Moments of inertia

A uniform lamina of mass M kg is modelled as the region which is bounded by the curve with equation y = x², the positive x-axis and the line with equation x = 2. The unit of length on both axes is the metre. Find the moment of inertia of the lamina about the x-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 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.

Find, using integration, the moment of inertia of a uniform cylindrical shell of radius r, height h and mass M, about a diameter of one end.

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 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.

Find, using integration, the moment of inertia of a uniform solid hemisphere, of mass m and radius a, about a diameter of its plane face. [You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass m and radius r, about a diameter is mr².]

Hence find the moment of inertia of a uniform solid sphere, of mass M and radius a, about a diameter.

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.

Show, using integration, that the moment of inertia of a uniform solid right circular cone of mass M, height h and base radius a, about an axis through the vertex, parallel to the base, is

[ You may assume without proof that the moment of inertia of a uniform circular disc, of radius r and mass m, about a diameter is mr.]

A uniform triangular lamina ABC, of mass M, has AB = AC and BC = 2 a. The mid-point of BC is D and AD = h, as shown in Figure 1. Show, using integration, that the moment of inertia of the lamina about an axis through A, perpendicular to the plane of the lamina, is

[You may assume without proof that the moment of inertia of a uniform rod, of length 2l and mass m, about an axis through its midpoint and perpendicular to the rod, is ml².]

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 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.

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 lamina of mass m is in the shape of a triangle ABC. The perpendicular distance of C from the line AB is h. Prove, using integration, that the moment of inertia of the lamina about AB is mh?. (7)

Deduce the radius of gyration of a uniform square lamina of side 2a, about a diagonal.(3) The points X and Y are the mid-points of the sides RO and RS respectively of a square PORS of side 2a. A uniform lamina of mass M is in the shape of POXYS.

Show that the moment of inertia of this lamina about XY is Ma’. (6)

Prove, using integration, that the moment of inertia of a uniform solid right circular cone, of mass M and base radius a, about its axis is Ma². [You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass m and radius r, about an axis through its centre and perpendicular to its plane is

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 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 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 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 solid right circular cylinder has mass M, height h and radius a. Find, using integration, its moment of inertia about a diameter of one of its circular ends. [You may assume without proof that the moment of inertia of a uniform circular disc, of mass m and radius a, about a diameter is .2ma ma² ]