Rotational kinetic energy

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.

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.

Figure 2 shows a yo-yo made of two discs separated by a cylindrical axle. Thin string is wrapped tightly around the axle.

Initially both the free end A of the string and the yo-yo are held stationary.

With A remaining stationary, the yo-yo is now released so that it falls vertically. As the yo-yo falls, the string unwinds from the axle so that the yo-yo spins about its centre of mass.

The linear velocity v of the centre of mass of the falling yo-yo is related to the angular velocity ω by v = rω where r is the radius of the axle.

The yo-yo accelerates uniformly as it falls from rest. The string remains taut and has negligible thickness.

When the yo-yo has fallen a distance of 0.50 m, its linear velocity is V.

Calculate V by considering the energy transfers that occur during the fall.

The yo-yo falls further until all the string is unwound. The yo-yo then ‘sleeps’. This means the yo-yo continues to rotate in a loose loop of string as shown in Figure 3.

The string applies a constant frictional torque of 8.3 × 10⁻⁴ N m to the axle. The angular velocity of the yo-yo at the start of the sleep is 145 rad s⁻¹.

Determine, in rad, the total angle turned through by the yo-yo during the first 10 s of sleeping.

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 one function of a flywheel.

A student does an experiment to determine the frictional torque acting at the bearings of a steel flywheel. The flywheel has a radius of 0.075 m and is perfectly balanced.

The student places a small magnet of mass 0.020 kg at point A on the circumference of the flywheel on a horizontal line through the axis of rotation as shown in Figure 2a The student releases the flywheel. The flywheel first comes to rest when it has moved through an angle of 3.00 rad (172°), with the magnet now in position B as shown in Figure 2b.

Figure 2a
Figure 2b

The loss in gravitational potential energy of the magnet equals the work done against the frictional torque acting at the bearings.

Show that the frictional torque is about 7 × 10^–4 N m

The student goes on to determine the moment of inertia of the flywheel.

The magnet is removed and the flywheel is made to spin. Measurements show that the flywheel makes 573 rotations as its angular speed reduces uniformly from 25.0 rad s^–1 to zero. Assume the frictional torque at the bearings is constant and the same as in question 02.2.

Determine the moment of inertia of the flywheel about its axis of rotation.