SL Paper 2
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.
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.
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 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.
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.
(i) Define gravitational field strength.
(ii) State the SI unit for gravitational field strength.
A planet orbits the Sun in a circular orbit with orbital period T and orbital radius R. The mass of the Sun is M.
(i) Show that .
(ii) The Earth’s orbit around the Sun is almost circular with radius 1.5×1011 m. Estimate the mass of the Sun.
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.
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 planet is in a circular orbit around a star. The speed of the planet is constant.
Explain why a centripetal force is needed for the planet to be in a circular orbit.
State the nature of this centripetal force.
Determine the gravitational field of the planet.
The following data are given:
Mass of planet kg
Radius of the planet m.
An electron moves in circular motion in a uniform magnetic field.
The velocity of the electron at point P is 6.8 × 105 m s–1 in the direction shown.
The magnitude of the magnetic field is 8.5 T.
State the direction of the magnetic field.
Calculate, in N, the magnitude of the magnetic force acting on the electron.
Explain why the electron moves at constant speed.
Explain why the electron moves on a circular path.
Titan is a moon of Saturn. The Titan-Sun distance is 9.3 times greater than the Earth-Sun distance.
Show that the intensity of the solar radiation at the location of Titan is 16 W m−2
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 average intensity of solar radiation absorbed by the whole surface of Titan is 3.1 W m−2
Show that the equilibrium surface temperature of Titan is about 90 K.
The orbital radius of Titan around Saturn is and the period of revolution is .
Show that where is the mass of Saturn.
The orbital radius of Titan around Saturn is 1.2 × 109 m and the orbital period is 15.9 days. Estimate the mass of Saturn.
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 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.
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.
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 satellite powered by solar cells directed towards the Sun is in a polar orbit about the Earth.
The satellite is orbiting the Earth at a distance of 6600 km from the centre of the Earth.
The satellite carries an experiment that measures the peak wavelength emitted by different objects. The Sun emits radiation that has a peak wavelength λS of 509 nm. The peak wavelength λE of the radiation emitted by the Earth is 10.1 μm.
Determine the orbital period for the satellite.
Mass of Earth = 6.0 x 1024 kg
Determine the mean temperature of the Earth.
Suggest how the difference between λS and λE helps to account for the greenhouse effect.
Not all scientists agree that global warming is caused by the activities of man.
Outline how scientists try to ensure agreement on a scientific issue.