HL Paper 2
Curling is a game played on a horizontal ice surface. A player pushes a large smooth stone across the ice for several seconds and then releases it. The stone moves until friction brings it to rest. The graph shows the variation of speed of the stone with time.
The total distance travelled by the stone in 17.5 s is 29.8 m.
Determine the coefficient of dynamic friction between the stone and the ice during the last 14.0 s of the stone’s motion.
The diagram shows the stone during its motion after release.
Label the diagram to show the forces acting on the stone. Your answer should include the name, the direction and point of application of each force.
Airboats are used for transport across a river. To move the boat forward, air is propelled from the back of the boat by a fan blade.
An airboat has a fan blade of radius 1.8 m. This fan can propel air with a maximum speed relative to the boat of 20 m s−1. The density of air is 1.2 kg m−3.
In a test the airboat is tied to the river bank with a rope normal to the bank. The fan propels the air at its maximum speed. There is no wind.
The rope is untied and the airboat moves away from the bank. The variation with time t of the speed v of the airboat is shown for the motion.
Outline why a force acts on the airboat due to the fan blade.
Show that a mass of about 240 kg of air moves through the fan every second.
Show that the tension in the rope is about 5 kN.
Explain why the airboat has a maximum speed under these conditions.
Estimate the distance the airboat travels to reach its maximum speed.
Deduce the mass of the airboat.
A small metal pendulum bob is suspended at rest from a fixed point with a length of thread of negligible mass. Air resistance is negligible.
The pendulum begins to oscillate. Assume that the motion of the system is simple harmonic, and in one vertical plane.
The graph shows the variation of kinetic energy of the pendulum bob with time.
When the 75 g bob is moving horizontally at 0.80 m s–1, it collides with a small stationary object also of mass 75 g. The object and the bob stick together.
Calculate, in m, the length of the thread. State your answer to an appropriate number of significant figures.
Label on the graph with the letter X a point where the speed of the pendulum is half that of its initial speed.
The mass of the pendulum bob is 75 g. Show that the maximum speed of the bob is about 0.7 m s–1.
Calculate the speed of the combined masses immediately after the collision.
Show that the collision is inelastic.
Sketch, on the axes, a graph to show the variation of gravitational potential energy with time for the bob and the object after the collision. The data from the graph used in (a) is shown as a dashed line for reference.
The speed after the collision of the bob and the object was measured using a sensor. This sensor emits a sound of frequency f and this sound is reflected from the moving bob. The sound is then detected by the sensor as frequency f′.
Explain why f and f′ are different.
A vertical wall carries a uniform positive charge on its surface. This produces a uniform horizontal electric field perpendicular to the wall. A small, positively-charged ball is suspended in equilibrium from the vertical wall by a thread of negligible mass.
The centre of the ball, still carrying a charge of 1.2 × 10−6 C, is now placed 0.40 m from a point charge Q. The charge on the ball acts as a point charge at the centre of the ball.
P is the point on the line joining the charges where the electric field strength is zero. The distance PQ is 0.22 m.
The charge per unit area on the surface of the wall is σ. It can be shown that the electric field strength E due to the charge on the wall is given by the equation
.
Demonstrate that the units of the quantities in this equation are consistent.
The thread makes an angle of 30° with the vertical wall. The ball has a mass of 0.025 kg.
Determine the horizontal force that acts on the ball.
The charge on the ball is 1.2 × 10−6 C. Determine σ.
The thread breaks. Explain the initial subsequent motion of the ball.
Calculate the charge on Q. State your answer to an appropriate number of significant figures.
Outline, without calculation, whether or not the electric potential at P is zero.
A non-uniform electric field, with field lines as shown, exists in a region where there is no gravitational field. X is a point in the electric field. The field lines and X lie in the plane of the paper.
Outline what is meant by electric field strength.
An electron is placed at X and released from rest. Draw, on the diagram, the direction of the force acting on the electron due to the field.
The electron is replaced by a proton which is also released from rest at X. Compare, without calculation, the motion of the electron with the motion of the proton after release. You may assume that no frictional forces act on the electron or the proton.
An elastic climbing rope is tested by fixing one end of the rope to the top of a crane. The other end of the rope is connected to a block which is initially at position A. The block is released from rest. The mass of the rope is negligible.
The unextended length of the rope is 60.0 m. From position A to position B, the block falls freely.
In another test, the block hangs in equilibrium at the end of the same elastic rope. The elastic constant of the rope is 400 Nm–1. The block is pulled 3.50 m vertically below the equilibrium position and is then released from rest.
An elastic climbing rope is tested by fixing one end of the rope to the top of a crane. The other end of the rope is connected to a block which is initially at position A. The block is released from rest. The mass of the rope is negligible.
The unextended length of the rope is 60.0 m. From position A to position B, the block falls freely.
At position C the speed of the block reaches zero. The time taken for the block to fall between B and C is 0.759 s. The mass of the block is 80.0 kg.
For the rope and block, describe the energy changes that take place
At position B the rope starts to extend. Calculate the speed of the block at position B.
Determine the magnitude of the average resultant force acting on the block between B and C.
Sketch on the diagram the average resultant force acting on the block between B and C. The arrow on the diagram represents the weight of the block.
Calculate the magnitude of the average force exerted by the rope on the block between B and C.
between A and B.
between B and C.
The length reached by the rope at C is 77.4 m. Suggest how energy considerations could be used to determine the elastic constant of the rope.
Calculate the time taken for the block to return to the equilibrium position for the first time.
Calculate the speed of the block as it passes the equilibrium position.
The ball is now displaced through a small distance x from the bottom of the bowl and is then released from rest.
The magnitude of the force on the ball towards the equilibrium position is given by
where R is the radius of the bowl.
A small ball of mass m is moving in a horizontal circle on the inside surface of a frictionless hemispherical bowl.
The normal reaction force N makes an angle θ to the horizontal.
State the direction of the resultant force on the ball.
On the diagram, construct an arrow of the correct length to represent the weight of the ball.
Show that the magnitude of the net force F on the ball is given by the following equation.
The radius of the bowl is 8.0 m and θ = 22°. Determine the speed of the ball.
Outline whether this ball can move on a horizontal circular path of radius equal to the radius of the bowl.
Outline why the ball will perform simple harmonic oscillations about the equilibrium position.
Show that the period of oscillation of the ball is about 6 s.
The amplitude of oscillation is 0.12 m. On the axes, draw a graph to show the variation with time t of the velocity v of the ball during one period.
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.
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.
A company delivers packages to customers using a small unmanned aircraft. Rotating horizontal blades exert a force on the surrounding air. The air above the aircraft is initially stationary.
The air is propelled vertically downwards with speed . The aircraft hovers motionless above the ground. A package is suspended from the aircraft on a string. The mass of the aircraft is and the combined mass of the package and string is . The mass of air pushed downwards by the blades in one second is .
State the value of the resultant force on the aircraft when hovering.
Outline, by reference to Newton’s third law, how the upward lift force on the aircraft is achieved.
Determine . State your answer to an appropriate number of significant figures.
Calculate the power transferred to the air by the aircraft.
The package and string are now released and fall to the ground. The lift force on the aircraft remains unchanged. Calculate the initial acceleration of the aircraft.
The moon Phobos moves around the planet Mars in a circular orbit.
Outline the origin of the force that acts on Phobos.
Outline why this force does no work on Phobos.
The orbital period T of a moon orbiting a planet of mass M is given by
where R is the average distance between the centre of the planet and the centre of the moon.
Show that
The following data for the Mars–Phobos system and the Earth–Moon system are available:
Mass of Earth = 5.97 × 1024 kg
The Earth–Moon distance is 41 times the Mars–Phobos distance.
The orbital period of the Moon is 86 times the orbital period of Phobos.
Calculate, in kg, the mass of Mars.
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.
Determine, using the graph, the mass of the Moon.
A chicken’s egg of mass 58 g is dropped onto grass from a height of 1.1 m. Assume that air resistance is negligible and that the egg does not bounce or break.
Define impulse.
Show that the kinetic energy of the egg just before impact is about 0.6 J.
The egg comes to rest in a time of 55 ms. Determine the magnitude of the average decelerating force that the ground exerts on the egg.
Explain why the egg is likely to break when dropped onto concrete from the same height.
Plutonium-238 (Pu) decays by alpha (α) decay into uranium (U).
The following data are available for binding energies per nucleon:
plutonium 7.568 MeV
uranium 7.600 MeV
alpha particle 7.074 MeV
The energy in b(i) can be transferred into electrical energy to run the instruments of a spacecraft. A spacecraft carries 33 kg of pure plutonium-238 at launch. The decay constant of plutonium is 2.50 × 10−10 s−1.
Solar radiation falls onto a metallic surface carried by the spacecraft causing the emission of photoelectrons. The radiation has passed through a filter so it is monochromatic. The spacecraft is moving away from the Sun.
State what is meant by the binding energy of a nucleus.
Draw, on the axes, a graph to show the variation with nucleon number of the binding energy per nucleon, . Numbers are not required on the vertical axis.
Identify, with a cross, on the graph in (a)(ii), the region of greatest stability.
Some unstable nuclei have many more neutrons than protons. Suggest the likely decay for these nuclei.
Show that the energy released in this decay is about 6 MeV.
The plutonium nucleus is at rest when it decays.
Calculate the ratio .
Estimate the power, in kW, that is available from the plutonium at launch.
The spacecraft will take 7.2 years (2.3 × 108 s) to reach a planet in the solar system. Estimate the power available to the spacecraft when it gets to the planet.
State and explain what happens to the kinetic energy of an emitted photoelectron.
State and explain what happens to the rate at which charge leaves the metallic surface.
Ion-thrust engines can power spacecraft. In this type of engine, ions are created in a chamber and expelled from the spacecraft. The spacecraft is in outer space when the propulsion system is turned on. The spacecraft starts from rest.
The mass of ions ejected each second is 6.6 × 10–6 kg and the speed of each ion is 5.2 × 104 m s–1. The initial total mass of the spacecraft and its fuel is 740 kg. Assume that the ions travel away from the spacecraft parallel to its direction of motion.
An initial mass of 60 kg of fuel is in the spacecraft for a journey to a planet. Half of the fuel will be required to slow down the spacecraft before arrival at the destination planet.
In practice, the ions leave the spacecraft at a range of angles as shown.
Determine the initial acceleration of the spacecraft.
(i) Estimate the maximum speed of the spacecraft.
(ii) Outline why the answer to (i) is an estimate.
Outline why scientists sometimes use estimates in making calculations.
Outline why the ions are likely to spread out.
Explain what effect, if any, this spreading of the ions has on the acceleration of the spacecraft.
A fixed horizontal coil is connected to an ideal voltmeter. A bar magnet is released from rest so that it falls vertically through the coil along the central axis of the coil.
The variation with time t of the emf induced in the coil is shown.
Write down the maximum magnitude of the rate of change of flux linked with the coil.
State the fundamental SI unit for your answer to (a)(i).
Explain why the graph becomes negative.
Part of the graph is above the t-axis and part is below. Outline why the areas between the t-axis and the curve for these two parts are likely to be the same.
Predict the changes to the graph when the magnet is dropped from a lower height above the coil.
Hydrogen atoms in an ultraviolet (UV) lamp make transitions from the first excited state to the ground state. Photons are emitted and are incident on a photoelectric surface as shown.
The photons cause the emission of electrons from the photoelectric surface. The work function of the photoelectric surface is 5.1 eV.
The electric potential of the photoelectric surface is 0 V. The variable voltage is adjusted so that the collecting plate is at –1.2 V.
Show that the energy of photons from the UV lamp is about 10 eV.
Calculate, in J, the maximum kinetic energy of the emitted electrons.
Suggest, with reference to conservation of energy, how the variable voltage source can be used to stop all emitted electrons from reaching the collecting plate.
The variable voltage can be adjusted so that no electrons reach the collecting plate. Write down the minimum value of the voltage for which no electrons reach the collecting plate.
On the diagram, draw and label the equipotential lines at –0.4 V and –0.8 V.
An electron is emitted from the photoelectric surface with kinetic energy 2.1 eV. Calculate the speed of the electron at the collecting plate.
A longitudinal wave travels in a medium with speed 340 m s−1. The graph shows the variation with time t of the displacement x of a particle P in the medium. Positive displacements on the graph correspond to displacements to the right for particle P.
Another wave travels in the medium. The graph shows the variation with time t of the displacement of each wave at the position of P.
A standing sound wave is established in a tube that is closed at one end and open at the other end. The period of the wave is . The diagram represents the standing wave at and at . The wavelength of the wave is 1.20 m. Positive displacements mean displacements to the right.
Calculate the wavelength of the wave.
Determine, for particle P, the magnitude and direction of the acceleration at t = 2.0 m s.
State the phase difference between the two waves.
Identify a time at which the displacement of P is zero.
Estimate the amplitude of the resultant wave.
Calculate the length of the tube.
A particle in the tube has its equilibrium position at the open end of the tube.
State and explain the direction of the velocity of this particle at time .
Draw on the diagram the standing wave at time .
A student strikes a tennis ball that is initially at rest so that it leaves the racquet at a speed of 64 m s–1. The ball has a mass of 0.058 kg and the contact between the ball and the racquet lasts for 25 ms.
The student strikes the tennis ball at point P. The tennis ball is initially directed at an angle of 7.00° to the horizontal.
The following data are available.
Height of P = 2.80 m
Distance of student from net = 11.9 m
Height of net = 0.910 m
Initial speed of tennis ball = 64 m s-1
Calculate the average force exerted by the racquet on the ball.
Calculate the average power delivered to the ball during the impact.
Calculate the time it takes the tennis ball to reach the net.
Show that the tennis ball passes over the net.
Determine the speed of the tennis ball as it strikes the ground.
A student models the bounce of the tennis ball to predict the angle θ at which the ball leaves a surface of clay and a surface of grass.
The model assumes
• during contact with the surface the ball slides.
• the sliding time is the same for both surfaces.
• the sliding frictional force is greater for clay than grass.
• the normal reaction force is the same for both surfaces.
Predict for the student’s model, without calculation, whether θ is greater for a clay surface or for a grass surface.
A metal sphere is charged positively and placed far away from other charged objects. The electric potential at a point on the surface of the sphere is 53.9 kV.
A small positively charged object moves towards the centre of the metal sphere. When the object is 2.8 m from the centre of the sphere, its speed is 3.1 m s−1. The mass of the object is 0.14 g and its charge is 2.4 × 10−8 C.
Outline what is meant by electric potential at a point.
The electric potential at a point a distance 2.8 m from the centre of the sphere is 7.71 kV. Determine the radius of the sphere.
Comment on the angle at which the object meets equipotential surfaces around the sphere.
Show that the kinetic energy of the object is about 0.7 mJ.
Determine whether the object will reach the surface of the sphere.
A student investigates how light can be used to measure the speed of a toy train.
Light from a laser is incident on a double slit. The light from the slits is detected by a light sensor attached to the train.
The graph shows the variation with time of the output voltage from the light sensor as the train moves parallel to the slits. The output voltage is proportional to the intensity of light incident on the sensor.
As the train continues to move, the first diffraction minimum is observed when the light sensor is at a distance of 0.13 m from the centre of the fringe pattern.
A student investigates how light can be used to measure the speed of a toy train.
Light from a laser is incident on a double slit. The light from the slits is detected by a light sensor attached to the train.
The graph shows the variation with time of the output voltage from the light sensor as the train moves parallel to the slits. The output voltage is proportional to the intensity of light incident on the sensor.
Explain, with reference to the light passing through the slits, why a series of voltage peaks occurs.
The slits are separated by 1.5 mm and the laser light has a wavelength of 6.3 x 10–7 m. The slits are 5.0 m from the train track. Calculate the separation between two adjacent positions of the train when the output voltage is at a maximum.
Estimate the speed of the train.
Determine the width of one of the slits.
Suggest the variation in the output voltage from the light sensor that will be observed as the train moves beyond the first diffraction minimum.
In another experiment the student replaces the light sensor with a sound sensor. The train travels away from a loudspeaker that is emitting sound waves of constant amplitude and frequency towards a reflecting barrier.
The graph shows the variation with time of the output voltage from the sounds sensor.
Explain how this effect arises.