Torque and angular acceleration

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.

Figure 3 shows an excavator.

The bucket position can be changed by moving the boom and arm and by rotating the platform about the vertical axis of rotation. For the purposes of this question, assume that the excavator tracks do not move.

The bucket is moved to a new position by rotating the platform about the axis of rotation. Figure 4 shows the variation in angular velocity ω with time t for the rotation of the platform about the vertical axis.

The total angular displacement of the platform is 2.52 rad during the movement of the bucket.

Show that ω_max is about 0.9 rad s⁻¹.

At the same time as the platform is rotating, the bucket is moved up and down, and away from and towards the cab.

The moment of inertia of the rotating parts of the excavator about the axis of rotation is I. Figure 5 shows how I varies with t for the same time period as Figure 4.

Torque must be applied to the platform to change its angular velocity and to overcome friction at the platform bearing.

Show that the torque applied to the platform is at a maximum at time t = 2.1 s.

Deduce whether the maximum power applied to the platform occurs at the same time of 2.1 s.

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.

Figure 3 shows a heavy stone grinding wheel used for sharpening tools.

The pedal is connected to the axle of the wheel by a connecting link and crank. The operator pushes the pedal downwards to accelerate the wheel from rest. The wheel begins to rotate in the direction shown.

Explain why the torque applied to the axle varies as the operator pushes downwards on the pedal.

The wheel is rotating at a high angular speed. The operator is told not to use the pedal to stop the rotation of the wheel suddenly.

Explain, with reference to angular impulse, why a sudden stop is likely to damage the mechanism.

The connecting link breaks. At this instant the angular speed of the wheel

speeds.

Calculate the moment of inertia of the wheel.

The arrangement is modified as shown in Figure 4. The pedal, connecting link and crank are removed and the grinding wheel is driven by an electric motor.

Figure 4 also shows a tool being sharpened by pressing it on the edge of the rotating wheel. The tool applies a tangential force F on the wheel. A torque of 3.10 N m is needed at the axle to drive the wheel at constant angular speed while the tool is being sharpened.

frictional torque at the axle bearings = 0.77 N m radius of wheel = 0.24 m

Calculate F.

Three motors E, F and G were available to drive the wheel in Figure 4. Figure 5 shows how the torque T varies with angular speed ω for each motor.

The no-load speed ω₀ is the angular speed of a motor when the torque applied is zero. The maximum power of each motor is developed at about 0.5ω₀. The required output power of the motor when a tool is being sharpened is 52 W. The required output power of the chosen motor should be about of its maximum power.

Deduce which motor E, F or G was chosen for this application.

Figure 1 shows a fairground ride.

The ride consists of a rotor that rotates in a vertical circle about a horizontal axis. The rotor has two rigid arms. A pod containing passengers is attached to each arm. The rotor is perfectly balanced. The direction of rotation of the rotor is reversed at times during the ride.

Figure 2 shows the variation of the angular velocity ω of the rotor with time t during a 12 s period.

Determine the mean angular velocity of the rotor during the 12 s period.

The moment of inertia of the rotor about its axis of rotation is 2.1 × 10⁴ kg m². A constant frictional torque of 390 N m acts at the bearings of the rotor.

Calculate the power output of the driving mechanism during the first 2 s shown in Figure 2.

Calculate the maximum torque applied by the driving mechanism to the rotor during the 12 s period.

Calculate the magnitude of the angular impulse on the rotor between t = 2.0 s and t = 7.0 s.

Which graph best shows the variation of the torque T applied to the rotor for the 12 s period? Tick () one box.

A copy of Figure 2 is provided to help you.

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.

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.

State the condition necessary so that the law of conservation of angular momentum applies to a rotating system.

A clutch is used to connect two rotating shafts together so that they rotate at the same speed.

Figure 3 shows two shafts, A and B, rotating freely about the same axis. Table 1 gives information about the two shafts.

The two shafts are connected by forcing the clutch discs together. Friction acts between the discs and slipping occurs for a short time until both shafts rotate at a common angular speed. The clutch is now said to be engaged.

Show that the common angular speed of the two shafts immediately after the clutch is engaged is about 9 rad s⁻¹.

State whether the direction of the common angular speed is clockwise or anticlockwise when viewed from the left.

Table 2 gives information about two clutches, C and D. C and D provide different constant frictional torques during slipping at the clutch discs.

The slipping time is to be kept between 1.0 s and 2.0 s with the same initial conditions shown in Table 1, and the same final common angular speed.

Deduce whether either or both clutches allow this.

Figure 4 shows some of the equipment used to investigate the thermal and mechanical efficiencies of a single-cylinder four-stroke petrol engine.

• Petrol is supplied to the engine from a calibrated reservoir.

• Sensors are used to measure the volume V and pressure p above the piston inside the cylinder.

• The dynamometer applies a load to the output shaft and measures the output torque of the engine.

• The tachometer measures the rotational speed of the engine in revolutions per second.

In one test the air intake valve (throttle) setting remains fixed and the load provided by the dynamometer is kept constant.

Describe how you would determine the input power, the indicated power, the brake power, the thermal efficiency and the mechanical efficiency.

In your answer you should

• describe the measurements that you would take

• show how you would use the measurements and any other necessary data.

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.

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.

Figure 3

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.

Calculate the input torque.

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.

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.

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.