SL Paper 3
An ideal nuclear power plant can be modelled as a heat engine that operates between a hot temperature of 612°C and a cold temperature of 349°C.
Calculate the Carnot efficiency of the nuclear power plant.
Explain, with a reason, why a real nuclear power plant operating between the stated temperatures cannot reach the efficiency calculated in (a).
The nuclear power plant works at 71.0% of the Carnot efficiency. The power produced is 1.33 GW. Calculate how much waste thermal energy is released per hour.
Discuss the production of waste heat by the power plant with reference to the first law and the second law of thermodynamics.
A flywheel consists of a solid cylinder, with a small radial axle protruding from its centre.
The following data are available for the flywheel.
| Flywheel mass M | = 1.22 kg |
| Small axle radius r | = 60.0 mm |
| Flywheel radius R | = 240 mm |
| Moment of inertia | = 0.5 MR2 |
An object of mass m is connected to the axle by a light string and allowed to fall vertically from rest, exerting a torque on the flywheel.
The velocity of the falling object is 1.89 m s–1 at 3.98 s. Calculate the average angular acceleration of the flywheel.
Show that the torque acting on the flywheel is about 0.3 Nm.
(i) Calculate the tension in the string.
(ii) Determine the mass m of the falling object.
A horizontal rigid bar of length 2R is pivoted at its centre. The bar is free to rotate in a horizontal plane about a vertical axis through the pivot. A point particle of mass M is attached to one end of the bar and a container is attached to the other end of the bar.
A point particle of mass moving with speed v at right angles to the rod collides with the container and gets stuck in the container. The system then starts to rotate about the vertical axis.
The mass of the rod and the container can be neglected.
A torque of 0.010 N m brings the system to rest after a number of revolutions. For this case R = 0.50 m, M = 0.70 kg and v = 2.1 m s–1.
Write down an expression, in terms of M, v and R, for the angular momentum of the system about the vertical axis just before the collision.
Just after the collision the system begins to rotate about the vertical axis with angular velocity ω. Show that the angular momentum of the system is equal to .
Hence, show that .
Determine in terms of M and v the energy lost during the collision.
Show that the angular deceleration of the system is 0.043 rads–2.
Calculate the number of revolutions made by the system before it comes to rest.
The P–V diagram of the Carnot cycle for a monatomic ideal gas is shown.
The system consists of 0.150 mol of a gas initially at A. The pressure at A is 512 k Pa and the volume is 1.20 × 10–3 m3.
At C the volume is VC and the temperature is TC.
State what is meant by an adiabatic process.
Identify the two isothermal processes.
Determine the temperature of the gas at A.
The volume at B is 2.30 × 10–3m3. Determine the pressure at B.
Show that
The volume at C is 2.90 × 10–3m3. Calculate the temperature at C.
State a reason why a Carnot cycle is of little use for a practical heat engine.
A bar rotates horizontally about its centre, reaching a maximum angular velocity in six complete rotations from rest. The bar has a constant angular acceleration of . The moment of inertia of the bar about the axis of rotation is .
Show that the final angular velocity of the bar is about .
Draw the variation with time of the angular displacement of the bar during the acceleration.
Calculate the torque acting on the bar while it is accelerating.
The torque is removed. The bar comes to rest in complete rotations with constant angular deceleration. Determine the time taken for the bar to come to rest.
A cylindrical space probe of mass 8.00 x 102 kg and diameter 12.0 m is at rest in outer space.
Rockets at opposite points on the probe are fired so that the probe rotates about its axis. Each rocket produces a force F = 9.60 x 103 N. The moment of inertia of the probe about its axis is 1.44 x 104 kgm2.
The diagram shows a satellite approaching the rotating probe with negligibly small speed. The satellite is not rotating initially, but after linking to the probe they both rotate together.
The moment of inertia of the satellite about its axis is 4.80 x 103 kgm2. The axes of the probe and of the satellite are the same.
Deduce the linear acceleration of the centre of mass of the probe.
Calculate the resultant torque about the axis of the probe.
The forces act for 2.00 s. Show that the final angular speed of the probe is about 16 rads–1.
Determine the final angular speed of the probe–satellite system.
Calculate the loss of rotational kinetic energy due to the linking of the probe with the satellite.
Two of the brightest objects in the night sky are the planet Jupiter and the star Vega.
The light observed from Jupiter has a similar brightness to that received from Vega.
Vega is found in the constellation Lyra. The stellar parallax angle of Vega is about 0.13 arc sec.
Identify the mechanism leading stars to produce the light they emit.
Outline why the light detected from Jupiter and Vega have a similar brightness, according to an observer on Earth.
Outline what is meant by a constellation.
Outline how the stellar parallax angle is measured.
Show that the distance to Vega from Earth is about 25 ly.
A train is passing through a tunnel of proper length 80 m. The proper length of the train is 100 m. According to an observer at rest relative to the tunnel, when the front of the train coincides with one end of the tunnel, the rear of the train coincides with the other end of the tunnel.
Explain what is meant by proper length.
Draw a spacetime diagram for this situation according to an observer at rest relative to the tunnel.
Calculate the velocity of the train, according to an observer at rest relative to the tunnel, at which the train fits the tunnel.
For an observer on the train, it is the tunnel that is moving and therefore will appear length contracted. This seems to contradict the observation made by the observer at rest to the tunnel, creating a paradox. Explain how this paradox is resolved. You may refer to your spacetime diagram in (b).
An astronomical reflecting telescope is being used to observe the night sky.
The diagram shows an incomplete reflecting telescope.
Complete the diagram, with a Newtonian mounting, continuing the two rays to show how they pass through the eyepiece.
When the Earth-Moon distance is 363 300 km, the Moon is observed using the telescope. The mean radius of the Moon is 1737 km. Determine the focal length of the mirror used in this telescope when the diameter of the Moon’s image formed by the main mirror is 1.20 cm.
The final image of the Moon is observed through the eyepiece. The focal length of the eyepiece is 5.0 cm. Calculate the magnification of the telescope.
The Hubble Space reflecting telescope has a Cassegrain mounting. Outline the main optical difference between a Cassegrain mounting and a Newtonian mounting.
A heat engine operates on the cycle shown in the pressure–volume diagram. The cycle consists of an isothermal expansion AB, an isovolumetric change BC and an adiabatic compression CA. The volume at B is double the volume at A. The gas is an ideal monatomic gas.
At A the pressure of the gas is 4.00 x 106 Pa, the temperature is 612 K and the volume is 1.50 x 10–4m3. The work done by the gas during the isothermal expansion is 416 J.
Justify why the thermal energy supplied during the expansion AB is 416 J.
Show that the temperature of the gas at C is 386 K.
Show that the thermal energy removed from the gas for the change BC is approximately 330 J.
Determine the efficiency of the heat engine.
State and explain at which point in the cycle ABCA the entropy of the gas is the largest.
A solid sphere of radius and mass is released from rest and rolls down a slope, without slipping. The vertical height of the slope is . The moment of inertia of this sphere about an axis through its centre is .
Show that the linear velocity of the sphere as it leaves the slope is .
The diagram shows two methods of pedalling a bicycle using a force F.
In method 1 the pedal is always horizontal to the ground. A student claims that method 2 is better because the pedal is always parallel to the crank arm. Explain why method 2 is more effective.
A constant force of 50.0 N is applied tangentially to the outer edge of a merry-go-round. The following diagram shows the view from above.
The merry-go-round has a moment of inertia of 450 kg m2 about a vertical axis. The merry-go-round has a diameter of 4.00 m.
A child of mass 30.0 kg is now placed onto the edge of the merry-go-round. No external torque acts on the system.
The child now moves towards the centre.
The merry-go-round starts from rest and the force is applied for one complete revolution.
Show that the angular acceleration of the merry-go-round is 0.2 rad s–2.
Calculate, for the merry-go-round after one revolution, the angular speed.
Calculate, for the merry-go-round after one revolution, the angular momentum.
Calculate the new angular speed of the rotating system.
Explain why the angular speed will increase.
Calculate the work done by the child in moving from the edge to the centre.
A hoop of mass m, radius r and moment of inertia mr2 rests on a rough plane inclined at an angle θ to the horizontal. It is released so that the hoop gains linear and angular acceleration by rolling, without slipping, down the plane.
On the diagram, draw and label the forces acting on the hoop.
Show that the linear acceleration a of the hoop is given by the equation shown.
a =
Calculate the acceleration of the hoop when θ = 20°. Assume that the hoop continues to roll without slipping.
State the relationship between the force of friction and the angle of the incline.
The angle of the incline is slowly increased from zero. Determine the angle, in terms of the coefficient of friction, at which the hoop will begin to slip.
A monatomic ideal gas is confined to a cylinder with volume 2.0 x 10–3 m3. The initial pressure of the gas is 100 kPa. The gas undergoes a three-step cycle. First, the gas pressure increases by a factor of five under constant volume. Then, the gas expands adiabatically to its initial pressure. Finally it is compressed at constant pressure to its initial volume.
Show that the volume of the gas at the end of the adiabatic expansion is approximately 5.3 x 10–3 m3.
Using the axes, sketch the three-step cycle.
The initial temperature of the gas is 290 K. Calculate the temperature of the gas at the start of the adiabatic expansion.
Using your sketched graph in (b), identify the feature that shows that net work is done by the gas in this three-step cycle.
The first diagram shows a person standing on a turntable which can rotate freely. The person is stationary and holding a bicycle wheel. The wheel rotates anticlockwise when seen from above.
© International Baccalaureate Organization 2020.
The wheel is flipped, as shown in the second diagram, so that it rotates clockwise when seen from above.
© International Baccalaureate Organization 2020.
Explain the direction in which the person-turntable system starts to rotate.
Explain the changes to the rotational kinetic energy in the person-turntable system.
The pV diagram of a heat engine using an ideal gas consists of an isothermal expansion A → B, an isobaric compression B → C and an adiabatic compression C → A.
The following data are available:
Temperature at A = 385 K
Pressure at A = 2.80 × 106 Pa
Volume at A = 1.00 × 10–4 m3
Volume at B = 2.80 × 10–4 m3
Volume at C = 1.85 × 10–4 m3
Show that at C the pressure is 1.00 × 106 Pa.
Show that at C the temperature is 254 K.
Show that the thermal energy transferred from the gas during the change B → C is 238 J.
The work done by the gas from A → B is 288 J. Calculate the efficiency of the cycle.
State, without calculation, during which change (A → B, B → C or C → A) the entropy of the gas decreases.
A uniform rod of weight 36.0 N and length 5.00 m rests horizontally. The rod is pivoted at its left-hand end and is supported at a distance of 4.00 m from the frictionless pivot.
The support is suddenly removed and the rod begins to rotate clockwise about the pivot point. The moment of inertia of the rod about the pivot point is 30.6 kg m2.
Calculate the force the support exerts on the rod.
Calculate, in rad s–2, the initial angular acceleration of the rod.
After time t the rod makes an angle θ with the horizontal. Outline why the equation cannot be used to find the time it takes θ to become (that is for the rod to become vertical for the first time).
At the instant the rod becomes vertical show that the angular speed is ω = 2.43 rad s–1.
At the instant the rod becomes vertical calculate the angular momentum of the rod.
A wheel of mass 0.25 kg consists of a cylinder mounted on a central shaft. The shaft has a radius of 1.2 cm and the cylinder has a radius of 4.0 cm. The shaft rests on two rails with the cylinder able to spin freely between the rails.
The stationary wheel is released from rest and rolls down a slope with the shaft rolling on the rails without slipping from point A to point B.
The wheel leaves the rails at point B and travels along the flat track to point C. For a short time the wheel slips and a frictional force F exists on the edge of the wheel as shown.
The moment of inertia of the wheel is 1.3 × 10–4 kg m2. Outline what is meant by the moment of inertia.
In moving from point A to point B, the centre of mass of the wheel falls through a vertical distance of 0.36 m. Show that the translational speed of the wheel is about 1 m s–1 after its displacement.
Determine the angular velocity of the wheel at B.
Describe the effect of F on the linear speed of the wheel.
Describe the effect of F on the angular speed of the wheel.
A cylinder is fitted with a piston. A fixed mass of an ideal gas fills the space above the piston.
The gas expands isobarically. The following data are available.
The gas returns to its original state by an adiabatic compression followed by cooling at constant volume.
Show that the final volume of the gas is about 53 m3.
Calculate, in J, the work done by the gas during this expansion.
Determine the thermal energy which enters the gas during this expansion.
Sketch, on the pV diagram, the complete cycle of changes for the gas, labelling the changes clearly. The expansion shown in (a) and (b) is drawn for you.
Outline the change in entropy of the gas during the cooling at constant volume.
There are various equivalent versions of the second law of thermodynamics. Outline the benefit gained by having alternative forms of a law.
A heat pump is modelled by the cycle A→B→C→A.
The heat pump transfers thermal energy to the interior of a building during processes C→A and A→B and absorbs thermal energy from the environment during process B→C. The working substance is an ideal gas.
Show that the work done on the gas for the isothermal process C→A is approximately 440 J.
Calculate the change in internal energy of the gas for the process A→B.
Calculate the temperature at A if the temperature at B is −40°C.
Determine, using the first law of thermodynamics, the total thermal energy transferred to the building during the processes C→A and A→B.
Suggest why this cycle is not a suitable model for a working heat pump.
An ideal gas consisting of 0.300 mol undergoes a process ABCD. AB is an adiabatic expansion from the initial volume VA to the volume 1.5 VA. BC is an isothermal compression. The pressures at C and D are the same as at A.
The following data are available.
Pressure at A = 250 kPa
Volume at C = 3.50 × 10–3 m3
Volume at D = 2.00 × 10–3 m3
The gas at C is further compressed to D at a constant pressure. During this compression the temperature decreases by 150 K.
For the compression CD,
Show that the pressure at B is about 130 kPa.
Calculate the ratio .
determine the thermal energy removed from the system.
explain why the entropy of the gas decreases.
state and explain whether the second law of thermodynamics is violated.
The pressure–volume (pV) diagram shows a cycle ABCA of a heat engine. The working substance of the engine is 0.221 mol of ideal monatomic gas.
At A the temperature of the gas is 295 K and the pressure of the gas is 1.10 × 105 Pa. The process from A to B is adiabatic.
The process from B to C is replaced by an isothermal process in which the initial state is the same and the final volume is 5.00 × 10–3 m3.
Show that the pressure at B is about 5 × 105 Pa.
For the process BC, calculate, in J, the work done by the gas.
For the process BC, calculate, in J, the change in the internal energy of the gas.
For the process BC, calculate, in J, the thermal energy transferred to the gas.
Explain, without any calculation, why the pressure after this change would belower if the process was isothermal.
Determine, without any calculation, whether the net work done by the engine during one full cycle would increase or decrease.
Outline why an efficiency calculation is important for an engineer designing a heat engine.
A flywheel is made of a solid disk with a mass M of 5.00 kg mounted on a small radial axle. The mass of the axle is negligible. The radius R of the disk is 6.00 cm and the radius r of the axle is 1.20 cm.
A string of negligible thickness is wound around the axle. The string is pulled by an electric motor that exerts a vertical tension force T on the flywheel. The diagram shows the forces acting on the flywheel. W is the weight and N is the normal reaction force from the support of the flywheel.
The moment of inertia of the flywheel about the axis is .
The flywheel is initially at rest. At time t = 0 the motor is switched on and a time-varying tension force acts on the flywheel. The torque exerted on the flywheel by the tension force in the string varies with t as shown on the graph.
At t = 5.00 s the string becomes fully unwound and it disconnects from the flywheel. The flywheel remains spinning around the axle.
State the torque provided by the force W about the axis of the flywheel.
Identify the physical quantity represented by the area under the graph.
Show that the angular velocity of the flywheel at t = 5.00 s is 200 rad s–1.
Calculate the maximum tension in the string.
The flywheel is in translational equilibrium. Distinguish between translational equilibrium and rotational equilibrium.
At t = 5.00 s the flywheel is spinning with angular velocity 200 rad s–1. The support bearings exert a constant frictional torque on the axle. The flywheel comes to rest after 8.00 × 103 revolutions. Calculate the magnitude of the frictional torque exerted on the flywheel.
The diagram represents an ideal, monatomic gas that first undergoes a compression, then an increase in pressure.
An adiabatic process then increases the volume of the gas to .
Calculate the work done during the compression.
Calculate the work done during the increase in pressure.
Calculate the pressure following this process.
Outline how an approximate adiabatic change can be achieved.
A uniform ladder of weight 50.0 N and length 4.00 m is placed against a frictionless wall making an angle of 60.0° with the ground.
Outline why the normal force acting on the ladder at the point of contact with the wall is equal to the frictional force F between the ladder and the ground.
Calculate F.
The coefficient of friction between the ladder and the ground is 0.400. Determine whether the ladder will slip.
The moment of inertia of a solid sphere is where m is the mass of the sphere and r is the radius.
Show that the total kinetic energy Ek of the sphere when it rolls, without slipping, at speed v is .
A solid sphere of mass 1.5 kg is rolling, without slipping, on a horizontal surface with a speed of 0.50 m s-1. The sphere then rolls, without slipping, down a ramp to reach a horizontal surface that is 45 cm lower.
Calculate the speed of the sphere at the bottom of the ramp.