SL Paper 2
A student uses a load to pull a box up a ramp inclined at 30°. A string of constant length and negligible mass connects the box to the load that falls vertically. The string passes over a pulley that runs on a frictionless axle. Friction acts between the base of the box and the ramp. Air resistance is negligible.
The load has a mass of 3.5 kg and is initially 0.95 m above the floor. The mass of the box is 1.5 kg.
The load is released and accelerates downwards.
Outline two differences between the momentum of the box and the momentum of the load at the same instant.
The vertical acceleration of the load downwards is 2.4 m s−2.
Calculate the tension in the string.
Show that the speed of the load when it hits the floor is about 2.1 m s−1.
The radius of the pulley is 2.5 cm. Calculate the angular speed of rotation of the pulley as the load hits the floor. State your answer to an appropriate number of significant figures.
After the load has hit the floor, the box travels a further 0.35 m along the ramp before coming to rest. Determine the average frictional force between the box and the surface of the ramp.
The student then makes the ramp horizontal and applies a constant horizontal force to the box. The force is just large enough to start the box moving. The force continues to be applied after the box begins to move.
Explain, with reference to the frictional force acting, why the box accelerates once it has started to move.
Two players are playing table tennis. Player A hits the ball at a height of 0.24 m above the edge of the table, measured from the top of the table to the bottom of the ball. The initial speed of the ball is 12.0 m s−1 horizontally. Assume that air resistance is negligible.
The ball bounces and then reaches a peak height of 0.18 m above the table with a horizontal speed of 10.5 m s−1. The mass of the ball is 2.7 g.
Show that the time taken for the ball to reach the surface of the table is about 0.2 s.
Sketch, on the axes, a graph showing the variation with time of the vertical component of velocity vv of the ball until it reaches the table surface. Take g to be +10 m s−2.
The net is stretched across the middle of the table. The table has a length of 2.74 m and the net has a height of 15.0 cm.
Show that the ball will go over the net.
Determine the kinetic energy of the ball immediately after the bounce.
Player B intercepts the ball when it is at its peak height. Player B holds a paddle (racket) stationary and vertical. The ball is in contact with the paddle for 0.010 s. Assume the collision is elastic.
Calculate the average force exerted by the ball on the paddle. State your answer to an appropriate number of significant figures.
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.
The 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.
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.
Estimate the distance the airboat travels to reach its maximum speed.
Deduce the mass of the airboat.
The fan is rotating at 120 revolutions every minute. Calculate the centripetal acceleration of the tip of a fan blade.
A girl on a sledge is moving down a snow slope at a uniform speed.
The sledge, without the girl on it, now travels up a snow slope that makes an angle of 6.5˚ to the horizontal. At the start of the slope, the speed of the sledge is 4.2 m s–1. The coefficient of dynamic friction of the sledge on the snow is 0.11.
Draw the free-body diagram for the sledge at the position shown on the snow slope.
After leaving the snow slope, the girl on the sledge moves over a horizontal region of snow. Explain, with reference to the physical origin of the forces, why the vertical forces on the girl must be in equilibrium as she moves over the horizontal region.
When the sledge is moving on the horizontal region of the snow, the girl jumps off the sledge. The girl has no horizontal velocity after the jump. The velocity of the sledge immediately after the girl jumps off is 4.2 m s–1. The mass of the girl is 55 kg and the mass of the sledge is 5.5 kg. Calculate the speed of the sledge immediately before the girl jumps from it.
The girl chooses to jump so that she lands on loosely-packed snow rather than frozen ice. Outline why she chooses to land on the snow.
Show that the acceleration of the sledge is about –2 m s–2.
Calculate the distance along the slope at which the sledge stops moving. Assume that the coefficient of dynamic friction is constant.
The coefficient of static friction between the sledge and the snow is 0.14. Outline, with a calculation, the subsequent motion of the sledge.
A football player kicks a stationary ball of mass 0.45 kg towards a wall. The initial speed of the ball after the kick is 19 m s−1 and the ball does not rotate. Air resistance is negligible and there is no wind.
The player’s foot is in contact with the ball for 55 ms. Calculate the average force that acts on the ball due to the football player.
The ball leaves the ground at an angle of 22°. The horizontal distance from the initial position of the edge of the ball to the wall is 11 m. Calculate the time taken for the ball to reach the wall.
The top of the wall is 2.4 m above the ground. Deduce whether the ball will hit the wall.
In practice, air resistance affects the ball. Outline the effect that air resistance has on the vertical acceleration of the ball. Take the direction of the acceleration due to gravity to be positive.
The player kicks the ball again. It rolls along the ground without sliding with a horizontal velocity of . The radius of the ball is . Calculate the angular velocity of the ball. State an appropriate SI unit for your answer.
The graph shows the variation with time t of the horizontal force F exerted on a tennis ball by a racket.
The tennis ball was stationary at the instant when it was hit. The mass of the tennis ball is 5.8 × 10–2 kg. The area under the curve is 0.84 N s.
Calculate the speed of the ball as it leaves the racket.
Show that the average force exerted on the ball by the racket is about 50 N.
Determine, with reference to the work done by the average force, the horizontal distance travelled by the ball while it was in contact with the racket.
Draw a graph to show the variation with t of the horizontal speed v of the ball while it was in contact with the racket. Numbers are not required on the axes.
An electron is placed at a distance of 0.40 m from a fixed point charge of –6.0 mC.
Show that the electric field strength due to the point charge at the position of the electron is 3.4 × 108 N C–1.
Calculate the magnitude of the initial acceleration of the electron.
Describe the subsequent motion of the electron.
A mass of 1.0 kg of water is brought to its boiling point of 100 °C using an electric heater of power 1.6 kW.
A mass of 0.86 kg of water remains after it has boiled for 200 s.
The electric heater has two identical resistors connected in parallel.
The circuit transfers 1.6 kW when switch A only is closed. The external voltage is 220 V.
The molar mass of water is 18 g mol−1. Estimate the average speed of the water molecules in the vapor produced. Assume the vapor behaves as an ideal gas.
State one assumption of the kinetic model of an ideal gas.
Estimate the specific latent heat of vaporization of water. State an appropriate unit for your answer.
Explain why the temperature of water remains at 100 °C during this time.
The heater is removed and a mass of 0.30 kg of pasta at −10 °C is added to the boiling water.
Determine the equilibrium temperature of the pasta and water after the pasta is added. Other heat transfers are negligible.
Specific heat capacity of pasta = 1.8 kJ kg−1 K−1
Specific heat capacity of water = 4.2 kJ kg−1 K−1
Show that each resistor has a resistance of about 30 Ω.
Calculate the power transferred by the heater when both switches are closed.
A girl rides a bicycle that is powered by an electric motor. A battery transfers energy to the electric motor. The emf of the battery is 16 V and it can deliver a charge of 43 kC when discharging completely from a full charge.
The maximum speed of the girl on a horizontal road is 7.0 m s–1 with energy from the battery alone. The maximum distance that the girl can travel under these conditions is 20 km.
The bicycle and the girl have a total mass of 66 kg. The girl rides up a slope that is at an angle of 3.0° to the horizontal.
The bicycle has a meter that displays the current and the terminal potential difference (pd) for the battery when the motor is running. The diagram shows the meter readings at one instant. The emf of the cell is 16 V.
The battery is made from an arrangement of 10 identical cells as shown.
Show that the time taken for the battery to discharge is about 3 × 103 s.
Deduce that the average power output of the battery is about 240 W.
Friction and air resistance act on the bicycle and the girl when they move. Assume that all the energy is transferred from the battery to the electric motor. Determine the total average resistive force that acts on the bicycle and the girl.
Calculate the component of weight for the bicycle and girl acting down the slope.
The battery continues to give an output power of 240 W. Assume that the resistive forces are the same as in (a)(iii).
Calculate the maximum speed of the bicycle and the girl up the slope.
On another journey up the slope, the girl carries an additional mass. Explain whether carrying this mass will change the maximum distance that the bicycle can travel along the slope.
Determine the internal resistance of the battery.
Calculate the emf of one cell.
Calculate the internal resistance of one cell.
The air in a kitchen has pressure 1.0 × 105 Pa and temperature 22°C. A refrigerator of internal volume 0.36 m3 is installed in the kitchen.
The refrigerator door is closed. The air in the refrigerator is cooled to 5.0°C and the number of air molecules in the refrigerator stays the same.
With the door open the air in the refrigerator is initially at the same temperature and pressure as the air in the kitchen. Calculate the number of molecules of air in the refrigerator.
Determine the pressure of the air inside the refrigerator.
The door of the refrigerator has an area of 0.72 m2. Show that the minimum force needed to open the refrigerator door is about 4 kN.
Comment on the magnitude of the force in (b)(ii).
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.
A small metal pendulum bob of mass 75 g is suspended at rest from a fixed point with a length of thread of negligible mass. Air resistance is negligible. The bob is then displaced to the left.
At time t = 0 the bob is moving horizontally to the right at 0.8 m s–1. It collides with a small stationary object also of mass 75 g. Both objects then move together with motion that is simple harmonic.
Calculate the speed of the combined masses immediately after the collision.
Show that the collision is inelastic.
Describe the changes in gravitational potential energy of the oscillating system from t = 0 as it oscillates through one cycle of its motion.
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.
On arrival at the planet, the spacecraft goes into orbit as it comes into the gravitational field of the planet.
Determine the initial acceleration of the spacecraft.
Estimate the maximum speed of the spacecraft.
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.
Outline what is meant by the gravitational field strength at a point.
Newton’s law of gravitation applies to point masses. Suggest why the law can be applied to a satellite orbiting a spherical planet of uniform density.
A proton is moving in a region of uniform magnetic field. The magnetic field is directed into the plane of the paper. The arrow shows the velocity of the proton at one instant and the dotted circle gives the path followed by the proton.
The speed of the proton is 2.0 × 106 m s–1 and the magnetic field strength B is 0.35 T.
Explain why the path of the proton is a circle.
Show that the radius of the path is about 6 cm.
Calculate the time for one complete revolution.
Explain why the kinetic energy of the proton is constant.
A glider is an aircraft with no engine. To be launched, a glider is uniformly accelerated from rest by a cable pulled by a motor that exerts a horizontal force on the glider throughout the launch.
The glider reaches its launch speed of 27.0 m s–1 after accelerating for 11.0 s. Assume that the glider moves horizontally until it leaves the ground. Calculate the total distance travelled by the glider before it leaves the ground.
The glider and pilot have a total mass of 492 kg. During the acceleration the glider is subject to an average resistive force of 160 N. Determine the average tension in the cable as the glider accelerates.
The cable is pulled by an electric motor. The motor has an overall efficiency of 23 %. Determine the average power input to the motor.
The cable is wound onto a cylinder of diameter 1.2 m. Calculate the angular velocity of the cylinder at the instant when the glider has a speed of 27 m s–1. Include an appropriate unit for your answer.
After takeoff the cable is released and the unpowered glider moves horizontally at constant speed. The wings of the glider provide a lift force. The diagram shows the lift force acting on the glider and the direction of motion of the glider.
Draw the forces acting on the glider to complete the free-body diagram. The dotted lines show the horizontal and vertical directions.
Explain, using appropriate laws of motion, how the forces acting on the glider maintain it in level flight.
At a particular instant in the flight the glider is losing 1.00 m of vertical height for every 6.00 m that it goes forward horizontally. At this instant, the horizontal speed of the glider is 12.5 m s–1. Calculate the velocity of the glider. Give your answer to an appropriate number of significant figures.
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.
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.
A ball of mass 0.250 kg is released from rest at time t = 0, from a height H above a horizontal floor.
The graph shows the variation with time t of the velocity v of the ball. Air resistance is negligible. Take g = −9.80 m s−2. The ball reaches the floor after 1.0 s.
Determine H.
Label the time and velocity graph, using the letter M, the point where the ball reaches the maximum rebound height.
State the acceleration of the ball at the maximum rebound height.
Draw, on the axes, a graph to show the variation with time of the height of the ball from the instant it rebounds from the floor until the instant it reaches the maximum rebound height. No numbers are required on the axes.
Estimate the loss in the mechanical energy of the ball as a result of the collision with the floor.
Determine the average force exerted on the floor by the ball.
Suggest why the momentum of the ball was not conserved during the collision with the floor.
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.
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.
The Rotor is an amusement park ride that can be modelled as a vertical cylinder of inner radius rotating about its axis. When the cylinder rotates sufficiently fast, the floor drops out and the passengers stay motionless against the inner surface of the cylinder. The diagram shows a person taking the Rotor ride. The floor of the Rotor has been lowered away from the person.
Draw and label the free-body diagram for the person.
The person must not slide down the wall. Show that the minimum angular velocity of the cylinder for this situation is
where is the coefficient of static friction between the person and the cylinder.
The coefficient of static friction between the person and the cylinder is . The radius of the cylinder is . The cylinder makes revolutions per minute. Deduce whether the person will slide down the inner surface of the cylinder.
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
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.
Show that the energy released in this decay is about 6 MeV.
The plutonium nucleus is at rest when it decays.
Calculate the ratio .
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.
Label with arrows on the diagram the magnetic force F on the proton.
Label with arrows on the velocity vector v of the proton.
The speed of the proton is 2.16 × 106 m s-1 and the magnetic field strength is 0.042 T. For this proton, determine, in m, the radius of the circular path. Give your answer to an appropriate number of significant figures.
The diagram below shows part of a downhill ski course which starts at point A, 50 m above level ground. Point B is 20 m above level ground.
A skier of mass 65 kg starts from rest at point A and during the ski course some of the gravitational potential energy transferred to kinetic energy.
At the side of the course flexible safety nets are used. Another skier of mass 76 kg falls normally into the safety net with speed 9.6 m s–1.
From A to B, 24 % of the gravitational potential energy transferred to kinetic energy. Show that the velocity at B is 12 m s–1.
Some of the gravitational potential energy transferred into internal energy of the skis, slightly increasing their temperature. Distinguish between internal energy and temperature.
The dot on the following diagram represents the skier as she passes point B.
Draw and label the vertical forces acting on the skier.
The hill at point B has a circular shape with a radius of 20 m. Determine whether the skier will lose contact with the ground at point B.
The skier reaches point C with a speed of 8.2 m s–1. She stops after a distance of 24 m at point D.
Determine the coefficient of dynamic friction between the base of the skis and the snow. Assume that the frictional force is constant and that air resistance can be neglected.
Calculate the impulse required from the net to stop the skier and state an appropriate unit for your answer.
Explain, with reference to change in momentum, why a flexible safety net is less likely to harm the skier than a rigid barrier.
A chicken’s egg of mass 58 g is dropped onto grass from a height of 1.1 m. The egg comes to rest in a time of 55 ms. Assume that air resistance is negligible and that the egg does not bounce or break.
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.
A stationary nucleus of uranium-238 undergoes alpha decay to form thorium-234.
The following data are available.
Energy released in decay 4.27 MeV
Binding energy per nucleon for helium 7.07 MeV
Binding energy per nucleon for thorium 7.60 MeV
Radioactive decay is said to be “random” and “spontaneous”. Outline what is meant by each of these terms.
Random:
Spontaneous:
Calculate the binding energy per nucleon for uranium-238.
Calculate the ratio .
A charged particle, P, of charge +68 μC is fixed in space. A second particle, Q, of charge +0.25 μC is held at a distance of 48 cm from P and is then released.
The diagram shows two parallel wires X and Y that carry equal currents into the page.
Point Q is equidistant from the two wires. The magnetic field at Q due to wire X alone is 15 mT.
The work done to move a particle of charge 0.25 μC from one point in an electric field to another is 4.5 μJ. Calculate the magnitude of the potential difference between the two points.
Determine the force on Q at the instant it is released.
Describe the motion of Q after release.
On the diagram draw an arrow to show the direction of the magnetic field at Q due to wire X alone.
Determine the magnitude and direction of the resultant magnetic field at Q.
The Moon has no atmosphere and orbits the Earth. The diagram shows the Moon with rays of light from the Sun that are incident at 90° to the axis of rotation of the Moon.
A black body is on the Moon’s surface at point A. Show that the maximum temperature that this body can reach is 400 K. Assume that the Earth and the Moon are the same distance from the Sun.
Another black body is on the Moon’s surface at point B.
Outline, without calculation, why the aximum temperature of the black body at point B is less than at point A.
The albedo of the Earth’s atmosphere is 0.28. Outline why the maximum temperature of a black body on the Earth when the Sun is overhead is less than that at point A on the Moon.
Outline why a force acts on the Moon.
Outline why this force does no work on the Moon.
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.
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 sound sensor gives a graph of the variation of output voltage with time along the track that is similar in shape to the graph shown in the resource. Explain how this effect arises.
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 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 centre of the ball, still carrying a charge of , is now placed 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 .
Calculate the charge on Q. State your answer to an appropriate number of significant figures.