11.1 engineering physics - rotational dynamic 1 - qs

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11.1 Engineering physics - Rotational Dynamic 1 – Questions

Q1. The diagram shows the mechanism of a dynamo torch for providing light without the use of batteries. When the handle is squeezed the gears rotate the flywheel increasing the kinetic energy stored in the flywheel. On the same axle as the flywheel is a magnet which generates an emf in surrounding coils as it rotates. The magnet, coils and electrical connections are not shown.

When the handle is released the mechanism disengages and a spring returns the handle to its original position. During this time the gears continue to turn. The energy of the flywheel enables it to continue to rotate for several seconds.

A designer wishes to enable the flywheel to store more energy and thus rotate for longer after the handle has been released. The radius of the flywheel is limited by the overall size of the torch and cannot be increased.

Describe and explain other changes that can be made to the mechanism and flywheel to store more energy. Your answer should include consideration of:

• the flywheel’s shape • the material from which it is made • changes to the mechanism.

The quality of your written communication will be assessed in your answer. (Total 6 marks)

Q2. A roundabout in a fairground requires an input power of 2.5 kW when operating at a constant angular velocity of 0.47 rad s–1.

(a) Show that the frictional torque in the system is about 5 kN m.

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(3)

(b) When the power is switched off, the roundabout decelerates uniformly because the frictional torque remains constant. The roundabout takes a time of 34 s to come to rest.

(i) Calculate the moment of inertia of the roundabout. Give an appropriate unit for your answer.

moment of inertia ____________________ unit __________ (3)

(ii) Calculate the number of revolutions that are made before the roundabout comes to rest.

number of revolutions ____________________ (3)

(c) An operator of mass 65 kg is standing on the roundabout when the roundabout is rotating at an angular velocity of 0.47 rad s–1. His centre of mass is 2.2 m from the axis of rotation. The diagram shows that his body leans towards the centre of the path.

(i) Calculate the centripetal force needed for the operator to remain at this radius on the roundabout.

centripetal force ____________________ N (2)

(ii) State the origin of this centripetal force and suggest why the operator has to

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incline his body towards the centre of rotation to avoid falling over.

You may draw the forces that act on the operator in the diagram to help your answer.

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______________________________________________________________ (2)

(iii) While the roundabout is moving, the operator drops a coin.

Which statement correctly describes and explains what happens to the coin? Tick (✔) the correct answer in the right-hand column.

Tick (✔)

There is no longer a centripetal force acting, so the coin falls vertically downwards and lands on the roundabout directly below the point at which it was dropped.

The centripetal force causes the coin to have a horizontal component of velocity towards the centre of the roundabout, so that it follows a trajectory towards the centre of the roundabout.

There is no longer a centripetal force acting, so there is a horizontal component of the coin’s velocity directed away from the centre of the roundabout and it follows a trajectory directly away from the centre.

There is no longer a centripetal force acting, so the coin has a horizontal component of its velocity tangential to its original path on the roundabout and it follows a trajectory along this tangent.

(1) (Total 14 marks)

Q3. The following figure shows a motor-driven winch for raising loads on a building site. As the motor turns the cable is wound around the drum, raising the load.

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The drum, axle and other rotating parts have a moment of inertia about the axis of rotation of 7.4 kg m2, and the mass of the load is 85 kg. The drum has a radius of 0.088 m.

The load is accelerated uniformly from rest to a speed of 2.2 m s–1. When it is accelerating it rises through a height of 3.5 m. It then continues at the constant speed of 2.2 m s–1.

(a) Show that the drum turns through 40 rad as the load accelerates.

(1)

(b) Calculate the angular speed of the drum when the load is moving at 2.2 m s–1.

angular speed ____________________ rad s–1

(1)

(c) (i) Show that for the time that the load is accelerating the total increase in energy of the load and the rotating parts is about 5400 J.

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(3)

(ii) A constant frictional torque of 5.2 N m acts at the bearings of the winch.

Calculate the total work done by the motor to accelerate the load.

Give your answer to an appropriate number of significant figures.

total work done ____________________ J (3)

(d) Calculate the maximum power developed by the motor.

maximum power ____________________ W (2)

(Total 10 marks)

Q4. The turntable of a microwave oven has a moment of inertia of 8.2 × 10–3 kg m2 about its vertical axis of rotation.

(a) With the drive disconnected, the turntable is set spinning. Starting at an angular speed of 6.4 rad s–1 it makes 8.3 revolutions before coming to rest.

(i) Calculate the angular deceleration of the turntable, assuming that the deceleration is uniform. State an appropriate unit for your answer.

angular deceleration ____________________ unit ___________ (4)

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(ii) Calculate the magnitude of the frictional torque acting at the turntable bearings.

torque ____________________ N m (1)

(b) The turntable drive is reconnected. A circular pie is placed centrally on the turntable. The power input to the microwave oven is 900 W, and to cook the pie the oven is switched on for 270 seconds. The turntable reaches its operating speed of 0.78 rad s–1 almost immediately, and the friction torque is the same as in part (a)(ii).

(i) Calculate the work done to keep the turntable rotating for 270 s at a constant angular speed of 0.78 rad s–1 as the pie cooks.

work done ____________________ J (2)

(ii) Show that the ratio

is of the order of 105. (2)

(Total 9 marks)

Q5. During the spin part of a washing machine programme the maximum rotational speed is 126 rad s–1 (1200 rev min–1). The moment of inertia of the drum and washing at the start of the spin part of the cycle is 0.565 kg m2. Assume that the wet washing is evenly distributed around the drum as shown in Figure 1.

Figure 1

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(a) Figure 2 shows the variation of angular speed with time for the spin part of the cycle. The graph is not drawn to scale.

Figure 2

The motor provides a torque of 8.80 N m to accelerate the drum to 126 rad s–1. The drum rotates at 126 rad s–1 until near the end of the spin cycle, when the drum decelerates uniformly to zero angular speed in 15 seconds. Friction at the bearings may be neglected. Assume that during acceleration the moment of inertia of the drum and washing remains constant.

(i) Show that the drum accelerates for about 8 s.

(2)

(ii) Calculate the total number of revolutions made by the drum during the 195 s

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shown in Figure 2.

number of revolutions = ____________________ (3)

(b) (i) In practice, at the start of the cycle the acceleration will not remain constant. Draw on Figure 2 a line to show how the initial part of the graph will change.

(1)

(ii) Explain your reasons for the line you have drawn.

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______________________________________________________________ (2)

(Total 8 marks)

Q6. A student carries out an experiment to determine the moment of inertia of a turntable. The diagram shows the turntable with a small lump of plasticine held above it. An optical sensor connected to a data recorder measures the angular speed of the turntable.

The turntable is made to rotate and then it rotates freely. The lump of plasticine is dropped from a small height above the turntable and sticks to it. Results from the experiment are as follows.

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mass of plasticine = 16.0 g radius at which plasticine sticks to the turntable = 125 mm angular speed of turntable immediately before plasticine is dropped = 3.46 rad s–1

angular speed of turntable immediately after plasticine is dropped = 3.31 rad s–1

The student treats the plasticine as a point mass.

(a) Explain why the turntable speed decreases when the plasticine sticks to it.

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___________________________________________________________________ (3)

(b) Use the results of the experiment to determine the moment of inertia of the turntable.

Give your answer to an appropriate number of significant figures.

moment of inertia ____________________ kg m2

(3)

(c) (i) Calculate the change in rotational kinetic energy of the turntable and plasticine from the instant before the plasticine is dropped until immediately after it sticks to the turntable.

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change in kinetic energy ____________________ J (2)

(ii) Explain the change in rotational kinetic energy.

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______________________________________________________________ (1)

(Total 9 marks)

Q7. A garden sprinkler consists of a fixed base and a rotating spinner having three arms as shown in Figure 1.

At the end of each arm is a nozzle at 90° to the arm and inclined at 45° to the horizontal. Water flows in jets at a constant rate from these nozzles when the hose water tap is turned on.

Figure 2 shows a side view of one of the nozzles viewed in the direction of arrow A in Figure 1.

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The water jets produce reaction forces that act on the arms which cause the spinner to rotate. The base remains fixed in position. In operation, the spinner rotates at a constant rate of 240 rev min−1. The nozzles rotate in a horizontal circle of radius 120 mm.

(a) Each water jet exerts a constant force of 0.11 N on its arm at 45° to the horizontal.

Show that the torque exerted on the spinner by the jets of water is about 3 × 10−2 N m.

(2)

(b) (i) Explain why, when the water tap is turned on, the spinner accelerates initially but then reaches a constant angular speed. Assume that, when the tap is turned on, the flow-rate of the water from the jets is constant.

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______________________________________________________________ (2)

(ii) Calculate the power dissipated by the frictional torque acting between the spinner and the fixed base when the sprinkler is rotating at 240 rev min–1.

power = ____________________ W (2)

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(c) When the water is suddenly turned off all the kinetic energy of the spinner and arms is dissipated as heat due to work done by the frictional torque and the spinner makes a further 13 rotations before coming to rest. Assume uniform deceleration.

(i) Calculate the time taken for the spinner to come to rest.

time = ____________________ s (2)

(ii) Show that the kinetic energy of the spinner when rotating at its operating speed is about 2 J.

(1)

(iii) Determine the moment of inertia of the spinner about its axis of rotation.

moment of inertia = ____________________ kg m2

(1) (Total 10 marks)

Q8. (a) There is an analogy between quantities in rotational and translational dynamics.

Complete the table, stating in words the quantities in rotational dynamics that are analogous to force and mass in translational dynamics.

Translational dynamics Rotational dynamics

force

mass

Figure 1 shows a side view of the jib of a tower crane. The load is supported by a trolley which can move along the jib. The jib consists of all the parts of the crane above the bearing, but excluding the trolley and load.

Figure 1

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The moment of inertia of the jib about the axis of rotation = 2.6 × 107 kg m2

Mass of trolley and load = 2.2 × 103 kg (2)

(b) The load is at a distance of 35 m from the axis of rotation.

Show that the total moment of inertia of the jib, and the trolley and load, about the axis of rotation is about 3 × 107 kg m2.

(1)

(c) Figure 2 shows the variation of angular speed of the jib as it turns through an angle of 4.7 rad (270°) in a total time of 95 s. The trolley and load remain at a distance of 35 m from the axis.

Figure 2

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Calculate the maximum angular speed ωmax of the jib.

maximum angular speed = ____________________ rad s–1

(2)

(d) At time X in Figure 2 the motor that is driving the jib is disengaged. A constant braking torque is then applied to bring the jib to a standstill from its maximum angular speed.

The crane driver repeats the movement of the jib with the same load at 35 m from the axis of rotation. Up to time X the motion is the same as before. From time X the trolley is driven at a steady speed away from the axis as the jib continues to rotate until the jib comes to a standstill.

Assume the braking torque remains the same as before.

Discuss how the motion of the trolley affects the time taken for the jib to come to a standstill.

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(Total 8 marks)

Q9. Figure 1 shows a satellite with three solar panels folded in close to the satellite’s axis for the journey into space in the hold of a cargo space craft.

Figure 1 Figure 2

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Just before it is released into space, the satellite is spun to rotate at 5.2 rad s–1. Once released, the solar panels are extended as shown in Figure 2.

moment of inertia of the satellite about its axis with panels folded = 110 kg m2

moment of inertia of the satellite about its axis with panels extended = 230 kg m2

(a) State the law of conservation of angular momentum.

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___________________________________________________________________ (1)

(b) The total mass of the satellite is 390 kg and the solar panels each have a mass of 16 kg.

State what is meant by moment of inertia and explain why extending the solar panels changes the moment of inertia of the satellite by a large factor.

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(c) Calculate the angular momentum of the satellite when it is rotating at 5.2 rad s–1 with the solar panels folded. State an appropriate unit for your answer.

angular momentum = ____________________ unit __________ (2)

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(d) Calculate the angular speed of the satellite after the solar panels have been fully extended.

angular speed = ____________________rad s–1

(2) (Total 8 marks)

Q10. Flywheels store energy very efficiently and are being considered as an alternative to battery power.

(a) A flywheel for an energy storage system has a moment of inertia of 0.60 kg m2 and a maximum safe angular speed of 22 000 rev min–1.

Show that the energy stored in the flywheel when rotating at its maximum safe speed is 1.6 MJ.

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___________________________________________________________________ (2)

(b) In a test the flywheel was taken up to maximum safe speed and then allowed to run freely until it came to rest. The average power dissipated in overcoming friction was 8.7 W.

Calculate

(i) the time taken for the flywheel to come to rest from its maximum speed,

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(ii) the average frictional torque acting on the flywheel.

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______________________________________________________________ (2)

(c) The energy storage capacity of the flywheel can be improved by adding solid discs to the flywheel as shown in cross-section in A in the figure below, or by adding a hoop or tyre to the rim of the flywheel as shown in B in the same diagram. The same

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mass of material is added in each case. State, with reasons, which arrangement stores the more energy when rotating at a given angular speed.

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(Total 6 marks)

Q11. ‘Low inertia’ motors are used in applications requiring rapid changes of speed and direction of rotation. These motors are designed so that the rotor has a very low moment of inertia about its axis of rotation.

(a) (i) Explain why a low moment of inertia is desirable when the speed and direction of rotation must be changed quickly.

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(ii) State, giving a reason in each case, two features of rotor design which would lead to a low moment of inertia about the axis of rotation.

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______________________________________________________________ (4)

(b) In one application, a rotor of moment of inertia 4.4 × 10–5 kg m2 about its axis of rotation is required to reverse direction from an angular speed of 120 rad s–1 to the same speed in the opposite direction in a time of 50 ms. Assuming that the torque acting is constant throughout the change, calculate

(i) the angular acceleration of the rotor,

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(ii) the torque needed to achieve this acceleration,

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(iii) the angular impulse given to the rotor during the time the torque is acting,

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(iv) the angle turned through by the rotor in coming to rest momentarily before reversing direction.

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______________________________________________________________ (4)

(Total 8 marks)

Q12. A student is told that if a small cup of coffee is placed near the edge of a table flap, the cup and the flap will lose contact if the flap support is suddenly removed. This is shown in Figure 1.

Figure 1

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The student does not believe this to be true so decides to model the arrangement using a metre ruler free to pivot about one end A, with a small mass X resting on the ruler at the other end B. The arrangement is shown in Figure 2. The mass of X is negligible compared to the mass of the ruler. The metre ruler is held in the horizontal position by a support P which is quickly removed. A video is taken of the subsequent motion of the ruler and mass.

Figure 2

Assume the ruler is a thin uniform beam of mass m and length l.

(a) Derive an expression for the torque T acting on the ruler at the moment of release.

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___________________________________________________________________ (1)

(b) The moment of inertia I of the metre ruler about the axis through A is given by

I = l2

Show that the angular acceleration α of the ruler at the moment of release is given by

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α = (2)

(c) The linear acceleration a of a point on a rotating rigid body at a distance r from the axis of rotation is related to the angular acceleration α by

a = r × α

Explain why this causes the small mass to lose contact with the metre ruler as soon as the ruler is released.

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___________________________________________________________________ (2)

(d) Estimate how far from A the small mass must be placed to ensure it just maintains contact with the ruler when the ruler is released.

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(Total 6 marks)

Q13. A grinding wheel is used to sharpen chisels in a school workshop. A chisel is forced against the edge of the grinding wheel so that the tangential force on the wheel is a steady 7.0 N as the wheel rotates at 120 rad s–1. The diameter of the grinding wheel is 0.15 m.

(a) (i) Calculate the torque on the grinding wheel, giving an appropriate unit.

answer = ______________________ (2)

(ii) Calculate the power required to keep the wheel rotating at 120 rad s–1.

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answer = ______________________ W (1)

(b) When the chisel is removed and the motor is switched off, it takes 6.2 s for the grinding wheel to come to rest.

Calculate the number of rotations the grinding wheel makes in this time.

answer = ______________________ (2)

(Total 5 marks)

Q14. The figure below shows a remote-control camera used in space for inspecting space stations. The camera can be moved into position and rotated by firing ‘thrusters’ which eject xenon gas at high speed. The camera is spherical with a diameter of 0.34 m.

In use, the camera develops a spin about its axis of rotation. In order to bring it to rest, the thrusters on opposite ends of a diameter are fired, as shown in the figure below.

(a) When fired, each thruster provides a constant force of 0.12 N.

(i) Calculate the torque on the camera provided by the thrusters.

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(ii) The moment of inertia of the camera about its axis of rotation is 0.17 kg m2. Show that the angular deceleration of the camera whilst the thrusters are firing is 0.24 rad s–2.

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(b) The initial rotational speed of the camera is 0.92 rad s–1. Calculate

(i) the time for which the thrusters have to be fired to bring the camera to rest,

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(ii) the angle turned through by the camera whilst the thrusters are firing. Express your answer in degrees.

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(Total 6 marks)

Q15. (a) State the law of conservation of angular momentum.

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(b) When a star undergoes a supernova explosion, the star’s core collapses into a very much smaller diameter forming an extremely dense neutron star as shown in the figure below.

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A star has a period of rotation about an axis through its centre of 44 days (3.8 × 106 s) and a core of radius 4.1 × 107 m. The star undergoes a supernova explosion and the core collapses into a neutron star of radius 1.2 × 104 m. You may assume that during the collapse no mass is lost from the core and that the star remains spherical.

Moment of inertia of a sphere of uniform mass m and radius R about an axis through its centre = 0.40mR2

(i) Explain why the period of rotation of the star decreases as it becomes a neutron star.

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______________________________________________________________ (2)

(ii) Determine the period of rotation of the neutron star. Give your answer to an appropriate number of significant figures.

answer = ______________________ s (4)

(Total 7 marks)

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Q16. (a) A playground roundabout has a moment of inertia about its vertical axis of rotation

of 82 kg m2. Two children are standing on the roundabout which is rotating freely at 35 revolutions per minute. The children can be considered to be point masses of 39 kg and 28 kg and their distances from the centre are as shown in the figure below.

(i) Calculate the total moment of inertia of the roundabout and children about the axis of rotation. Give your answer to an appropriate number of significant figures.

answer = ______________________ kg m2

(3)

(ii) Calculate the total rotational kinetic energy of the roundabout and children.

answer = ______________________ J (2)

(b) The children move closer to the centre of the roundabout so that they are both at a distance of 0.36 m from the centre. This changes the total moment of inertia to 91 kg m2.

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(i) Explain why the roundabout speeds up as the children move to the centre of the roundabout.

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(ii) Calculate the new angular speed of the roundabout. You may assume that the frictional torque at the roundabout bearing is negligible.

answer = ______________________ rad s–1

(2)

(iii) Calculate the new rotational kinetic energy of the roundabout and children.

answer = ______________________ J (1)

(c) Explain where the increase of rotational kinetic energy of the roundabout and children has come from.

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___________________________________________________________________ (1)

(Total 11 marks)

Q17. The figure below shows a type of circular saw. The blade is driven by an electric motor and rotates at 2600 rev min–1 when cutting a piece of wood. A constant frictional torque of 1.2 Nm acts at the bearings of the motor and axle.

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A horizontal force of 32 N is needed to push a piece of wood into the saw. The force acts on the blade at an effective radius of 0.15 m.

(a) (i) Calculate the torque on the saw blade resulting from the horizontal force on the wood.

answer = ____________________ Nm (1)

(ii) Calculate the output power of the motor when the saw is cutting the wood.

answer = ____________________ W (3)

(b) Immediately after cutting the wood the motor is switched off. The time taken for the saw blade to come to rest is 8.5 s. Calculate the moment of inertia of the rotating parts (i.e. the motor rotor, axle and blade). State an appropriate unit.

answer = ____________________ unit = __________ (3)

(c) If the blade is accidentally touched when it is rotating, an electronic safety brake stops the blade in 5.0 ms. This is fast enough to prevent serious injury. The safety brake works by forcing a block of aluminium into the saw teeth.

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Estimate the rate at which the rotational kinetic energy is dissipated as heat and in deforming the aluminium when the brake operates.

answer = ____________________ W (2)

(Total 9 marks)

Q18. The graph below shows how the moment of inertia I of a diver performing a reverse dive varies with time t from just after he has left the springboard until he enters the water.

The diver starts with his arms extended above his head (position 1), and then brings his legs towards his chest as he rotates (position 2). After somersaulting in mid-air, he extends his arms and legs before entering the water (position 3).

(a) Explain how moving the legs towards the chest causes the moment of inertia of the diver about the axis of rotation to decrease.

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(b) (i) Explain in terms of angular momentum why the angular velocity of the diver varies during the dive.

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(ii) Describe how the angular velocity of the diver varies throughout the dive.

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(c) At time t = 0 the angular velocity of the diver is 4.4 rad s–1 and his moment of inertia about the axis of rotation is 10.9 kg m2.

With reference to the graph above calculate the maximum angular velocity of the diver during the dive.

angular velocity ____________________ rad s–1

(3) (Total 8 marks)

Q19. The diagram shows the basic principle of operation of a hand-operated salad spinner used to dry washed salads.

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The salad is placed in the basket and the lid is attached.

When handle A is turned the basket and its contents spin rapidly. Water on the salad is driven through holes in the basket into the stationary water collecting bowl. The pivot for gear B is fixed to the lid. This pivot and the lid do not move. When gear B rotates, gear C also rotates but at a greater angular speed. Gear C is fixed to the basket and rotates it.

A force of 6.0 N is applied to handle A as shown. Handle A is at a radius of 36 mm from its centre of rotation.

(a) Calculate the input torque.

torque = ____________________ N m (1)

(b) Gear C rotates four times for every one revolution of gear B.

Deduce whether it is possible for the torque on gear C to be greater than one quarter of the input torque.

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(c) It takes 2.1 s for the empty basket to reach an angular speed of 76 rad s–1. The torque on gear C is a constant 0.054 N m during this time. Frictional losses are negligible.

Calculate the moment of inertia of the basket about its axis of rotation.

moment of inertia = ____________________ kg m2

(2)

(d) The gears are made from polymer (plastic). An early version of this salad spinner suffered from damaged gear teeth.

Explain with reference to angular impulse why a great force is put on the gear teeth if the user tries to stop the loaded basket too quickly using the handle.

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(Total 8 marks)

Q20. (a) A metal flywheel is rotating on frictionless bearings. The temperature is increased so

that the flywheel expands.

Consider each of the following statements and indicate with a tick (✔) if it is correct.

✔ if correct

The moment of inertia will decrease.

The angular velocity will decrease.

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The angular momentum will be unaltered.

(1)

An electric motor drives a machine which stamps out shapes from sheet steel. The machine is fitted with a flywheel of moment of inertia 25 kg m2 which is accelerated uniformly until it is rotating at 640 rev min−1. The machine then starts a stamping operation which reduces the flywheel's angular speed to 360 rev min−1.

(b) Explain why a flywheel is fitted between the motor and the stamping machine.

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(c) Calculate the energy needed for the stamping operation.

energy _________________ J (2)

(d) Immediately after the stamping operation the flywheel is accelerated to its initial speed of 640 rev min−1 in a time of 5.0 s. The next stamping operation then begins.

Calculate the constant torque provided by the motor during this 5.0 s. Assume that the bearing frictional torque is negligible.

torque = ______________ N m (2)

(e) Calculate the minimum power output of the electric motor required.

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power __________________ W (2)

(f) The flywheel is a solid disc. It is to be replaced with a flywheel which gives a smaller angular speed change for each stamping operation.

Two replacement flywheels, A and B, are available and information about them and the original flywheel is given in the table below.

flywheel density of

material / kg m−3

thickness of disc / m outer radius / m

original 7800 0.10 0.38

A 8800 0.20 0.30

B 2900 0.10 0.50

Deduce which flywheel, A or B, would be more suitable. Explain your choice.

The moment of inertia I of a solid disc of mass m and outer radius r about an axis through the centre is given by

I = m r2

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(Total 12 marks)

Q21. The figure below shows an experiment to determine the moment of inertia of a bicycle wheel. One end of a length of strong thread is attached to the tyre. The thread is wrapped around the wheel and a 0.200 kg mass is attached to the free end. The wheel is held so that the mass is at a height of 1.50 m above the floor. The wheel is released and the time taken for the mass to reach the floor is measured.

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(a) State the energy transfers that take place from the moment the wheel is released until the mass hits the floor.

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___________________________________________________________________ (2)

(b) Calculations based on the measurements made show that at the instant the mass hits the floor:

• the speed of the mass is 2.22 m s–1

• the wheel is rotating at 6.73 rad s–1

• the wheel has turned through an angle of 4.55 rad from the point of release.

A separate experiment shows that a constant frictional torque of 7.50 × 10–3 N m acts on the wheel when it is rotating.

By considering the energy changes in the system, show that the moment of inertia of the wheel about its axis is approximately 0.1 kg m2.

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(3)

(c) When the mass hits the floor the thread is released from the wheel.

Calculate the angle turned through by the wheel before it comes to rest after the thread is released.

angle = ____________________ rad (2)

(Total 7 marks)

11.1 engineering physics - rotational dynamic 1 - qs

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