VKp+NtnrzEyIr93dJH4swngkwQWP+hjA6GfD997tkj86bcHvDN8eSZO/Du7ccAST/4qoYVbjx3Y8lxzwJm+ni2qfwRLfnCV3z192WMbn67SIvEHU3soaKmtzJHjWY9yaWnx1wnfiszx0M5vMSJ/fs/Twy0EktsthK75PHm9Jm8zeh3l0rLiz4mfoHu/Qc5ItvcpIt+Py+/XQyC5vR5nLR4nry0o5ZgWBKJcWlb8W0C5y5iI/LvEv3Kcye2a7Cava/I6Iqool1L8R7Bx8poR+Se7kct1QCC57QDqBCaT1wlIWMSFKJdS/BchuBZGRH5tTj67BgLJ7TV42utl8roXsRwfIRDlUop/hNhC/RH5C4V421CS2zWpT17X5HVEVFEupfiPYOPkNSPyT3Yjl+uAQHLbAdQJTCavE5CwiAtRLqX4L0JwLYyI/NqcfHYNBJLba/C018vkdS9iOT5CIMqlFP8IsYX6I/IXCvG2oSS3a1KfvK7J64ioolz6QvwZlJ/EIHMgcyBzIHMgc2CdHCgdOr4Q/9KA7Ls+AhRxtjURSG6T1zURyKiOQiDaI1L8j0J4YjsR+RO7nK41IpDcNgJ1sWHJ68UIm9jdKJdS/Ccm7SjXIvKPsp92xiGQ3I7DvufKyWtPdO9lO8qlFP8b5EFE/g1CXz7E5HZNipPXNXkdEVWUSyn+I9g4ec2I/JPdyOU6IJDcdgB1ApPJ6wQkLOJClEsp/osQXAsjIr82J59dA4Hk9ho87fUyed2LWI6PEIhyKcU/Qmyh/oj8hUK8bSjJ7ZrUJ69r8joiqiiXUvxHsHHymhH5J7uRy3VAILntAOoEJpPXCUhYxIUol1L8FyG4FkZEfm1OPisj8Ntvv37+Q1jff//d66A///zjc9+HD7+UJ3bqTW77AAuPYPv111+9/P33X30WqVhNXivgTPKI+oenN2++efn48eMkXv3bjSiXUvz/jdVyPRH5ywV6QkCIAWL/6dOn16L//ff/vLx9++0LhwI2ALDm2VktuT0eacQeTmkcAn788YfjF9mwmLxuADT4MXVvD/8jcqQVgiiXphZ/NtN37376/FZFEHzfu7lSyO/f//yKFaRhBxt8uN/ztrZnbCs5vcdF5PdeF/72vjk9MufROMiLEp8UMpjxwf/SGNZkPvmkhmgwfm9+av4j11HcEqvefPABLPa8IYMp83pjxX5R2pi1Pj7wYZwaz3ibo1+HPT0768raV2kcfMGppYEtY4nP1w/zqX+bV5aXFvuPjHk0R+Sb9qxH1j5jTpRL04q/NlIKVxuEEoPCVF8LeCRZSfxb5tox2MDW1VpEfs844EobaKsoPDLn0Ri0wbAZ2SaO5bNEgrd92/huc0H/HKC3ATu25/0Ibqk9uLWiykbYevCx/0yyp4734giX4GP9xAYHNnyFM5r2GuUCV+0xPLM87/Xh0fEjeH3EV71MtYg/Y4Ura5EHPmfAWjwwhnrS3v2If1tzHs0RYlFeaY/YWmvU8yiXphV/kkAnKwuaNp7SMzvO3mNLCaRkfWTTwcaIjcDG8sh9RP4jtlrmUAzCnLUlpLW5j8yp2YueseGzAalgudqmZ7U+CUIph9isvE1r6+j7s7nFfx10fPxs5Fuxq37BGd+9jSPwgR+JButok5Zt9g5/SLN9xKD9RQdSzT3rOoLXPbHBG7iCI9cW8Wec58LmDLz5/QIuWmzv8Z2xz+YINogdf9nryLNZW5RLU4q/3gy4lprvJ0G0mRAoyaJTPfMlRNxb8SeBGa8NS78s0MeHeYyn6YSoZyQPza+tTeP14ST/g89nNooCbIW1sKr5sHcOm4h4s3bBv9SvMTzHH8+9npM74lx9rIV/NGyTF8xXY45ijPzS2KOvZ3Nb8x8catgzF/zBknH4bnH0thkj3O0zarvUrzHMkx9w5QWHuXoZsHPwn8b+os2c+9pamn/0dSZeS7GR78IVLIVdaaz6GK856rM5g03iVi0xBh5rtkflCHuEzZEZ931hHOXSlOK/RbiC4iqBsQcCitUmTKv4M88WOoRaO/7NHz8BVgcNkpaE8Alu/R1xH5Hf2xdxY4t5a83WOYgGvMKBGny1FmFJ/Dn8gRU+2CbeNYcx+sA9uafvNn+sjV73rDtDE2+2Dr1fYEV9gKNqh/tag0+LKTb84as2vyT++ODFX/7LH9YFW+p/T/7WfNnzbBZeW3xuFX8wFv/YJVfAl7pT8zUN91s1PSpH8A2e8NnGoFhmuUa5dHnxLwGsjUWEQI6K3Ra5NnMJCIlpNxpvWyKgfsb7xGRzAmytrbEjrxH5vX0S1ns2z71z4Av+WjYJG6/nnmf4CVb4YJvn3T4bfT+KWx83NeZrwY6hHtjoha1qVGJrx/p78cuVdVrmyAbj/WEcP7QfaJzybo9tze1xnYXXltjAEkxbmgST+Ph4HuxBmufYbeHkjjnSgjdjolxaQvxJDpKIj00uiU6r+GsDACzs+MTkO7ZorMk4ks62SEDsmLPvI/J7+yE8xUPLeo/MIT7x0rIGY0r8IVDYwgfbLO+2f4b7Udza2KmVLfx5bg8H1A2+t2zsrKW6bh0v/1jXi3/Lm7/mj7rOwGtr7NRHi/jDAzyqUW9woT20tHdimzEt7W450oIJY6JcmlL8dfqLfkIkWZQwGgvxJIqeEbBEhw2AZzSJC5tISQAYgw3GY4Ok1ls8fdrkormlBH5deOD/ROT3dklYi4eW9fbOgXf44mrFZWutiD/45tcb27CL/RnbKG6FBbhQE+AZNdUEvpY+/rDl7Yhfrltr+bmM9+JfyhU4bxEwb7/X99G87omLfXELO9Wbry27p3LvhV7zMkf2MPLl2CiXphR/XKdoo82cBFFB+9Mkc0kiApboYIs+msSFpFJisalEjaTWXK7YUsMP72P+7C90/sFaPPzzJL4TP1tz4A4uLHdw4fmIVoq4h1Nrk/mIhe+L7J7dHxV2bz/gB/z5gOXeBp74vjUXPu3Bi/rasyZjtVfIR2+Tfvyx62jsqOsoXh+Jl32xVfy9iNs91d7LD9Wpn6fnXD2fd8kRi0HtPsqlacWfzYWEonC1QfAGToGyQettHOL5rjEQzzwCloCwAZBYNIkL45VY2tixYzcKjVXiYYMxahJ6rjTW8zY0duQ1Ir+3T8JPPLSs1zoHnsSbtUs+lPrtGO4993rOXDiUz+I4+hVK80ZdR3BL7VFj1NWjDZzxXXVbssOYkiDDia3T0lz1lcRfvxb6um3JG9ntfR3B66MxsS9uiT+24Qw+xTlXak24U3PYES/M4Zndc72Pd84Rj0X0PcqlacWfQNhkSBic14fNXcnDGO5JKD3nnoTgu5KoVfxJPmvLJqb80cFCBwLWYJzW1yEjImJEP76NaJGQix8JrPUtmmPHHHFP3oALvtjm86k0xo4ffT+CW2qQdUsfiTLc1rBTDtha7oFlSfxZhzq1/svvHj48YhPfrtIi8QdTfyiwucMzX3/UP5yJG/Zjvej1wuOqOdKKR5RLU4t/a3A5ro5ARH59Vj69AgLJ7RVY2u9j8rofs5xRRiDKpRT/Ml5L9UbkLxXkTYNJbtckPnldk9cRUUW5lOI/go2T14zIP9mNXK4DAsltB1AnMJm8TkDCIi5EuZTivwjBtTAi8mtz8tk1EEhur8HTXi+T172I5fgIgSiXUvwjxBbqj8hfKMTbhpLcrkl98romryOiinIpxX8EGyevGZF/shu5XAcEktsOoE5gMnmdgIRFXIhyKcV/EYJrYUTk1+bks2sgkNxeg6e9XiavexHL8RECUS6l+EeILdQfkb9QiLcNJbldk/rkdU1eR0QV5VKK/wg2Tl4zIv9kN3K5Dggktx1AncBk8joBCYu4EOVSiv8iBNfCiMivzcln10Agub0GT3u9TF73IpbjIwSiXPpC/BmUn8QgcyBzIHMgcyBzYJ0cKB0MPot/6WH2JQKJQCKQCCQCicB6CKT4r8dpRpQIJAKJQCKQCFQRSPGvwpMPE4FEIBFIBBKB9RBI8V+P04woEUgEEoFEIBGoIpDiX4UnHyYCiUAikAgkAushkOK/HqcZUSKQCCQCiUAiUEUgxb8KTz5MBBKBRCARSATWQ+B/AdhOJMx6FPxkAAAAAElFTkSuQmCC"></p>
</div>
<div class="specification">
<p>A spacecraft is to be sent from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Io</mtext></math> to infinity.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate, for the surface of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Io</mtext></math>, the gravitational field strength <em>g</em><sub>Io</sub> due to the mass of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Io</mtext></math>. State an appropriate unit for your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>gravitational</mi><mo> </mo><mi>potential</mi><mo> </mo><mi>due</mi><mo> </mo><mi>to</mi><mo> </mo><mi>Jupiter</mi><mo> </mo><mi>at</mi><mo> </mo><mi>the</mi><mo> </mo><mi>orbit</mi><mo> </mo><mi>of</mi><mo> </mo><mi>Io</mi></mrow><mrow><mo> </mo><mi>gravitational</mi><mo> </mo><mi>potential</mi><mo> </mo><mi>due</mi><mo> </mo><mi>to</mi><mo> </mo><mi>Io</mi><mo> </mo><mi>at</mi><mo> </mo><mi>the</mi><mo> </mo><mi>surface</mi><mo> </mo><mi>of</mi><mo> </mo><mi>Io</mi></mrow></mfrac></math> is about 80.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Outline, using (b)(i), why it is not correct to use the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><mrow><mn>2</mn><mi>G</mi><mo>×</mo><mtext>mass of Io</mtext></mrow><mtext>radius of Io</mtext></mfrac></msqrt></math> to calculate the speed required for the spacecraft to reach infinity from the surface of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Io</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>An engineer needs to move a space probe of mass 3600 kg from Ganymede to Callisto. Calculate the energy required to move the probe from the orbital radius of Ganymede to the orbital radius of Callisto. Ignore the mass of the moons in your calculation. </p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>«</mo><mfrac><mrow><mi>G</mi><mi>M</mi></mrow><msup><mi>r</mi><mn>2</mn></msup></mfrac><mo>=</mo><mfrac><mrow><mn>6</mn><mo>.</mo><mn>67</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>11</mn></mrow></msup><mo>×</mo><mn>8</mn><mo>.</mo><mn>9</mn><mo>×</mo><msup><mn>10</mn><mn>22</mn></msup></mrow><msup><mfenced><mrow><mn>1</mn><mo>.</mo><mn>8</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow></mfenced><mn>2</mn></msup></mfrac><mo>=</mo><mo>»</mo><mn>1</mn><mo>.</mo><mn>8</mn></math><strong> ✓</strong></p>
<p>N kg<sup>−1 </sup><em><strong>OR</strong> </em>m s<sup>−2</sup><strong> ✓</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>1</mn><mo>.</mo><mn>9</mn><mo>×</mo><msup><mn>10</mn><mn>27</mn></msup></mrow><mrow><mn>4</mn><mo>.</mo><mn>9</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup></mrow></mfrac></math><strong> <em>AND </em> </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>8</mn><mo>.</mo><mn>9</mn><mo>×</mo><msup><mn>10</mn><mn>22</mn></msup></mrow><mrow><mn>1</mn><mo>.</mo><mn>8</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow></mfrac></math><strong> </strong>seen<strong> ✓</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>«</mo><mfrac><mrow><mn>1</mn><mo>.</mo><mn>9</mn><mo>×</mo><msup><mn>10</mn><mn>27</mn></msup><mo>×</mo><mn>1</mn><mo>.</mo><mn>8</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow><mrow><mn>4</mn><mo>.</mo><mn>9</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><mo>×</mo><mn>8</mn><mo>.</mo><mn>9</mn><mo>×</mo><msup><mn>10</mn><mn>22</mn></msup></mrow></mfrac><mo>=</mo><mo>»</mo><mn>78</mn></math><strong> ✓</strong></p>
<p><em><br>For <strong>MP1</strong>, potentials can be seen individually or as a ratio.</em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>«this is the escape speed for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Io</mtext></math> alone but» gravitational potential / field of Jupiter must be taken into account<strong> ✓</strong></p>
<p><em><strong><br>OWTTE</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>G</mi><msub><mi>M</mi><mtext>Jupiter</mtext></msub><mfenced><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>.</mo><mn>88</mn><mo>×</mo><msup><mn>10</mn><mn>9</mn></msup></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>.</mo><mn>06</mn><mo>×</mo><msup><mn>10</mn><mn>9</mn></msup></mrow></mfrac></mrow></mfenced><mo>=</mo><mo>«</mo><mn>5</mn><mo>.</mo><mn>21</mn><mo>×</mo><msup><mn>10</mn><mn>7</mn></msup><mo> </mo><msup><mtext>J kg</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>»</mo></math><strong> ✓</strong></p>
<p>« multiplies by 3600 kg to get » 1.9 × 10<sup>11 </sup>«J» <strong>✓</strong></p>
<p><em><br>Award <strong>[2]</strong> marks if factor of ½ used, taking into account orbital kinetic energies, leading to a final answer of 9.4 x 10<sup>10 </sup>«J».</em></p>
<p><em>Allow <strong>ECF</strong> from <strong>MP1</strong></em></p>
<p><em>Award <strong>[2] marks</strong> for a bald correct answer.</em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Rhodium-106 (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_{\,\,\,45}^{106}{\text{Rh}}">
<msubsup>
<mi></mi>
<mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>45</mn>
</mrow>
<mrow>
<mn>106</mn>
</mrow>
</msubsup>
<mrow>
<mtext>Rh</mtext>
</mrow>
</math></span>) decays into palladium-106 (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_{\,\,\,46}^{106}{\text{Pd}}">
<msubsup>
<mi></mi>
<mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>46</mn>
</mrow>
<mrow>
<mn>106</mn>
</mrow>
</msubsup>
<mrow>
<mtext>Pd</mtext>
</mrow>
</math></span>) by beta minus (<em>β</em><sup>–</sup>) decay. The diagram shows some of the nuclear energy levels of rhodium-106 and palladium-106. The arrow represents the <em>β</em><sup>–</sup> decay.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-08-14_om_06.42.36.png" alt="M18/4/PHYSI/HP2/ENG/TZ2/09.d"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Bohr modified the Rutherford model by introducing the condition <em>mvr </em>= <em>n</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{h}{{2\pi }}">
<mfrac>
<mi>h</mi>
<mrow>
<mn>2</mn>
<mi>π</mi>
</mrow>
</mfrac>
</math></span>. Outline the reason for this modification.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the speed <em>v </em>of an electron in the hydrogen atom is related to the radius <em>r </em>of the orbit by the expression</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="v = \sqrt {\frac{{k{e^2}}}{{{m_{\text{e}}}r}}} ">
<mi>v</mi>
<mo>=</mo>
<msqrt>
<mfrac>
<mrow>
<mi>k</mi>
<mrow>
<msup>
<mi>e</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msub>
<mi>m</mi>
<mrow>
<mtext>e</mtext>
</mrow>
</msub>
</mrow>
<mi>r</mi>
</mrow>
</mfrac>
</msqrt>
</math></span></p>
<p>where <em>k </em>is the Coulomb constant.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using the answer in (b) and (c)(i), deduce that the radius <em>r </em>of the electron’s orbit in the ground state of hydrogen is given by the following expression.</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="r = \frac{{{h^2}}}{{4{\pi ^2}k{m_{\text{e}}}{e^2}}}">
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>k</mi>
<mrow>
<msub>
<mi>m</mi>
<mrow>
<mtext>e</mtext>
</mrow>
</msub>
</mrow>
<mrow>
<msup>
<mi>e</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the electron’s orbital radius in (c)(ii).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain what may be deduced about the energy of the electron in the <em>β</em><sup>–</sup> decay.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Suggest why the <em>β</em><sup>–</sup> decay is followed by the emission of a gamma ray photon.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the wavelength of the gamma ray photon in (d)(ii).</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>the electrons accelerate and so radiate energy</p>
<p>they would therefore spiral into the nucleus/atoms would be unstable</p>
<p>electrons have discrete/only certain energy levels</p>
<p>the only orbits where electrons do not radiate are those that satisfy the Bohr condition <strong>«</strong><em>mvr</em> = <em>n</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{h}{{2\pi }}">
<mfrac>
<mi>h</mi>
<mrow>
<mn>2</mn>
<mi>π</mi>
</mrow>
</mfrac>
</math></span><strong>»</strong></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{m_{\text{e}}}{v^2}}}{r} = \frac{{k{e^2}}}{{{r^2}}}">
<mfrac>
<mrow>
<mrow>
<msub>
<mi>m</mi>
<mrow>
<mtext>e</mtext>
</mrow>
</msub>
</mrow>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mi>r</mi>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mrow>
<msup>
<mi>e</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p><strong><em>OR</em></strong></p>
<p>KE = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span>PE hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span><em>m</em><sub>e</sub><em>v</em><sup>2</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\frac{{k{e^2}}}{r}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mfrac>
<mrow>
<mi>k</mi>
<mrow>
<msup>
<mi>e</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mi>r</mi>
</mfrac>
</math></span></p>
<p><strong>«</strong>solving for <em>v </em>to get answer<strong>»</strong></p>
<p> </p>
<p><em>Answer given – look for correct working</em></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>combining <em>v</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {\frac{{k{e^2}}}{{{m_{\text{e}}}r}}} ">
<msqrt>
<mfrac>
<mrow>
<mi>k</mi>
<mrow>
<msup>
<mi>e</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msub>
<mi>m</mi>
<mrow>
<mtext>e</mtext>
</mrow>
</msub>
</mrow>
<mi>r</mi>
</mrow>
</mfrac>
</msqrt>
</math></span> with <em>m</em><sub>e</sub><em>vr</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{h}{{2\pi }}">
<mfrac>
<mi>h</mi>
<mrow>
<mn>2</mn>
<mi>π</mi>
</mrow>
</mfrac>
</math></span> using correct substitution</p>
<p><strong>«</strong><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_e}^2\frac{{k{e^2}}}{{{m_{\text{e}}}r}}{r^2} = \frac{{{h^2}}}{{4{\pi ^2}}}">
<msup>
<mrow>
<msub>
<mi>m</mi>
<mi>e</mi>
</msub>
</mrow>
<mn>2</mn>
</msup>
<mfrac>
<mrow>
<mi>k</mi>
<mrow>
<msup>
<mi>e</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msub>
<mi>m</mi>
<mrow>
<mtext>e</mtext>
</mrow>
</msub>
</mrow>
<mi>r</mi>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span><strong>»</strong></p>
<p>correct algebraic manipulation to gain the answer</p>
<p> </p>
<p><em>Answer given – look for correct working</em></p>
<p><em>Do not allow a bald statement of the answer for MP2. Some further working eg cancellation of m or r must be shown</em></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>«</strong> <em>r</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{(6.63 \times {{10}^{ - 34}})}^2}}}{{4{\pi ^2} \times 8.99 \times {{10}^9} \times 9.11 \times {{10}^{ - 31}} \times {{(1.6 \times {{10}^{ - 19}})}^2}}}">
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>6.63</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>34</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>×</mo>
<mn>8.99</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>9</mn>
</msup>
</mrow>
<mo>×</mo>
<mn>9.11</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>31</mn>
</mrow>
</msup>
</mrow>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>1.6</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>19</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span><strong>»</strong></p>
<p><em>r</em> = 5.3 × 10<sup>–11</sup> <strong>«</strong>m<strong>»</strong></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the energy released is 3.54 – 0.48 = 3.06 <strong>«</strong>MeV<strong>»</strong></p>
<p>this is shared by the electron and the antineutrino</p>
<p>so the electron’s energy varies from 0 to 3.06 <strong>«</strong>MeV<strong>»</strong></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the palladium nucleus emits the photon when it decays into the ground state <strong>«</strong>from the excited state<strong>»</strong></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Photon energy</p>
<p><em>E</em> = 0.48 × 10<sup>6</sup> × 1.6 × 10<sup>–19</sup> = <strong>«</strong>7.68 × 10<sup>–14</sup> <em>J</em><strong>»</strong></p>
<p><em>λ</em> = <strong>«</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{hc}}{E} = \frac{{6.63 \times {{10}^{ - 34}} \times 3 \times {{10}^8}}}{{7.68 \times {{10}^{ - 14}}}}">
<mfrac>
<mrow>
<mi>h</mi>
<mi>c</mi>
</mrow>
<mi>E</mi>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>6.63</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>34</mn>
</mrow>
</msup>
</mrow>
<mo>×</mo>
<mn>3</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>8</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>7.68</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>14</mn>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> =<strong>»</strong> 2.6 × 10<sup>–12</sup><strong> «</strong>m<strong>»</strong></p>
<p> </p>
<p><em>Award </em><strong><em>[2] </em></strong><em>for a bald correct answer</em></p>
<p><em>Allow ECF from incorrect energy</em></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>The moon Phobos moves around the planet Mars in a circular orbit.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Outline the origin of the force that acts on Phobos.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Outline why this force does no work on Phobos.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The orbital period <em>T</em> of a moon orbiting a planet of mass <em>M</em> is given by</p>
<p style="text-align:center;"><span style="background-color:#ffffff;"><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\frac{{{R^3}}}{{{T^2}}} = kM">
<mfrac>
<mrow>
<mrow>
<msup>
<mi>R</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>T</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mi>k</mi>
<mi>M</mi>
</math></span></span></p>
<p>where <em>R</em> is the average distance between the centre of the planet and the centre of the moon.</p>
<p>Show that <span style="background-color:#ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{G}{{4{\pi ^2}}}">
<mi>k</mi>
<mo>=</mo>
<mfrac>
<mi>G</mi>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The following data for the Mars–Phobos system and the Earth–Moon system are available:</p>
<p>Mass of Earth = 5.97 × 10<sup>24</sup> kg</p>
<p>The Earth–Moon distance is 41 times the Mars–Phobos distance.</p>
<p>The orbital period of the Moon is 86 times the orbital period of Phobos.</p>
<p>Calculate, in kg, the mass of Mars.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The graph shows the variation of the gravitational potential between the Earth and Moon with distance from the centre of the Earth. The distance from the Earth is expressed as a fraction of the total distance between the centre of the Earth and the centre of the Moon.</p>
<p style="text-align:center;"><img 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"></p>
<p style="text-align:left;">Determine, using the graph, the mass of the Moon.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>gravitational attraction/force/field «of the planet/Mars» ✔</p>
<p><em>Do not accept “gravity”</em>.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the force/field and the velocity/displacement are at 90° to each other <strong><em>OR</em></strong></p>
<p>there is no change in GPE of the moon/Phobos ✔</p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>ALTERNATE 1</strong></em></p>
<p>«using fundamental equations»</p>
<p>use of Universal gravitational force/acceleration/orbital velocity equations ✔</p>
<p>equating to centripetal force or acceleration. ✔</p>
<p>rearranges to get <span style="display:inline !important;float:none;background-color:#ffffff;color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{G}{{4{\pi ^2}}}">
<mi>k</mi>
<mo>=</mo>
<mfrac>
<mi>G</mi>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></span> ✔</p>
<p><em><strong>ALTERNATE 2</strong></em></p>
<p>«starting with <span style="display:inline !important;float:none;background-color:#ffffff;color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:center;text-decoration:none;text-indent:0px;white-space:normal;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{R^3}}}{{{T^2}}} = kM">
<mfrac>
<mrow>
<mrow>
<msup>
<mi>R</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>T</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mi>k</mi>
<mi>M</mi>
</math></span><span style="display:inline !important;float:none;background-color:#ffffff;color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;">»</span></span></p>
<p>substitution of proper equation for T from orbital motion equations ✔</p>
<p>substitution of proper equation for M <em><strong>OR</strong></em> R from orbital motion equations ✔</p>
<p>rearranges to get <span style="display:inline !important;float:none;background-color:#ffffff;color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{G}{{4{\pi ^2}}}">
<mi>k</mi>
<mo>=</mo>
<mfrac>
<mi>G</mi>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></span> ✔</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="background-color:#ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_{{\text{Mars}}}} = {\left( {\frac{{{R_{{\text{Mars}}}}}}{{{R_{{\text{Earth}}}}}}} \right)^3}{\left( {\frac{{{T_{{\text{Earth}}}}}}{{{T_{Mars}}}}} \right)^2}{m_{{\text{Earth}}}}">
<mrow>
<msub>
<mi>m</mi>
<mrow>
<mrow>
<mtext>Mars</mtext>
</mrow>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>R</mi>
<mrow>
<mrow>
<mtext>Mars</mtext>
</mrow>
</mrow>
</msub>
</mrow>
</mrow>
<mrow>
<mrow>
<msub>
<mi>R</mi>
<mrow>
<mrow>
<mtext>Earth</mtext>
</mrow>
</mrow>
</msub>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mrow>
<mtext>Earth</mtext>
</mrow>
</mrow>
</msub>
</mrow>
</mrow>
<mrow>
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mi>M</mi>
<mi>a</mi>
<mi>r</mi>
<mi>s</mi>
</mrow>
</msub>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msub>
<mi>m</mi>
<mrow>
<mrow>
<mtext>Earth</mtext>
</mrow>
</mrow>
</msub>
</mrow>
</math></span></span> or other consistent re-arrangement ✔</p>
<p>6.4 × 10<sup>23</sup> «kg» ✔</p>
<p> </p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>read off separation at maximum potential 0.9 ✔</p>
<p>equating of gravitational field strength of earth and moon at that location <em><strong>OR <img src="data:image/png;base64,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">✔</strong></em></p>
<p>7.4 × 10<sup>22</sup> «kg» ✔</p>
<p><em>Allow ECF from MP1</em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This was generally well answered, although some candidates simply used the vague term “gravity” rather than specifying that it is a gravitational force or a gravitational field. Candidates need to be reminded about using proper physics terms and not more general, “every day” terms on the exam.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Some candidates connected the idea that the gravitational force is perpendicular to the velocity (and hence the displacement) for the mark. It was also allowed to discuss that there is no change in gravitational potential energy, so therefore no work was being done. It was not acceptable to simply state that the net displacement over one full orbit is zero. Unfortunately, some candidates suggested that there is no net force on the moon so there is no work done, or that the moon is so much smaller so no work could be done on it.</p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was another “show that” derivation. Many candidates attempted to work with universal gravitation equations, either from memory or the data booklet, to perform this derivation. The variety of correct solution paths was quite impressive, and many candidates who attempted this question were able to receive some marks. Candidates should be reminded on “show that” questions that it is never allowed to work backwards from the given answer. Some candidates also made up equations (such as T = 2𝝿r) to force the derivation to work out.</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This question was challenging for candidates. The candidates who started down the correct path of using the given derived value from 5bi often simply forgot that the multiplication factors had to be squared and cubed as well as the variables.</p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This question was left blank by many candidates, and very few who attempted it were able to successfully recognize that the gravitational fields of the Earth and Moon balance at 0.9r and then use the proper equation to calculate the mass of the Moon.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A planet is in a circular orbit around a star. The speed of the planet is constant. The following data are given:</p>
<p style="padding-left: 120px;">Mass of planet <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn><mo>.</mo><mn>0</mn><mo>×</mo><msup><mn>10</mn><mn>24</mn></msup><mo> </mo></math>kg<br>Mass of star <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo>.</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>30</mn></msup><mo> </mo></math>kg<br>Distance from the star to the planet <em>R</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>.</mo><mn>4</mn><mo>×</mo><msup><mn>10</mn><mn>10</mn></msup><mo> </mo></math>m.</p>
</div>
<div class="specification">
<p>A spacecraft is to be launched from the surface of the planet to escape from the star system. The radius of the planet is 9.1 × 10<sup>3</sup> km.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why a centripetal force is needed for the planet to be in a circular orbit.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the value of the centripetal force.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the gravitational potential due to the planet and the star at the surface of the planet is about −5 × 10<sup>9 </sup>J kg<sup>−1</sup>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Estimate the escape speed of the spacecraft from the planet–star system.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>«circular motion» involves a changing velocity <strong>✓</strong></p>
<p>«Tangential velocity» is «always» perpendicular to centripetal force/acceleration <strong>✓</strong></p>
<p>there must be a force/acceleration towards centre/star <strong>✓</strong></p>
<p>without a centripetal force the planet will move in a straight line <strong>✓</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mfrac><mrow><mo>(</mo><mn>6</mn><mo>.</mo><mn>67</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>11</mn></mrow></msup><mo>)</mo><mo>(</mo><mn>8</mn><mo>×</mo><msup><mn>10</mn><mn>24</mn></msup><mo>)</mo><mo>(</mo><mn>3</mn><mo>.</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>30</mn></msup><mo>)</mo></mrow><mrow><mo>(</mo><mn>4</mn><mo>.</mo><mn>4</mn><mo>×</mo><msup><mn>10</mn><mn>10</mn></msup><msup><mo>)</mo><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>8</mn><mo>.</mo><mn>8</mn><mo>×</mo><msup><mn>10</mn><mn>23</mn></msup></math> «N» <strong>✓</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>V</em><sub>planet</sub> = «−»<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>(</mo><mn>6</mn><mo>.</mo><mn>67</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>11</mn></mrow></msup><mo>)</mo><mo>(</mo><mn>8</mn><mo>×</mo><msup><mn>10</mn><mn>24</mn></msup><mo>)</mo></mrow><mrow><mn>9</mn><mo>.</mo><mn>1</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow></mfrac><mo>=</mo></math>«−» 5.9 × 10<sup>7 </sup>«J kg<sup>−1</sup>» <strong>✓</strong></p>
<p><em>V</em><sub>star</sub> = «−»<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>(</mo><mn>6</mn><mo>.</mo><mn>67</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>11</mn></mrow></msup><mo>)</mo><mo>(</mo><mn>3</mn><mo>.</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>30</mn></msup><mo>)</mo></mrow><mrow><mn>4</mn><mo>.</mo><mn>4</mn><mo>×</mo><msup><mn>10</mn><mn>10</mn></msup></mrow></mfrac><mo>=</mo></math>«−» 4.9 × 10<sup>9 </sup>«J kg<sup>−1</sup>» <strong>✓</strong></p>
<p><em>V</em><sub>planet</sub> + <em>V</em><sub>star </sub>= «−» 4.9 «09» × 10<sup>9 </sup>«J kg<sup>−1</sup>» <strong>✓</strong></p>
<p><em><br></em><em>Must see substitutions and not just equations.</em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of <em>v</em><sub>esc</sub> = <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn><mi>V</mi></msqrt></math> <strong>✓</strong></p>
<p><em>v = </em>9.91 × 10<sup>4</sup> «m s<sup>−1</sup>» <strong>✓</strong></p>
<p> </p>
<p> </p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A planet has radius <em>R</em>. At a distance <em>h </em>above the surface of the planet the gravitational field strength is <em>g </em>and the gravitational potential is <em>V</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State what is meant by gravitational field strength.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <em>V </em>= –<em>g</em>(<em>R </em>+ <em>h</em>).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Draw a graph, on the axes, to show the variation of the gravitational potential <em>V</em> of the planet with height <em>h </em>above the surface of the planet.</p>
<p><img src="data:image/png;base64,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"></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A planet has a radius of 3.1 × 10<sup>6</sup> m. At a point P a distance 2.4 × 10<sup>7</sup> m above the surface of the planet the gravitational field strength is 2.2 N kg<sup>–1</sup>. Calculate the gravitational potential at point P, include an appropriate unit for your answer.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The diagram shows the path of an asteroid as it moves past the planet.</p>
<p> <img src="images/Schermafbeelding_2018-08-14_om_07.33.17.png" alt="M18/4/PHYSI/HP2/ENG/TZ2/06.c"></p>
<p>When the asteroid was far away from the planet it had negligible speed. Estimate the speed of the asteroid at point P as defined in (b).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The mass of the asteroid is 6.2 × 10<sup>12</sup> kg. Calculate the gravitational force experienced by the <strong>planet </strong>when the asteroid is at point P.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>the <strong>«</strong>gravitational<strong>» </strong>force per unit mass exerted on a point/small/test mass</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>at height <em>h </em>potential is <em>V</em> = –<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{GM}}{{(R + h)}}">
<mfrac>
<mrow>
<mi>G</mi>
<mi>M</mi>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>R</mi>
<mo>+</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math></span></p>
<p>field is <em>g </em>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{GM}}{{{{(R + h)}^2}}}">
<mfrac>
<mrow>
<mi>G</mi>
<mi>M</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>R</mi>
<mo>+</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p><strong>«</strong>dividing gives answer<strong>»</strong></p>
<p> </p>
<p><em>Do not allow an answer that starts with g = –</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\Delta V}}{{\Delta r}}">
<mfrac>
<mrow>
<mi mathvariant="normal">Δ</mi>
<mi>V</mi>
</mrow>
<mrow>
<mi mathvariant="normal">Δ</mi>
<mi>r</mi>
</mrow>
</mfrac>
</math></span><em> and then cancels the deltas and substitutes </em><em>R </em>+ <em>h</em></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct shape and sign</p>
<p>non-zero negative vertical intercept</p>
<p> </p>
<p><img src="images/Schermafbeelding_2018-08-14_om_07.26.11.png" alt="M18/4/PHYSI/HP2/ENG/TZ2/06.a.iii/M"></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>V</em> = <strong>«</strong>–2.2 × (3.1 × 10<sup>6</sup> + 2.4 × 10<sup>7</sup>) =<strong>»</strong> <strong>«</strong>–<strong>»</strong> 6.0 × 10<sup>7</sup> J kg<sup>–1</sup></p>
<p> </p>
<p><em>Unit is essential</em></p>
<p><em>Allow eg MJ kg<sup>–</sup></em><em><sup>1</sup> </em><em>if power of 10 is correct</em></p>
<p><em>Allow other correct SI units eg m</em><sup><em>2</em></sup><em>s<sup>–</sup></em><sup><em>2</em></sup><em>, N m kg<sup>–</sup></em><sup><em>1</em></sup></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>total energy at P = 0 / KE gained = GPE lost</p>
<p><strong>«</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span><em>mv</em><sup>2</sup> + <em>mV</em> = 0 ⇒<strong>»</strong> <em>v</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt { - 2V} ">
<msqrt>
<mo>−</mo>
<mn>2</mn>
<mi>V</mi>
</msqrt>
</math></span></p>
<p><em>v</em> = <strong>«</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {2 \times 6.0 \times {{10}^7}} ">
<msqrt>
<mn>2</mn>
<mo>×</mo>
<mn>6.0</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>7</mn>
</msup>
</mrow>
</msqrt>
</math></span> =<strong>»</strong> 1.1 × 10<sup>4</sup> <strong>«</strong>ms<sup>–1</sup><strong>»</strong></p>
<p> </p>
<p> </p>
<p><em>Award </em><strong><em>[3] </em></strong><em>for a bald correct answer</em></p>
<p><em>Ignore negative sign errors in the workings</em></p>
<p><em>Allow ECF from 6(b)</em></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>ALTERNATIVE 1</em></strong></p>
<p>force on asteroid is <strong>«</strong>6.2 × 10<sup>12</sup> × 2.2 =<strong>»</strong> 1.4 × 10<sup>13</sup> <strong>«</strong>N<strong>»</strong></p>
<p><strong>«</strong>by Newton’s third law<strong>» </strong>this is also the force on the planet</p>
<p><strong><em>ALTERNATIVE 2</em></strong></p>
<p>mass of planet = 2.4 x 10<sup>25</sup> <strong>«</strong>kg<strong>» «</strong>from <em>V</em> = –<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{GM}}{{(R + h)}}">
<mfrac>
<mrow>
<mi>G</mi>
<mi>M</mi>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>R</mi>
<mo>+</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math></span><strong>»</strong></p>
<p>force on planet <strong>«</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{GMm}}{{{{(R + h)}^2}}}">
<mfrac>
<mrow>
<mi>G</mi>
<mi>M</mi>
<mi>m</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>R</mi>
<mo>+</mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span><strong>»</strong> = 1.4 × 10<sup>13</sup> <strong>«</strong>N<strong>»</strong></p>
<p> </p>
<p><em>MP2 must be explicit</em></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>There is a proposal to place a satellite in orbit around planet Mars.</p>
</div>
<div class="specification">
<p>The satellite is to have an orbital time <em>T</em> equal to the length of a day on Mars. It can be shown that</p>
<p style="text-align: center;"><em>T</em><sup>2</sup> = <em>kR</em><sup>3</sup></p>
<p>where<em> R</em> is the orbital radius of the satellite and <em>k</em> is a constant.</p>
</div>
<div class="specification">
<p>The ratio <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{distance of Mars from the Sun}}}}{{{\text{distance of Earth from the Sun}}}}">
<mfrac>
<mrow>
<mrow>
<mtext>distance of Mars from the Sun</mtext>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>distance of Earth from the Sun</mtext>
</mrow>
</mrow>
</mfrac>
</math></span> = 1.5.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Outline what is meant by gravitational field strength at a point.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Newton’s law of gravitation applies to point masses. Suggest why the law can be applied to a satellite orbiting Mars.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Mars has a mass of 6.4 × 10<sup>23</sup> kg. Show that, for Mars, <em>k</em> is about 9 × 10<sup>–13 </sup>s<sup>2 </sup>m<sup>–3</sup>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The time taken for Mars to revolve on its axis is 8.9 × 10<sup>4</sup> s. Calculate, in m s<sup>–1</sup>, the orbital speed of the satellite.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the intensity of solar radiation at the orbit of Mars is about 600 W m<sup>–2</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine, in K, the mean surface temperature of Mars. Assume that Mars acts as a black body.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The atmosphere of Mars is composed mainly of carbon dioxide and has a pressure less than 1 % of that on the Earth. Outline why the mean temperature of Earth is strongly affected by gases in its atmosphere but that of Mars is not.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>force per unit mass ✔</p>
<p>acting on a small/test/point mass «placed at the point in the field» ✔</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Mars is spherical/a sphere «and of uniform density so behaves as a point mass» ✔</p>
<p>satellite has a much smaller mass/diameter/size than Mars «so approximates to a point mass» ✔</p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>«<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{m{v^2}}}{r} = \frac{{GMm}}{{{r^2}}}">
<mfrac>
<mrow>
<mi>m</mi>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mi>r</mi>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>G</mi>
<mi>M</mi>
<mi>m</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> hence» <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = \sqrt {\frac{{GM}}{R}} ">
<mi>v</mi>
<mo>=</mo>
<msqrt>
<mfrac>
<mrow>
<mi>G</mi>
<mi>M</mi>
</mrow>
<mi>R</mi>
</mfrac>
</msqrt>
</math></span>. Also <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = \frac{{2\pi R}}{T}">
<mi>v</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>π</mi>
<mi>R</mi>
</mrow>
<mi>T</mi>
</mfrac>
</math></span></p>
<p><em><strong>OR</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m{\omega ^2}r = \frac{{GMm}}{{{r^2}}}">
<mi>m</mi>
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>G</mi>
<mi>M</mi>
<mi>m</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\omega ^2} = \frac{{GM}}{{{R^3}}}">
<mrow>
<msup>
<mi>ω</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>G</mi>
<mi>M</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>R</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> ✔</p>
<p> </p>
<p>uses either of the above to get <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T^2} = \frac{{4{\pi ^2}}}{{GM}}{R^3}">
<mrow>
<msup>
<mi>T</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mi>G</mi>
<mi>M</mi>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>R</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span></p>
<p><em><strong>OR</strong></em></p>
<p>uses <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{{4{\pi ^2}}}{{GM}}">
<mi>k</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mi>G</mi>
<mi>M</mi>
</mrow>
</mfrac>
</math></span> ✔</p>
<p> </p>
<p><em>k</em> = 9.2 × 10<sup>−13</sup> / 9.3 × 10<sup>−13</sup></p>
<p> </p>
<p> </p>
<p><em>Unit not required</em></p>
<p> </p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{R^3} = \frac{{{T^2}}}{k} = \frac{{{{\left( {8.9 \times {{10}^4}} \right)}^2}}}{{9.25 \times {{10}^{ - 13}}}}">
<mrow>
<msup>
<mi>R</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>T</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mi>k</mi>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>8.9</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>9.25</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>13</mn>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <em>R</em> = 2.04 × 10<sup>7</sup> «m» ✔</p>
<p> </p>
<p><em>v</em> = «<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\omega r = \frac{{2\pi \times 2.04 \times {{10}^7}}}{{89000}} = ">
<mi>ω</mi>
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>π</mi>
<mo>×</mo>
<mn>2.04</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>7</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>89000</mn>
</mrow>
</mfrac>
<mo>=</mo>
</math></span>» 1.4 × 10<sup>3</sup> «m s<sup>–1</sup>»</p>
<p><em><strong>OR</strong></em></p>
<p><em>v</em> = «<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {\frac{{GM}}{R}} = \sqrt {\frac{{6.67 \times {{10}^{ - 11}} \times 6.4 \times {{10}^{23}}}}{{2.04 \times {{10}^7}}}} = ">
<msqrt>
<mfrac>
<mrow>
<mi>G</mi>
<mi>M</mi>
</mrow>
<mi>R</mi>
</mfrac>
</msqrt>
<mo>=</mo>
<msqrt>
<mfrac>
<mrow>
<mn>6.67</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>11</mn>
</mrow>
</msup>
</mrow>
<mo>×</mo>
<mn>6.4</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mn>23</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2.04</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>7</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</msqrt>
<mo>=</mo>
</math></span>» 1.4 × 10<sup>3</sup> «m s<sup>–1</sup>» ✔</p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="I \propto \frac{1}{{{r^2}}}">
<mi>I</mi>
<mo>∝</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> «1.36 × 10<sup>3</sup> × <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{{{1.5}^2}}}">
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mn>1.5</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>» ✔</p>
<p>604 «W m<sup>–2</sup>» ✔</p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{600}}{4}">
<mfrac>
<mrow>
<mn>600</mn>
</mrow>
<mn>4</mn>
</mfrac>
</math></span> for mean intensity ✔</p>
<p>temperature/K = «<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt[4]{{\frac{{600}}{{4 \times 5.67 \times {{10}^{ - 8}}}}}} = ">
<mroot>
<mrow>
<mfrac>
<mrow>
<mn>600</mn>
</mrow>
<mrow>
<mn>4</mn>
<mo>×</mo>
<mn>5.67</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mn>4</mn>
</mroot>
<mo>=</mo>
</math></span>» 230 ✔</p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>reference to greenhouse gas/effect ✔</p>
<p>recognize the link between molecular density/concentration and pressure ✔</p>
<p>low pressure means too few molecules to produce a significant heating effect</p>
<p><em><strong>OR</strong></em></p>
<p>low pressure means too little radiation re-radiated back to Mars ✔</p>
<p> </p>
<p><em>The greenhouse effect can be described, it doesn’t have to be named</em></p>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>Titan is a moon of Saturn. The Titan-Sun distance is 9.3 times greater than the Earth-Sun distance.</p>
</div>
<div class="specification">
<p>The molar mass of nitrogen is 28 g mol<sup>−1</sup>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the intensity of the solar radiation at the location of Titan is 16 W m<sup>−2</sup>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Titan has an atmosphere of nitrogen. The albedo of the atmosphere is 0.22. The surface of Titan may be assumed to be a black body. Explain why the <strong>average </strong>intensity of solar radiation <strong>absorbed</strong> by the whole surface of Titan is 3.1 W m<sup>−2</sup>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the equilibrium surface temperature of Titan is about 90 K.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The mass of Titan is 0.025 times the mass of the Earth and its radius is 0.404 times the radius of the Earth. The escape speed from Earth is 11.2 km s<sup>−1</sup>. Show that the escape speed from Titan is 2.8 km s<sup>−1</sup>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The orbital radius of Titan around Saturn is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> and the period of revolution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</p>
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>4</mn><msup><mi mathvariant="normal">π</mi><mn>2</mn></msup><msup><mi>R</mi><mrow><mo> </mo><mn>3</mn></mrow></msup></mrow><mrow><mi>G</mi><mi>M</mi></mrow></mfrac></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> is the mass of Saturn.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The orbital radius of Titan around Saturn is 1.2 × 10<sup>9 </sup>m and the orbital period is 15.9 days. Estimate the mass of Saturn.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the mass of a nitrogen molecule is 4.7 × 10<sup>−26</sup> kg.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Estimate the root mean square speed of nitrogen molecules in the Titan atmosphere. Assume an atmosphere temperature of 90 K.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Discuss, by reference to the answer in (b), whether it is likely that Titan will lose its atmosphere of nitrogen.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>incident intensity <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1360</mn><mrow><mn>9</mn><mo>.</mo><msup><mn>3</mn><mn>2</mn></msup></mrow></mfrac></math> <em><strong>OR </strong></em><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mo>.</mo><mn>7</mn><mo>≈</mo><mn>16</mn></math> «W m<sup>−2</sup>» ✓</p>
<p> </p>
<p><em>Allow the use of 1400 for the solar constant.</em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>exposed surface is ¼ of the total surface ✓</p>
<p>absorbed intensity = (1−0.22) × incident intensity ✓</p>
<p>0.78 × 0.25 × 15.7 <em><strong>OR </strong> </em>3.07 «W m<sup>−2</sup>» ✓</p>
<p> </p>
<p><em>Allow 3.06 from rounding and 3.12 if they use 16</em> W m<sup>−2</sup>.</p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>σT </em><sup>4</sup> = 3.07</p>
<p><em><strong>OR</strong></em></p>
<p><em>T</em> = 86 «K» ✓</p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mo>«</mo><msqrt><mfrac><mrow><mn>2</mn><mi>G</mi><mi>M</mi></mrow><mi>R</mi></mfrac></msqrt><mo>=</mo><mo>»</mo><msqrt><mfrac><mrow><mn>0</mn><mo>.</mo><mn>025</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>404</mn></mrow></mfrac></msqrt><mo>×</mo><mn>11</mn><mo>.</mo><mn>2</mn></math></p>
<p><em><strong>OR</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>79</mn></math> «km s<sup>−1</sup>» ✓</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct equating of gravitational force / acceleration to centripetal force / acceleration ✓</p>
<p>correct rearrangement to reach the expression given ✓</p>
<p> </p>
<p><em>Allow use of <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><mrow><mi>G</mi><mi>M</mi></mrow><mi>R</mi></mfrac></msqrt><mo>=</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi><mi>R</mi></mrow><mi>T</mi></mfrac></math> for <strong>MP1</strong>.</em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>15</mn><mo>.</mo><mn>9</mn><mo>×</mo><mn>24</mn><mo>×</mo><mn>3600</mn></math> «s» ✓</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mfrac><mrow><mn>4</mn><msup><mi mathvariant="normal">π</mi><mn>2</mn></msup><msup><mfenced><mrow><mn>1</mn><mo>.</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>9</mn></msup></mrow></mfenced><mn>3</mn></msup></mrow><mrow><mn>6</mn><mo>.</mo><mn>67</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>11</mn></mrow></msup><mo>×</mo><msup><mfenced><mrow><mn>15</mn><mo>.</mo><mn>9</mn><mo>×</mo><mn>24</mn><mo>×</mo><mn>3600</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>5</mn><mo>.</mo><mn>4</mn><mo>×</mo><msup><mn>10</mn><mn>26</mn></msup><mo> </mo></math>«kg» ✓</p>
<p> </p>
<p><em>Award <strong>[2]</strong> marks for a bald correct answer.</em></p>
<p><em>Allow <strong>ECF</strong> from <strong>MP1</strong>.</em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mrow><mn>28</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow><mrow><mn>6</mn><mo>.</mo><mn>02</mn><mo>×</mo><msup><mn>10</mn><mn>23</mn></msup></mrow></mfrac></math></p>
<p><em><strong>OR</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>65</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>26</mn></mrow></msup></math> «kg» ✓</p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>«</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>k</mi><mi>T</mi><mo>⇒</mo><mo>»</mo><mi>v</mi><mo>=</mo><msqrt><mfrac><mrow><mn>3</mn><mi>k</mi><mi>T</mi></mrow><mi>m</mi></mfrac></msqrt></math> ✓</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mo>«</mo><msqrt><mfrac><mrow><mn>3</mn><mo>×</mo><mn>1</mn><mo>.</mo><mn>38</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>23</mn></mrow></msup><mo>×</mo><mn>90</mn></mrow><mrow><mn>4</mn><mo>.</mo><mn>651</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>26</mn></mrow></msup></mrow></mfrac></msqrt><mo>=</mo><mo>»</mo><mn>283</mn><mo>≈</mo><mn>300</mn></math> «ms<sup>−1</sup>» ✓</p>
<p> </p>
<p><em>Award <strong>[2]</strong> marks for a bald correct answer.</em></p>
<p><em>Allow 282 from a rounded mass.</em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>no, molecular speeds much less than escape speed ✓</p>
<p> </p>
<p><em>Allow <strong>ECF</strong> from incorrect <strong>(d)(ii)</strong>.</em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="background-color: #ffffff;">A proton moves along a circular path in a region of a uniform magnetic field. The magnetic field is directed into the plane of the page.</span></p>
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"></span></p>
</div>
<div class="specification">
<p><span style="background-color: #ffffff;">The speed of the proton is 2.16 × 10<sup>6</sup> m s<sup>-1</sup> and the magnetic field strength is 0.042 T.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color:#ffffff;">Label with arrows on the diagram the magnetic force <em>F</em> on the proton.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">ai.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color:#ffffff;">Label with arrows on the diagram the velocity vector <em>v</em> of the proton.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">aii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color:#ffffff;">For this proton, determine, in m, the radius of the circular path. Give your answer to an appropriate number of significant figures.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">bi.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color:#ffffff;">For this proton, calculate, in s, the time for one full revolution.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">bii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;"><span style="background-color:#ffffff;"><em>F</em> towards centre ✔</span><span style="background-color:#ffffff;"><br></span></p>
<div class="question_part_label">ai.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;"><span style="background-color:#ffffff;"><em>v</em> tangent to circle and in the direction shown in the diagram ✔<br></span><span style="background-color:#ffffff;"><br></span></p>
<p style="color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;"><span style="background-color:#ffffff;"><img 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" width="305" height="248"></span></p>
<p style="color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;"> </p>
<div class="question_part_label">aii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;">«<span style="background-color:#ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="qvB = \frac{{m{v^2}}}{R} \Rightarrow » R = \frac{{mv}}{{qB}}/\frac{{1.673 \times {{10}^{ - 27}} \times 2.16 \times {{10}^6}}}{{1.60 \times {{10}^{ - 19}} \times 0.042}}">
<mi>q</mi>
<mi>v</mi>
<mi>B</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>m</mi>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mi>R</mi>
</mfrac>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>»</mo>
</mrow>
<mi>R</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>m</mi>
<mi>v</mi>
</mrow>
<mrow>
<mi>q</mi>
<mi>B</mi>
</mrow>
</mfrac>
<mrow>
<mo>/</mo>
</mrow>
<mfrac>
<mrow>
<mn>1.673</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>27</mn>
</mrow>
</msup>
</mrow>
<mo>×</mo>
<mn>2.16</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>6</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>1.60</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>19</mn>
</mrow>
</msup>
</mrow>
<mo>×</mo>
<mn>0.042</mn>
</mrow>
</mfrac>
</math></span> ✔</span></p>
<p style="color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;"><em><span style="background-color:#ffffff;">R = </span></em><span style="background-color:#ffffff;">0.538«m» ✔</span></p>
<p style="color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;"><span style="background-color:#ffffff;"><em>R</em> = 0.54«m» ✔</span></p>
<div class="question_part_label">bi.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;"><span style="background-color:#ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T = \frac{{2\pi R}}{v}/\frac{{2\pi \times 0.54}}{{2.16 \times {{10}^6}}}">
<mi>T</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>π</mi>
<mi>R</mi>
</mrow>
<mi>v</mi>
</mfrac>
<mrow>
<mo>/</mo>
</mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mi>π</mi>
<mo>×</mo>
<mn>0.54</mn>
</mrow>
<mrow>
<mn>2.16</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>6</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> ✔<br></span></p>
<p style="color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;"><span style="background-color:#ffffff;"><span style="background-color:#ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T = 1.6 \times {10^{ - 6}}">
<mi>T</mi>
<mo>=</mo>
<mn>1.6</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</msup>
</mrow>
</math></span>«s» <span style="display:inline !important;float:none;background-color:#ffffff;color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;">✔</span></span></span></p>
<div class="question_part_label">bii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Examiners were requested to be lenient here and as a result most candidates scored both marks. Had we insisted on <em>e.g.</em> straight lines drawn with a ruler or a force arrow passing exactly through the centre of the circle very few marks would have been scored. For those who didn’t know which way the arrows were supposed to be the common guesses were to the left and up the page. Some candidates neglected to label the arrows.</p>
<div class="question_part_label">ai.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Examiners were requested to be lenient here and as a result most candidates scored both marks. Had we insisted on <em>e.g.</em> straight lines drawn with a ruler or a force arrow passing exactly through the centre of the circle very few marks would have been scored. For those who didn’t know which way the arrows were supposed to be the common guesses were to the left and up the page. Some candidates neglected to label the arrows.</p>
<div class="question_part_label">aii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was generally well answered although usually to 3 sf. Common mistakes were to substitute 0.042 for F and 1 for q. Also some candidates tried to answer in terms of electric fields.</p>
<div class="question_part_label">bi.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was well answered with many candidates scoring ECF from the previous part.</p>
<div class="question_part_label">bii.</div>
</div>
<br><hr><br><div class="specification">
<p>The ball is now displaced through a small distance <em>x </em>from the bottom of the bowl and is then released from rest.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-08-14_om_06.19.20.png" alt="M18/4/PHYSI/HP2/ENG/TZ2/01.d"></p>
<p>The magnitude of the force on the ball towards the equilibrium position is given by</p>
<p style="text-align: left;"><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\frac{{mgx}}{R}">
<mfrac>
<mrow>
<mi>m</mi>
<mi>g</mi>
<mi>x</mi>
</mrow>
<mi>R</mi>
</mfrac>
</math></span></p>
<p>where <em>R </em>is the radius of the bowl.</p>
</div>
<div class="specification">
<p>A small ball of mass <em>m </em>is moving in a horizontal circle on the inside surface of a frictionless hemispherical bowl.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-08-12_om_12.45.38.png" alt="M18/4/PHYSI/SP2/ENG/TZ2/01.a"></p>
<p>The normal reaction force <em>N </em>makes an angle <em>θ</em> to the horizontal.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the direction of the resultant force on the ball.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the diagram, construct an arrow of the correct length to represent the weight of the ball.</p>
<p><img src="data:image/png;base64,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"></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the magnitude of the net force <em>F </em>on the ball is given by the following equation.</p>
<p> <span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="F = \frac{{mg}}{{\tan \theta }}">
<mi>F</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>m</mi>
<mi>g</mi>
</mrow>
<mrow>
<mi>tan</mi>
<mo></mo>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The radius of the bowl is 8.0 m and <em>θ</em> = 22°. Determine the speed of the ball.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Outline whether this ball can move on a horizontal circular path of radius equal to the radius of the bowl.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Outline why the ball will perform simple harmonic oscillations about the equilibrium position.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the period of oscillation of the ball is about 6 s.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The amplitude of oscillation is 0.12 m. On the axes, draw a graph to show the variation with time <em>t </em>of the velocity <strong><em>v </em></strong>of the ball during one period.</p>
<p><img src="data:image/png;base64,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"></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A second identical ball is placed at the bottom of the bowl and the first ball is displaced so that its height from the horizontal is equal to 8.0 m.</p>
<p> <img src="images/Schermafbeelding_2018-08-12_om_13.41.19.png" alt="M18/4/PHYSI/SP2/ENG/TZ2/01.d"></p>
<p>The first ball is released and eventually strikes the second ball. The two balls remain in contact. Determine, in m, the maximum height reached by the two balls.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>towards the centre <strong>«</strong>of the circle<strong>» </strong>/ horizontally to the right</p>
<p> </p>
<p><em>Do not accept towards the centre of the bowl</em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>downward vertical arrow of any length</p>
<p>arrow of correct length</p>
<p> </p>
<p><em>Judge the length of the vertical arrow by eye. The construction lines are not required. A label is not required</em></p>
<p><em>eg</em>: <img src="images/Schermafbeelding_2018-08-12_om_13.22.33.png" alt="M18/4/PHYSI/SP2/ENG/TZ2/01.a.ii"></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>ALTERNATIVE 1</em></strong></p>
<p><em>F</em> = <em>N</em> cos <em>θ</em></p>
<p><em>mg</em> = <em>N</em> sin <em>θ</em></p>
<p>dividing/substituting to get result</p>
<p> </p>
<p><strong><em>ALTERNATIVE 2</em></strong></p>
<p>right angle triangle drawn with <em>F</em>, <em>N </em>and <em>W/mg </em>labelled</p>
<p>angle correctly labelled and arrows on forces in correct directions</p>
<p>correct use of trigonometry leading to the required relationship</p>
<p> </p>
<p><img src="images/Schermafbeelding_2018-08-12_om_13.28.39.png" alt="M18/4/PHYSI/SP2/ENG/TZ2/01.a.ii"></p>
<p><em>tan θ</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\text{O}}}{A} = \frac{{mg}}{F}">
<mfrac>
<mrow>
<mtext>O</mtext>
</mrow>
<mi>A</mi>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>m</mi>
<mi>g</mi>
</mrow>
<mi>F</mi>
</mfrac>
</math></span></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{mg}}{{\tan \theta }}">
<mfrac>
<mrow>
<mi>m</mi>
<mi>g</mi>
</mrow>
<mrow>
<mi>tan</mi>
<mo></mo>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span> = <em>m</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{v^2}}}{r}">
<mfrac>
<mrow>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mi>r</mi>
</mfrac>
</math></span></p>
<p><em>r</em> = <em>R</em> cos <em>θ</em></p>
<p><em>v</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {\frac{{gR{{\cos }^2}\theta }}{{\sin \theta }}} /\sqrt {\frac{{gR\cos \theta }}{{\tan \theta }}} /\sqrt {\frac{{9.81 \times 8.0\cos 22}}{{\tan 22}}} ">
<msqrt>
<mfrac>
<mrow>
<mi>g</mi>
<mi>R</mi>
<mrow>
<msup>
<mrow>
<mi>cos</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
</mrow>
</mfrac>
</msqrt>
<mrow>
<mo>/</mo>
</mrow>
<msqrt>
<mfrac>
<mrow>
<mi>g</mi>
<mi>R</mi>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
</mrow>
<mrow>
<mi>tan</mi>
<mo></mo>
<mi>θ</mi>
</mrow>
</mfrac>
</msqrt>
<mrow>
<mo>/</mo>
</mrow>
<msqrt>
<mfrac>
<mrow>
<mn>9.81</mn>
<mo>×</mo>
<mn>8.0</mn>
<mi>cos</mi>
<mo></mo>
<mn>22</mn>
</mrow>
<mrow>
<mi>tan</mi>
<mo></mo>
<mn>22</mn>
</mrow>
</mfrac>
</msqrt>
</math></span></p>
<p><em>v</em> = 13.4/13 <strong>«</strong><em>ms <sup>–</sup></em><em><sup>1</sup></em><strong>»</strong></p>
<p> </p>
<p><em>Award </em><strong><em>[4] </em></strong><em>for a bald correct answer </em></p>
<p><em>Award </em><strong><em>[3] </em></strong><em>for an answer of 13.9/14 </em><strong>«</strong><em>ms <sup>–</sup></em><em><sup>1</sup></em><strong>»</strong><em>. MP2 omitted</em></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>there is no force to balance the weight/N is horizontal</p>
<p>so no / it is not possible</p>
<p> </p>
<p><em>Must see correct justification to award MP2</em></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the <strong>«</strong>restoring<strong>» </strong>force/acceleration is proportional to displacement</p>
<p> </p>
<p><em>Direction is not required</em></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>ω</em> = <strong>«</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {\frac{g}{R}} ">
<msqrt>
<mfrac>
<mi>g</mi>
<mi>R</mi>
</mfrac>
</msqrt>
</math></span><strong>»</strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {\frac{{9.81}}{{8.0}}} ">
<msqrt>
<mfrac>
<mrow>
<mn>9.81</mn>
</mrow>
<mrow>
<mn>8.0</mn>
</mrow>
</mfrac>
</msqrt>
</math></span> <strong>«</strong>= 1.107 s<sup>–1</sup><strong>»</strong></p>
<p><em>T</em> = <strong>«</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{\omega }">
<mfrac>
<mrow>
<mn>2</mn>
<mi>π</mi>
</mrow>
<mi>ω</mi>
</mfrac>
</math></span> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{{1.107}}">
<mfrac>
<mrow>
<mn>2</mn>
<mi>π</mi>
</mrow>
<mrow>
<mn>1.107</mn>
</mrow>
</mfrac>
</math></span> =<strong>»</strong> 5.7 <strong>«</strong>s<strong>»</strong></p>
<p> </p>
<p><em>Allow use of </em>or <em>g = 9.8 or 10</em></p>
<p><em>Award </em><strong><em>[0] </em></strong><em>for a substitution into T = 2π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {\frac{I}{g}} ">
<msqrt>
<mfrac>
<mi>I</mi>
<mi>g</mi>
</mfrac>
</msqrt>
</math></span></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>sine graph</p>
<p>correct amplitude <strong>«</strong>0.13 m s<sup>–1</sup><strong>»</strong></p>
<p>correct period and only 1 period shown</p>
<p> </p>
<p><em>Accept ± sine for shape of the graph. Accept 5.7 s or 6.0 s for the correct period.</em></p>
<p><em>Amplitude should be correct to ±</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <em>square for MP2</em></p>
<p><em>eg: v /</em>m s<sup>–1 </sup> <img src="images/Schermafbeelding_2018-08-14_om_06.59.06.png" alt="M18/4/PHYSI/HP2/ENG/TZ2/01.d.iii"></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>speed before collision <em>v</em> = <strong>«</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {2gR} "> <msqrt> <mn>2</mn> <mi>g</mi> <mi>R</mi> </msqrt> </math></span> =<strong>»</strong> 12.5 <strong>«</strong>ms<sup>–1</sup><strong>»</strong></p>
<p><strong>«</strong>from conservation of momentum<strong>» </strong>common speed after collision is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> initial speed <strong>«</strong><em>v<sub>c</sub></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{12.5}}{2}"> <mfrac> <mrow> <mn>12.5</mn> </mrow> <mn>2</mn> </mfrac> </math></span> = 6.25 ms<sup>–1</sup><strong>»</strong></p>
<p><em>h = </em><strong>«</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{v_c}^2}}{{2g}} = \frac{{{{6.25}^2}}}{{2 \times 9.81}}"> <mfrac> <mrow> <msup> <mrow> <msub> <mi>v</mi> <mi>c</mi> </msub> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn> <mi>g</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>6.25</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mn>9.81</mn> </mrow> </mfrac> </math></span><strong>»</strong> 2.0 <strong>«</strong>m<strong>»</strong></p>
<p> </p>
<p><em>Allow 12.5 from incorrect use of kinematics equations</em></p>
<p><em>Award </em><strong><em>[3] </em></strong><em>for a bald correct answer</em></p>
<p><em>Award </em><strong><em>[0] </em></strong><em>for mg(8) = 2mgh leading to h = 4 m if done in one step.</em></p>
<p><em>Allow ECF from MP1</em></p>
<p><em>Allow ECF from MP2</em></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The gravitational potential due to the Sun at its surface is –1.9 x 10<sup>11</sup> J kg<sup>–1</sup>. The following data are available.</p>
<table style="width: 441.4px; margin-left: 120px;">
<tbody>
<tr>
<td style="width: 422px;">Mass of Earth</td>
<td style="width: 558.4px;">= 6.0 x 10<sup>24</sup> kg</td>
</tr>
<tr>
<td style="width: 422px;">Distance from Earth to Sun</td>
<td style="width: 558.4px;">= 1.5 x 10<sup>11</sup> m</td>
</tr>
<tr>
<td style="width: 422px;">Radius of Sun</td>
<td style="width: 558.4px;">= 7.0 x 10<sup>8</sup> m</td>
</tr>
</tbody>
</table>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Outline why the gravitational potential is negative.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The gravitational potential due to the Sun at a distance <em>r</em> from its centre is <em>V</em><sub>S</sub>. Show that</p>
<p><em>rV</em><sub>S</sub> = constant.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the gravitational potential energy of the Earth in its orbit around the Sun. Give your answer to an appropriate number of significant figures.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the total energy of the Earth in its orbit.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>An asteroid strikes the Earth and causes the orbital speed of the Earth to suddenly decrease. Suggest the ways in which the orbit of the Earth will change.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iv.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Outline, in terms of the force acting on it, why the Earth remains in a circular orbit around the Sun.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>potential is defined to be zero at infinity</p>
<p>so a positive amount of work needs to be supplied for a mass to reach infinity</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>V</em><sub>S</sub> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{{GM}}{r}">
<mo>−</mo>
<mfrac>
<mrow>
<mi>G</mi>
<mi>M</mi>
</mrow>
<mi>r</mi>
</mfrac>
</math></span> so <em>r</em> x <em>V</em><sub>S</sub> «= –<em>GM</em>» = constant because<em> G</em> and<em> M</em> are constants</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>GM</em> = 1.33 x 10<sup>20</sup> «J m kg<sup>–1</sup>»</p>
<p>GPE at Earth orbit «= –<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1.33 \times {{10}^{20}} \times 6.0 \times {{10}^{24}}}}{{1.5 \times {{10}^{11}}}}">
<mfrac>
<mrow>
<mn>1.33</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mn>20</mn>
</mrow>
</msup>
</mrow>
<mo>×</mo>
<mn>6.0</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mn>24</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mn>1.5</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mn>11</mn>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>» = «–» 5.3 x 10<sup>33</sup> «J»</p>
<p> </p>
<p><em>Award <strong>[1 max]</strong> unless answer is to 2 sf.</em></p>
<p><em>Ignore addition of Sun radius to radius of Earth orbit.</em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>ALTERNATIVE 1</strong></em><br>work leading to statement that kinetic energy <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{GMm}}{{2r}}">
<mfrac>
<mrow>
<mi>G</mi>
<mi>M</mi>
<mi>m</mi>
</mrow>
<mrow>
<mn>2</mn>
<mi>r</mi>
</mrow>
</mfrac>
</math></span>, <em><strong>AND</strong></em> kinetic energy evaluated to be «+» 2.7 x 10<sup>33</sup> «J»</p>
<p>energy «= PE + KE = answer to (b)(ii) + 2.7 x 10<sup>33</sup>» = «–» 2.7 x 10<sup>33</sup> «J»</p>
<p> </p>
<p><em><strong>ALTERNATIVE 2</strong></em><br>statement that kinetic energy is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - \frac{1}{2}">
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> gravitational potential energy in orbit</p>
<p>so energy «<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{answer to (b)(ii)}}}}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>answer to (b)(ii)</mtext>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span>» = «–» 2.7 x 10<sup>33</sup> «J»</p>
<p> </p>
<p><em>Various approaches possible.</em></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>«KE will initially decrease so» total energy decreases<br><em><strong>OR</strong></em><br>«KE will initially decrease so» total energy becomes more negative</p>
<p>Earth moves closer to Sun</p>
<p>new orbit with greater speed «but lower total energy»</p>
<p>changes ellipticity of orbit</p>
<div class="question_part_label">b.iv.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>centripetal force is required</p>
<p>and is provided by gravitational force between Earth and Sun</p>
<p> </p>
<p><em>Award <strong>[1 max]</strong> for statement that there is a “centripetal force of gravity” without further qualification.</em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iv.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br>