Angular momentum

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. This axis is perpendicular to the plane of the disc and passes through a point A on the circumference of the disc. The disc is held with AB horizontal, where AB is a diameter of the disc, and released from rest.

Find the magnitude of

the horizontal component,

the vertical component of the force exerted on the disc by the axis immediately after the disc is released.

When AB is vertical the disc is instantaneously brought to rest by a horizontal impulse which acts in the plane of the disc and is applied to the disc at B.

Find the magnitude of the impulse.

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

A uniform rod AB of mass 4 m is free to rotate in a vertical plane about a fixed smooth horizontal axis, L, through A. The rod is hanging vertically at rest when it is struck at its end B by a particle of mass m. The particle is moving with speed u, in a direction which is horizontal and perpendicular to L, and after striking the rod it rebounds in the opposite direction with speed v. The coefficient of restitution between the particle and the rod is 1. Show that u = 7 v.

A uniform circular disc has mass m, centre O and radius 2 a. It is free to rotate about a fixed smooth horizontal axis L which lies in the same plane as the disc and which is tangential to the disc at the point A. The disc is hanging at rest in equilibrium with O vertically below A when it is struck at O by a particle of mass m. Immediately before the impact the particle is moving perpendicular to the plane of the disc with speed 3(ag). The particle adheres to the disc at O.

Find the angular speed of the disc immediately after the impact.

Find the magnitude of the force exerted on the disc by the axis immediately after the impact.

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.