Rotation of a rigid body

A pendulum consists of a uniform circular disc, of radius a and mass 4 m, whose centre is fixed to the end B of a uniform rod AB. The rod has mass 3 m and length 4 l, where 2l a. The rod lies in the same plane as the disc. The pendulum is free to rotate about a fixed smooth horizontal axis L which passes through A and is perpendicular to the plane of the disc. The moment of inertia of the pendulum about L is 2m(a² + 40l²).

Find the approximate period of small oscillations of the pendulum about its position of stable equilibrium.

The pendulum is held with B vertically above A and is then slightly displaced from rest. In the subsequent motion the midpoint of AB strikes a small peg, which is fixed at the same horizontal level as A, and the pendulum rebounds upwards. Immediately before it strikes the peg, the angular speed of the pendulum is .

Immediately after it strikes the peg, the angular speed of the pendulum is ω.

Find, in terms of m, g, a and l, the magnitude of the impulse exerted on the peg by the pendulum.

Show that the size of the angle turned through by the pendulum, between it hitting the peg and it next coming to rest, is arcsin

A uniform rod AB, of mass M and length 2 L, is free to rotate in a vertical plane about a smooth fixed horizontal axis through A. The rod is hanging vertically at rest, with B below A, when it is struck at its midpoint by a particle of mass M. Immediately before this impact, the particle is moving with speed u, in a direction which is horizontal and perpendicular to the axis. The particle is brought to rest by the impact and immediately after the impact the rod moves with angular speed .

Immediately after the impact, the magnitude of the vertical component of the force exerted on the rod at A by the axis is

Find u in terms of L and g.

Show that the magnitude of the horizontal component of the force exerted on the rod at A by the axis, immediately after the impact, is zero.

The rod first comes to instantaneous rest after it has turned through an angle

Find the size of

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A uniform square lamina PQRS, of mass m and side 2 a, is free to rotate about a fixed smooth horizontal axis which passes through P and Q. The lamina hangs at rest in a vertical plane with SR below PQ and is given a horizontal impulse of magnitude J at the midpoint of SR. The impulse is perpendicular to SR.

Find the initial angular speed of the lamina.

Find the magnitude of the angular deceleration of the lamina at the instant when the lamina has turned through radians.

Find the magnitude of the component of the force exerted on the lamina by the axis, in a direction perpendicular to the lamina, at the instant when the lamina has turned

A pendulum is modelled as a uniform rod AB, of mass 3 m and length 2 a, which has a particle of mass 2 m attached at B. The pendulum is free to rotate in a vertical plane about a fixed smooth horizontal axis L which passes through A. The vertical plane is perpendicular to the axis L.

Find the period of small oscillations of the pendulum about its position of stable equilibrium.

The pendulum is hanging at rest in a vertical position, with B below A, when it is given a horizontal impulse of magnitude J. The impulse acts at B in a vertical plane which is perpendicular to the axis L. Given that the pendulum turns through an angle of 60° before first coming to instantaneous rest,

find J.

A uniform circular disc, of radius r and mass m, is free to rotate in a vertical plane about a fixed smooth horizontal axis L which is perpendicular to the plane of the disc and passes through a point which is from the centre of the disc. The disc is held at rest with its centre vertically above the axis. The disc is then slightly disturbed from its rest position. You may assume without proof that the moment of inertia of the disc about L is

Find the magnitude of the force exerted on the disc by the axis when the disc has

Auniform rod PQ, ofmass m and length 3a, is free to rotate about a fixed smooth horizontal axis L, which passes through the end P of the rod and is perpendicular to the rod. The rod hangs at rest in equilibrium with Q vertically below P. One end of a light inextensible string of length 2a is attached to the rod at P and the other end is attached to a particle of mass 3m. The particle is held with the string taut, and horizontal and perpendicular to L, and is then released. After colliding, the particle sticks to the rod forming a body B.

Show that the moment of inertia of B about L is 15ma?. (2)

Show that B first comes to instantaneous rest after it has turned through an angle

circular pulley, of mass 4m and radius r, is free to rotate about a fixed smooth horizontal axis which passes through the centre of the pulley and is perpendicular to the plane of the pulley. A light inextensible string passes over the pulley and hasa particle of mass 2m attached to one end anda particle of mass 3m attached to the other end. The particles hang with the string vertical and taut on each side of the pulley. The rim of the pulley is sufficiently rough to prevent the string slipping. The system is released from rest.

Find the angular acceleration of the pulley. (8) When the angular speed of the pulley is Q, the string breaks and a constant braking couple of magnitude G is applied to the pulley which brings it to rest.

Find an expression for the angle turned through by the pulley from the instant when the string breaks to the instant when the pulley first comes to rest. (4)

A pendulum consists of a uniform rod PQ, of mass 3 m and length 2 a, which is rigidly fixed at its end Q to the centre of a uniform circular disc of mass m and radius a. The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis L which passes through the end P of the rod and is perpendicular to the rod.

Show that the moment of inertia of the pendulum about L is ma².

The pendulum is released from rest in the position where PQ makes an angle with the downward vertical. At time t, PQ makes an angle with the downward vertical.

Show that the angular speed, , of the pendulum satisfies²

Hence, or otherwise, find the angular acceleration of the pendulum.

Given that α and that PQ has length m,

find, to 3 significant figures, an approximate value for the angular speed of the pendulum 0.2 s after it has been released from rest.

A uniform circular disc has mass 4 m, centre O and radius 4 a. The line POQ is a diameter of the disc. A circular hole of radius 2 a is made in the disc with the centre of the hole at the point R on PQ where QR = 5 a, as shown in Figure 1. The resulting lamina is free to rotate about a fixed smooth horizontal axis L which passes through Q and is perpendicular to the plane of the lamina.

Show that the moment of inertia of the lamina about L is 69ma².

The lamina is hanging at rest with P vertically below Q when it is given an angular velocity . Given that the lamina turns through an angle before it first comes to instantaneous rest,

find in terms of g and a.

A uniform lamina of mass M is in the shape of a right-angled triangle OAB. The angle OAB is 90°, OA = a and AB = 2 a, as shown in Figure 1.

Prove, using integration, that the moment of inertia of the lamina OAB about the edge

(You may assume without proof that the moment of inertia of a uniform rod of mass m and length 2l about an axis through one end and perpendicular to the rod

The lamina OAB is free to rotate about a fixed smooth horizontal axis along the edge OA and hangs at rest with B vertically below A. The lamina is then given a horizontal impulse of magnitude J. The impulse is applied to the lamina at the point B, in a direction which is perpendicular to the plane of the lamina. Given that the lamina first comes to instantaneous rest after rotating through an angle of 120°,

find an expression for J, in terms of M, a and g.

A uniform square lamina ABCD, of mass 2m and side 3a√2, is free to rotate in a vertical plane about a fixed smooth horizontal axis L which passes through A and is perpendicular to the plane of the lamina. The moment of inertia of the lamina about L is 24ma². The lamina is at rest with C vertically above A. At time t = 0 the lamina is slightly displaced. At time t the lamina has rotated through an angle.

Show that

Show that, at time t, the magnitude of the component of the force acting on the lamina

When the lamina reaches the position with C vertically below A, it receives an impulse which acts at C, in the plane of the lamina and in a direction which is perpendicular to the line AC. As a result of this impulse the lamina is brought immediately to rest.

Find the magnitude of the impulse.