Rotation about fixed axis: angular acceleration and velocity

1 A uniform disc with centre \(O\) has mass \(m\) and radius \(a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through \(O\). One end of a light inextensible string is attached to a point on the circumference and is wrapped several times round the circumference. A particle \(P\), of mass \(2 m\), is attached to the free end of the string and the disc is held at rest with \(P\) hanging freely. The system is released from rest. Assuming that resistances may be neglected, find the acceleration of \(P\).

3 \includegraphics[max width=\textwidth, alt={}, center]{28e7fb78-e2b6-4f6e-92dc-a06eb87fe1ef-2_582_513_1292_815} A uniform disc, of mass \(m\) and radius \(a\), is free to rotate without resistance in a vertical plane about a horizontal axis through its centre. A light inextensible string has one end fixed to the rim of the disc, and is wrapped round the rim. A block of mass \(2 m\) is attached to the other end of the string (see diagram). The system is released from rest with the block hanging vertically. While the block moves it experiences a constant resistance to motion of magnitude \(\frac { 1 } { 10 } m g\). Find the angular acceleration of the disc, and find also the angular speed of the disc when it has turned through one complete revolution.
[0pt] [9]

3 A uniform rod \(X Y\), of mass 5 kg and length 1.8 m , is free to rotate in a vertical plane about a fixed horizontal axis through \(X\). The rod is at rest with \(Y\) vertically below \(X\) when a couple of constant moment is applied to the rod. It then rotates, and comes instantaneously to rest when \(X Y\) is horizontal.

1. Find the moment of the couple.
2. Find the angular acceleration of the rod
   1. immediately after the couple is first applied,
   2. when \(X Y\) is horizontal.

5 A uniform rod of mass 4 kg and length 2.4 m can rotate in a vertical plane about a fixed horizontal axis through one end of the rod. The rod is released from rest in a horizontal position and a frictional couple of constant moment 20 Nm opposes the motion.

1. Find the angular acceleration of the rod immediately after it is released.
2. Find the angle that the rod makes with the horizontal when its angular acceleration is zero.
3. Find the maximum angular speed of the rod.
4. The rod first comes to instantaneous rest after rotating through an angle \(\theta\) radians from its initial position. Find an equation for \(\theta\), and verify that \(2.0 < \theta < 2.1\).

6. \begin{figure}[h] \end{figure} 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 \(2 m g a\) 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 rod \(AB\) of mass \(m\) and length \(4a\) is free to rotate in a vertical plane about a horizontal axis through the point \(O\) of the rod, where \(OA = a\). The rod is slightly disturbed from rest when \(B\) is vertically above \(A\).

1. Find the magnitude of the angular acceleration of the rod when it is horizontal. [4]
2. Find the angular speed of the rod when it is horizontal. [2]
3. Calculate the magnitude of the force acting on the rod at \(O\) when the rod is horizontal. [5]