Rotational dynamics

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.

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.

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.

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.