Angular momentum

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.

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.

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.

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 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.

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.