Rigid Body Mechanics (DP IB Physics: HL): Exam Questions

A block of mass = 10 g is attached to a light inextensible string hooked around a cylindrical pulley of mass = 5.0 kg and radius = 0.30 m.

The moment of inertia of a solid cylinder about a central axis is .

When the block is released, the pulley begins to turn as the block falls.

Calculate the moment of inertia of the cylindrical pulley. 

The linear acceleration of the block as it begins to fall is 2.4 m s⁻².

Calculate the angular acceleration of the pulley as it begins to turn.

Determine the magnitude of the tension in the string. 

Calculate the angular velocity of the pulley after it has rotated 180°.

The moment of inertia of a hollow sphere is where m is the mass of the sphere and r  is the radius.

Show that the total kinetic energy of the sphere E_k  when it rolls without slipping at an angular velocity ω  is .

The hollow sphere has a mass of 0.80 kg and a radius of 30 cm. It rolls without slipping for 20 seconds making 50 rotations.

Calculate the angular velocity of the sphere.

After making 50 rotations the sphere encounters a ramp in its path and starts rolling without slipping up its smooth inclined surface. 

Determine the maximum height the sphere reaches as it travels up the ramp. 

It takes the sphere 15 seconds to reach its maximum height on the ramp. 

Calculate the angular acceleration of the sphere as it rolls up the ramp. 

A force of = 50.0 N is applied at an angle of = 43° to a spanner of length = 25 cm. 

Calculate the torque produced by this force. 

The person applying the force to the spanner changes their grip to apply a larger torque. The magnitude of the force applied remains the same.

State and explain the angle at which the force should be applied to the spanner to exert the greatest torque.

Turning a uniform metal bar of length 23.5 cm rotates the mechanism that opens and closes a mains water tap. A couple of forces of magnitude 175 N each are applied at an angle of 75.0° to the bar.

Calculate the torque provided by the couple. 

A 3.5 kg mass hangs vertically from a string fixed 2.8 m above. The mass is free to swing from side to side. 

Calculate the moment of inertia of the mass. 

State the assumptions made in your calculation of the moment of inertia in part (a). 

The mass is released from rest, and it makes one complete oscillation of  radians in 2.4 seconds.

Determine the maximum angular speed of the mass.

Calculate the maximum angular momentum of the mass. 

A ball bearing, modelled as a uniform solid sphere, moves along a horizontal surface at a constant angular velocity. It then rolls without slipping down a rough inclined plane. It moves with a constant angular velocity again when it reaches a second horizontal surface at the bottom of the inclined plane.

Sketch a graph to show the variation with time of the angular velocity during its journey. 

The ball bearing is now placed at the top of the slope and released from rest. It rolls without slipping to the bottom of the slope.

Assuming no energy is lost, determine an equation for the total energy of the ball bearing at the bottom of the slope.

The ball bearing has a diameter of 5.0 mm and a mass of 4.0 g. At the bottom of the slope its linear velocity is 12 m s⁻¹.

Moment of inertia of a solid sphere: 

Calculate the rotational kinetic energy of the ball bearing at the bottom of the slope. 

Calculate the height of the slope.

A factory delivery system uses a flywheel made of a cylinder with a mass of 0.60 kg mounted on a small central shaft. The mass of the central shaft is negligible. The radius of the cylinder is 5.0 cm, and the radius of the central shaft is 1.5 cm. The outer edge of the central shaft makes contact with the rails as the flywheel rolls without slipping. The centre of mass of the flywheel moves with a linear velocity of 10 m s⁻¹.

Calculate the angular velocity of the flywheel.

Explain why all points on the cylinder and central shaft have the same angular velocity, but points on their outer rims have different linear speeds.

Both the cylinder and the central shaft can be modelled as solid disks.

The moment of inertia of a solid disk about a central axis is .

Calculate

(i) the moment of inertia of the flywheel

[2]

(ii) the angular momentum of the flywheel.

[2]

A piece of clay is dropped onto the outer rim of the rotating cylinder and sticks to it.

Explain, without any calculation, how the angular velocity of the flywheel changes immediately after the clay sticks to the cylinder.

A hollow sphere of mass and radius = 650 mm rolls, without slipping, down a slope of length 5.0 m in 4.3 seconds. The moment of inertia of this sphere about an axis through its centre is .

Calculate the linear velocity of the sphere as it reaches the bottom of the plane.

Show that the total kinetic energy of the sphere at the bottom of the plane is .

Determine the angle of the inclined plane.

Discuss whether the assumption that the sphere does not slip is valid in this situation.

A carousel ride consists of two rings of seats at distances and  from the centre of the carousel. The following diagram shows the view from above.

A motor applies a constant torque to rotate the carousel from rest up to a maximum angular speed in one complete revolution. The torque from the motor is then reduced, and the carousel continues to rotate at a constant angular speed, completing 15 revolutions in 5 minutes.

Calculate

(i) the angular acceleration of the carousel

[3]

(ii) the time taken for the carousel to reach its maximum angular speed.

[1]

Compare the linear speed of a passenger seated in the outer ring at to that of a passenger seated in the inner ring at .

The torque exerted by the motor varies to maintain a constant angular speed. During the ride, a passenger seated in the inner ring moves to a seat in the outer ring.

Explain how the torque exerted by the motor must change to maintain the constant angular speed.

After the carousel completes 15 revolutions, the motor stops exerting torque. The carousel eventually comes to rest due to a constant frictional torque exerted on the central axle of the carousel.

Sketch the variation with time of the resultant torque acting on the carousel for the duration of the ride from the time it starts rotating until it comes to rest.

Two wooden blocks of mass 3.0 kg and 6.0 kg are connected by a light, inextensible string that passes over a smooth pulley.

At time the blocks are released from rest. The string exerts tension forces of magnitude and on either side of the pulley.

Explain why the pulley can be considered to be in translational equilibrium, but not in rotational equilibrium.

At the linear acceleration of the blocks is 1.55 m s⁻².

The following data are available for the pulley.

• Pulley mass = 20 kg

• Pulley radius = 40 cm

• Moment of inertia =

Calculate the angular velocity of the pulley at .

(i) Calculate the tensions and in the string.

[2]

(ii) Show that the torque acting on the pulley at is about 6 N m.

[2]

(i) Sketch the variation with time of the torque exerted on the pulley between and .

[3]

(ii) Identify the physical quantity represented by the area under the graph.

[1]

A pool ball of mass 0.21 kg and diameter 7.5 cm is hit horizontally with a cue and rolls without slipping along the smooth horizontal pool table before rolling up a ramp. The pool ball has a moment of inertia of . The average acceleration of the pool ball during the impact is 163 m s⁻². 

Calculate the average net torque applied to the pool ball during the impact. 

The pool ball is in contact with the cue for 3.2 ms.

Calculate the total kinetic energy of the pool ball after impact. 

Immediately following the impact the pool ball starts to roll up a smooth uniform ramp placed directly in its line of motion. Air resistance is negligible.

Determine the maximum vertical height reached by the pool ball. 

After reaching its maximum height the ball rolls back down the ramp.

Determine the angular velocity of the ball at the bottom of the ramp.