Explain, in terms of the energy of its molecules, why the temperature of a pure substance does not change during melting.

Three ice cubes at a temperature of 0°C are dropped into a container of water at a temperature of 22°C. The mass of each ice cube is 25 g and the mass of the water is 330 g. The ice melts, so that the temperature of the water decreases. The thermal capacity of the container is negligible.

The following data are available.

Specific latent heat of fusion of ice \( = \) 3.3 \( \times \) 10⁵J kg^–1

Specific heat capacity of water \( = \) 4.2 \( \times \) 10³ J kg^–1 K^–1

Calculate the final temperature of the water when all of the ice has melted. Assume that no thermal energy is exchanged between the water and the surroundings.

For particle P,

(i) state how graph 1 shows that its oscillations are not damped.

(ii) calculate the magnitude of its maximum acceleration.

(iii) calculate its speed at t=0.12 s.

(iv) state its direction of motion at t=0.12 s.

Graph 2 shows the variation with position d of the displacement x of particles in the medium at a particular instant of time.

Graph 2

Determine for the longitudinal wave, using graph 1 and graph 2,

(i) the frequency.

(ii) the speed.

Graph 2 – reproduced to assist with answering (c)(i).

(c) The diagram shows the equilibrium positions of six particles in the medium.

(i) On the diagram above, draw crosses to indicate the positions of these six particles at the instant of time when the displacement is given by graph 2.

(ii) On the diagram above, label with the letter C a particle that is at the centre of a compression.

(i) the amplitude is constant;

(ii) period is 0.20s;

\({a_{\max }} = \left( {{{\left[ {\frac{{2\pi }}{T}} \right]}^2}{x_0} = {{31.4}^2} \times 2.0 \times {{10}^{ - 2}}} \right) = 19.7 \approx 20{\rm{m}}{{\rm{s}}^{ - 2}}\)

Award [2] for correct bald answer and ignore any negative signs in answer.

(iii) displacement at t = 0.12cm is (−)1.62cm;
\(v\left( { = \frac{{2\pi }}{T}\sqrt {{x_0} - {x^2}} } \right) = 31.4\sqrt {\left( {2.0 \times {{10}^{ - 2}}} \right)^2 - {{\left( {1.62 \times {{10}^{ - 2}}} \right)}^2}}  = 0.37{\rm{m}}{{\rm{s}}^{ - 1}}\);
Accept displacement in range 1.60 to 1.70 cm for an answer in range 0.33ms⁻¹ to 0.38ms⁻¹.

or

\({v_0} = \frac{{2\pi }}{T}{x_0} = 0.628{\rm{m}}{{\rm{s}}^{ - 1}}\);
\(\left| {\left. v \right|} \right. = \left( {\left| {\left. { - {v_0}\sin \left[ {\frac{{2\pi }}{T}t} \right]} \right|} \right. \Rightarrow \left| {\left. v \right|} \right. = \left| {\left. { - 0.628\sin \left[ {31.4 \times 0.12} \right]} \right|} \right. = \left| {\left. {0.37} \right|} \right.} \right) = 0.37{\rm{m}}{{\rm{s}}^{ - 1}}\);

or

drawing a tangent at 0.12s;
measurement of slope of tangent;
Accept answer in range 0.33ms⁻¹ to 0.38ms⁻¹ .

(i) use of \(f = \frac{1}{T}\);
and so \(f\left( { = \frac{1}{{0.20}}} \right) = 5.0{\rm{Hz}}\);

(ii) wavelength is 16cm;
and so speed is v(=fλ=5.0×0.16)=0.80ms⁻¹;

(i) points at 0, 8 and 16 cm stay in the same place;
points at 4 and 20 cm move 2 cm to the right;
point at 12 cm moves 2 cm to the left;

(ii) the point at 8 cm;

State the Stefan-Boltzmann law for a black body.

Deduce that the solar power incident per unit area at distance d from the Sun is given by

\[\frac{{\sigma {R^2}{T^4}}}{{{d^2}}}\]

Calculate, using the data given, the solar power incident per unit area at distance d from the Sun.

State two reasons why the solar power incident per unit area at a point on the surface of the Earth is likely to be different from your answer in (c).

The average power absorbed per unit area at the Earth’s surface is 240Wm^–2. By treating the Earth’s surface as a black body, show that the average surface temperature of the Earth is approximately 250K.

Explain why the actual surface temperature of the Earth is greater than the value in (e).

Outline the conditions necessary for the object to execute simple harmonic motion.

The sketch graph below shows how the displacement of the object from point O varies with time over three time periods.

(i) Label with the letter A a point at which the magnitude of the acceleration of the object is a maximum.

(ii) Label with the letter V a point at which the speed of the object is a maximum.

(iii) Sketch on the same axes a graph of how the displacement varies with time if a small frictional force acts on the object.

Point P now begins to move from side to side with a small amplitude and at a variable driving frequency f. The frictional force is still small.

At each value of f, the object eventually reaches a constant amplitude A.

The graph shows the variation with f of A.

(i) With reference to resonance and resonant frequency, comment on the shape of the graph.

(ii) On the same axes, draw a graph to show the variation with f of A when the frictional force acting on the object is increased.

power/energy per second emitted proportional to surface area;
and proportional to fourth power of absolute temperature / temperature in K;
Accept equation with symbols defined.

solar power given by 4πR2σT 4 ;
spreads out over sphere of surface area 4πd 2 ;
Hence equation given.

\(\left( {\frac{{\sigma {R^2}{T^4}}}{{{d^2}}} = } \right)\frac{{5.7 \times {{10}^{ - 8}} \times {{\left[ {7.0 \times {{10}^8}} \right]}^2} \times {{\left[ {5.8 \times {{10}^3}} \right]}^4}}}{{{{\left[ {1.5 \times {{10}^{11}}} \right]}^2}}}\);
=1.4×10³(Wm^-2 );
Award [2] for a bald correct answer.

power radiated = power absorbed;
\(T = {}^4\sqrt {\frac{{240}}{{5.7 \times {{10}^{ - 8}}}}}  = \left( {250{\rm{K}}} \right)\);
Accept answers given as 260 (K).

radiation from Sun is re-emitted from Earth at longer wavelengths; greenhouse gases in the atmosphere absorb some of this energy; and radiate some of it back to the surface of the Earth;