Rotation about a fixed axis

Three equal uniform rods, each of mass m and length 2 a, form the sides of a rigid equilateral triangular frame ABC. The frame 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 frame.

Show that the moment of inertia of the frame about L is 6ma².

The frame is held with AB horizontal and C below AB, and released from rest. Given that the centre of mass of the frame is two thirds of the way along a median from a vertex,

find the magnitude of the force exerted by the axis on the frame at A at the instant when the frame is released.

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 lamina PQR of mass m is in the shape of an isosceles triangle, with PQ = PR = 5 a and QR = 6 a. The midpoint of QR is T.

Show, using integration, that the moment of inertia of the lamina about an axis which passes through P and is parallel to QR, is 8ma².

Show, using integration, that the moment of inertia of the lamina about an axis which passes through P and T, is 1.5ma². [ You may assume without proof that the moment of inertia of a uniform rod, of mass m and length 2l, about an axis perpendicular to the rod through its midpoint is ml₂]

The lamina is now free to rotate in a vertical plane about a fixed smooth horizontal axis A which passes through P and is perpendicular to the plane of the lamina. The lamina makes small oscillations about its position of stable equilibrium.

By writing down an equation of rotational motion for the lamina as it rotates about A, find the approximate period of these small oscillations.

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 piece of wire ABC, of mass 2 m and length 4 a, is bent into two straight equal portions, AB and BC, which are at right angles to each other, as shown in Figure 1. The wire rotates freely in a vertical plane about a fixed smooth horizontal axis L which passes through A and is perpendicular to the plane of the wire.

Show that the moment of inertia of the wire about L is

By writing down an equation of rotational motion for the wire as it rotates about L, find the period of small oscillations of the wire about its position of stable equilibrium.

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 uniform circular disc, of mass m and radius a, is free to rotate about a fixed smooth horizontal axis L. The axis L is a tangent to the disc at the point A. The centre O of the disc moves in a vertical plane that is perpendicular to L. The disc is held at rest with its plane horizontal and released.

Find the angular acceleration of the disc when it has turned through an angle of

Find the magnitude of the component, in a direction perpendicular to the disc, of the force of the axis L acting on the disc at A, when the disc has turned through an angle

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 square lamina S has side 2 a. The radius of gyration of S about an axis through a vertex, perpendicular to S, is k.

The lamina S is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to S and passes through a vertex.

By writing down an equation of rotational motion for S, find the period of small oscillations of S about its position of stable equilibrium.

A uniform circular lamina has radius 2 a and centre C. The points P, Q, R and S on the lamina are the vertices of a square with centre C and CP = a. Four circular discs, each of radius , with centres P, Q, R and S, are removed from the lamina. The remaining lamina forms a template T, as shown in Figure 1. The radius of gyration of T about an axis through C, perpendicular to T, is k.

The template T is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to T and passes through a point on its outer rim.

Write down an equation of rotational motion for T and deduce that the period of small oscillations of T about its stable equilibrium position is

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

A light inextensible string has a particle of mass m attached to one end and a particle of mass 4 m attached to the other end. The string passes over a rough pulley which is modelled as a uniform circular disc of radius a and mass 2 m, as shown in Figure 2. The pulley can rotate in a vertical plane about a fixed horizontal axis which passes through the centre of the pulley and is perpendicular to the plane of the pulley. As the pulley rotates, a frictional couple of constant magnitude 2 mga acts on it. The system is held with the string vertical and taut on each side of the pulley and released from rest. Given that the string does not slip on the pulley, find the initial angular acceleration of the pulley.

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.

A body consists of a uniform plane circular disc, of radius r and mass 2m, with a particle of mass 3m attached to the circumference of the disc at the point P. The line PQ is a diameter of the disc. The body 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 Q. The body is held with OP making an angle £ with the downward

component, perpendicular to PQ, of the force acting on the body at Q at the instant when it is released. [You may assume that the momentof inertia of the body about L is (6)

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 uniform rod PQ, of mass and length 2a, is made to rotate in a vertical plane with constant angular speed about a fixed smooth horizontal axis through the end P of the rod. Show that, when the rod is inclined at an angle to the downward vertical, the magnitude of the force exerted on the axis by the rod is 2mg .

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 lamina ABC of mass m is in the shape of an isosceles triangle with AB = AC = 5 a and BC = 8 a.

Show, using integration, that the moment of inertia of the lamina about an axis

The foot of the perpendicular from A to BC is D. The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through D and is perpendicular to the plane of the lamina. The lamina is released from rest when DA makes an angle α with the downward vertical. It is given that the moment of inertia of the lamina about an axis through A, perpendicular to BC and in the plane of the lamina, is ma².

Find the angular acceleration of the lamina when DA makes an angle θ with the downward vertical.

Given that α is small,

find an approximate value for the period of oscillation of the lamina about the vertical.

A pendulum consists of a uniform rod AB, of length 4 a and mass 2 m, whose end A is rigidly attached to the centre O of a uniform square lamina PQRS, of mass 4 m and side a. The rod AB is perpendicular to the plane of the lamina. The pendulum is free to rotate about a fixed smooth horizontal axis L which passes through B. The axis L is perpendicular to AB and parallel to the edge PQ of the square.

Show that the moment of inertia of the pendulum about L is 75ma².

The pendulum is released from rest when BA makes an angle α with the downward vertical through B, where tan α = . When BA makes an angle θ with the downward vertical through B, the magnitude of the component, in the direction AB, of the force exerted by the axis L on the pendulum is X.

Find an expression for X in terms of m, g and θ.

Using the approximation sin,

find an estimate of the time for the pendulum to rotate through an angle α from its initial rest position.

A pendulum P is modelled as a uniform rod AB, of length 9 a and mass m, rigidly fixed to a uniform circular disc of radius a and mass 2 m. The end B of the rod is attached to the centre of the disc, and the rod lies in the plane of the disc, as shown in Figure 1. The pendulum is free to rotate in a vertical plane about a fixed smooth horizontal axis L which passes through the end A and is perpendicular to the plane of the disc.

Show that the moment of inertia of P about L is 190ma².

The pendulum makes small oscillations about L.

By writing down an equation of motion for P, find the approximate period of these small oscillations.

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.