Moment of inertia

The rotating part of an electric motor is called the rotor.

Figure 1 shows an end view of a rotor turning clockwise due to a driving torque from the motor. In this question, the clockwise direction is treated as positive.

Figure 1
Figure 2

The rotor can be brought to rest rapidly by reversing the electrical supply connections to the motor. Figure 2 shows the rotor at time t = 0 when the supply connections are reversed. The rotor then slows down due to a constant anticlockwise retarding torque so that it

The angular velocity of the rotor at t = 0 is 98.0 rad s⁻¹ clockwise. The applied torque on the rotor at t = 0 is anticlockwise. The applied torque produces a constant angular acceleration of −303 rad s⁻². Friction torque is negligible.

Determine t₁.

The electrical supply remains connected and the rotor now accelerates uniformly anticlockwise with an acceleration of magnitude 303 rad s⁻². At a later time t = t₂, the angular velocity of the rotor is −120 rad s⁻¹.

Determine the number of anticlockwise revolutions made by the rotor between

Another way of quickly stopping the motor is to use a disc brake. The motor is brought to rest by switching off the power supply and then forcing two brake pads against the disc.

Figure 3 shows the two brake pads each applying a constant retarding force of 182 N to the disc.

moment of inertia of rotor and disc about the axis of rotation = 0.101 kg m² diameter of disc = 0.160 m diameter of pad = 22 mm

Compare the angular deceleration produced by the disc brake system with the angular deceleration produced by reversing the electrical connections.

The moment of inertia I of any rolling object about its axis of rotation is given by

where A is a constant that depends on the shape of the object m is the mass of the object r is the radius of the object.

Figure 4 shows an object that starts from rest and rolls down a slope without slipping.

The object falls a vertical distance h. It then has a linear velocity v and angular velocity ω. For a rolling object the linear velocity v of the centre of mass is related to the angular velocity ω by v = rω.

Show, by considering the energy transfers that occur, that

Table 1 gives the moments of inertia of a solid sphere P, a hollow sphere Q, a solid cylinder R and a hollow cylinder S about their axes of rotation.

P, Q, R and S have the same mass m and outer radius r.

Explain why the hollow sphere Q has a greater moment of inertia about its axis of rotation than the solid sphere P.

Figure 5 shows all four objects at rest at the top of a slope. They are released at the same time and roll down the slope without slipping.

Deduce the order in which the objects reach the bottom of the slope.

A heavy turntable is mounted on a fixed base. The turntable can rotate freely on a low-friction bearing.

Figure 1 shows a propeller unit fixed to the centre of the turntable. The propeller unit consists of a motor-driven propeller and a battery. The propeller and the turntable have a common axis of rotation.

At first, the turntable and the propeller are at rest. The propeller motor is switched on and the propeller quickly reaches a high final angular speed. The propeller rotates clockwise when viewed from above.

Compare, with reference to angular momentum, the motions of the turntable and the propeller.

Figure 2 shows an arrangement used to determine the moment of inertia of the turntable.

A small steel mass is held by an electromagnet above the top surface of the turntable. The diameter of the turntable is about half a metre. The turntable rotates freely at an initial angular speed ω₁.

The switch is opened so that the mass falls and sticks to the surface of the turntable. This changes the angular speed of the turntable to ω₂. The steel mass can be considered to be a point mass.

Describe how to determine the moment of inertia of the turntable using observations

In your answer you should:

• suggest how ω₁ and ω₂ are measured

• state any other measurements needed and name the equipment used to make them

• explain how the moment of inertia of the turntable is determined from the measurements.

One equation used in translational dynamics is:

force = mass × acceleration

State in words the equivalent equation used in rotational dynamics.

Figure 1 shows two identical uniform plates A and B. The axis of rotation of each plate is shown.

State and explain which plate has the greater moment of inertia about its axis of rotation.

An electric motor drives a machine that punches out plates from a long strip of sheet metal. The motor runs continuously and is fitted with a flywheel. Figure 2 shows how the angular velocity ω of the flywheel varies with time t.

Table 1 describes the sequence for the machine after it has been brought up to

A new flywheel with a greater moment of inertia is fitted in place of the original flywheel. The motor torque is constant and the same as before.

Sketch on Figure 2 a graph showing how the angular velocity varies with time for the machine fitted with the new flywheel.

Assume that:

• the punching operation starts at the same angular speed ω_A

• the same quantity of energy is transferred when punching the metal sheet.

Explain one difference between your graph and the original graph.

A moving tram is powered by energy stored in a rapidly spinning flywheel.

When the tram is at a tram stop, the flywheel is ‘charged’ by being accelerated to a high rotational speed.

The mass of the tram, flywheel and passengers is 1.46 × 10⁴ kg.

The distance between tram stops is 500 m.

Figure 3 shows that between stops A and B the track is inclined at a constant 5.0° to the horizontal.

The tram must travel 500 m along this incline on one charge of energy.

The total resistive force on the tram due to its motion is constant at 1.18 kN.

The flywheel is a solid steel disc of diameter 1.00 m. It has a moment of inertia of

Calculate the minimum angular speed of the flywheel when the tram leaves stop A so that the tram reaches stop B using only energy stored in the flywheel.

Between stops C and D the tram travels downhill.

Suggest two advantages of keeping the flywheel connected to the driving wheels when the tram travels downhill.

The same tram is to be used on a track where the stops are further apart, so the flywheel system needs to be modified.

Discuss the design features of the flywheel that will enable it to store more energy without increasing the mass of the tram.

In your answer you should consider:

• the design of the flywheel

• how the choice of materials used to make the flywheel is influenced by its design and maximum angular speed

• other design aspects that allow for high angular speeds of the flywheel.

A gymnast dismounts from an exercise in which he swings on a high bar. The gymnast rotates in the air before landing.

Figure 1 shows the gymnast in three positions during the dismount.

The arrows show the direction of rotation of the gymnast.

In position 1 the gymnast has just let go of the bar. His body is fully extended.

Position 2 shows the rotating gymnast a short time later. His knees have been brought close to his chest into a ‘tuck’.

Position 3 is at the end of the dismount as the gymnast lands on the mat. His body is once again fully extended.

Explain why the moment of inertia about the axis of rotation decreases when his knees are moved towards his chest. Go on to explain the effect this has on his angular speed.

Table 1 gives some data about the gymnast in position 1 and in position 2.

Calculate the angular speed ω of the gymnast in position 1.

The gymnast stays in the tuck for 1.2 s.

Determine the number of complete rotations performed by the gymnast when in the tuck during the dismount.

The gymnast repeats the exercise. The height of the bar remains unchanged.

State and explain two actions the gymnast can take to complete more rotations during the dismount.

State what is meant by the moment of inertia of an object about an axis.

A student does an experiment using the apparatus shown in Figure 1.

A solid disc pulley of mass M and radius R is supported in bearings which have negligible friction.

A string of negligible mass is wrapped around the circumference of the pulley. A load of mass 0.5 M is fixed to the free end of the string. The string does not slip on the pulley. The moment of inertia of the pulley about the axis of rotation is 0.5MR². The student holds the pulley stationary.

When the student releases the pulley, the load accelerates downwards uniformly and is at a velocity v after moving a distance h.

Show that the acceleration of the load is 0.5 g.

The student repeats the experiment using a spoked pulley of the same mass and radius, as shown in Figure 2.

Compare the acceleration of the load in this experiment with its acceleration in the previous experiment. A calculation is not required.

The fly-press shown in Figure 1 is used by a jeweller to punch shapes out of a thin metal sheet.

The frame holds a screw and punch. Two arms are attached to the screw, each loaded with a heavy steel ball. The screw is driven downwards when the arms are rotated. Kinetic energy is stored in the rotating parts: the balls, arms, screw and punch. This energy is used to punch the shape out of the metal sheet.

When the punch reaches the metal sheet, the rotational speed of the arms is 2.9 rev s⁻¹. At this speed the rotational kinetic energy of the rotating parts is 10.3 J.

Calculate the moment of inertia of the rotating parts about the axis of rotation.

The total mass of the screw, punch and arms is the same as the total mass of the two balls.

Explain why the moment of inertia of the screw, punch and arms about the axis of rotation is much smaller than the moment of inertia of the steel balls about the same axis.

During the punching of the metal sheet, the rotating parts of the fly-press are brought uniformly to rest from an initial rotational speed of 2.9 rev s⁻¹ in a time of 89 ms.

Determine

• the angular deceleration

• the angle turned through by the rotating parts.

Which of the following is the SI unit for angular impulse? Tick (✓) one box.

A turning moment diagram is a graph showing the variation of torque produced by an engine or motor with the angle of rotation of the output shaft.

Figure 2 shows the turning moment diagrams for a single-cylinder diesel engine and an electric motor that have the same output power.

State what is represented by the area between the curve and the angle axis for a turning moment diagram.

The diesel engine or the electric motor may be used to drive a machine that has a low moment of inertia and that requires an almost constant torque.

Discuss why, to drive this machine, the diesel engine would need to be fitted with a flywheel.

In your answer you should explain

• why the electric motor does not require a flywheel

• why the torque of the diesel engine varies over one cycle, including why there are points where the torque is zero

• how the moment of inertia of the flywheel influences the motion of the output shaft of the diesel engine.

State the law of conservation of angular momentum.

Figure 1 shows an orbiting satellite fitted with two small instrument pods attached to the ends of telescopic arms. The arms can be extended or retracted by a motor in the body of the satellite.

With the telescopic arms fully extended, the centre of mass of each instrument pod is at a radius of 4.1 m from the axis of rotation.

moment of inertia of satellite body about axis = 71 kg m² mass of each instrument pod = 5.0 kg

The mass of the telescopic arms is negligible.

Show that the total moment of inertia of the satellite with the arms fully extended is

The satellite is initially rotating slowly about its axis with the arms fully extended. The arms are slowly retracted so that the instrument pods move closer to the body of the satellite.

State and explain the change in the angular speed of the satellite as the arms are retracted.

The satellite is initially rotating at 1.3 rad s⁻¹ with the telescopic arms fully extended. When fully retracted the instrument pods are at a radius of 0.74 m from the axis. The satellite contains sensitive equipment that may be damaged if the rotational speed exceeds 4.2 rad s⁻¹

Deduce whether the arms can be retracted fully without the satellite exceeding its maximum permitted angular speed.

There is an analogy between quantities in rotational and translational dynamics.

Complete Table 1, stating in words the quantities in rotational dynamics that are analogous to force and mass in translational dynamics.

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.

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

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.

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.

Calculate the maximum angular speed ω_max of the jib.

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.