OCR M4 (Mechanics 4) 2016 June

1 A uniform square lamina, of mass 5 kg and side 0.2 m , is rotating about a fixed vertical axis that is perpendicular to the lamina and that passes through its centre. A couple of constant moment 0.06 Nm is applied to the lamina. The lamina turns through an angle of 155 radians while its angular speed increases from \(8 \mathrm { rads } ^ { - 1 }\) to \(\omega \mathrm { rads } ^ { - 1 }\). Find \(\omega\).

2
\includegraphics[max width=\textwidth, alt={}, center]{27b790da-800f-4f5e-8f63-d52159efb48e-2_959_1166_609_450} Boat \(A\) is travelling with constant speed \(7.9 \mathrm {~ms} ^ { - 1 }\) on a course with bearing \(035 ^ { \circ }\). Boat \(B\) is travelling with constant speed \(10.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a course with bearing \(330 ^ { \circ }\). At one instant, the boats are 1500 m apart with \(B\) on a bearing of \(125 ^ { \circ }\) from \(A\) (see diagram).

1. Find the magnitude and the bearing of the velocity of \(B\) relative to \(A\).
2. Find the shortest distance between \(A\) and \(B\) in the subsequent motion.
3. Find the time taken from the instant when \(A\) and \(B\) are 1500 m apart to the instant when \(A\) and \(B\) are at the point of closest approach.
   \includegraphics[max width=\textwidth, alt={}, center]{27b790da-800f-4f5e-8f63-d52159efb48e-3_1057_1047_248_511} Two uniform rods \(A B\) and \(B C\), each of length \(a\) and mass \(m\), are rigidly joined together so that \(A B\) is perpendicular to \(B C\). The rod \(A B\) is freely hinged to a fixed point at \(A\). The rods can rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda m g\) is attached to \(B\). The other end of the string is attached to a fixed point \(D\) vertically above \(A\), where \(A D = a\). The string \(B D\) makes an angle \(\theta\) radians with the downward vertical (see diagram).
4. Taking \(D\) as the reference level for gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$\mathrm { V } = \frac { 1 } { 2 } \mathrm { mga } ( \sin 2 \theta - 3 \cos 2 \theta ) + \frac { 1 } { 2 } \lambda \mathrm { mga } ( 2 \cos \theta - 1 ) ^ { 2 } - 2 \mathrm { mga } .$$
5. Given that \(\theta = \frac { 1 } { 4 } \pi\) is a position of equilibrium, find the exact value of \(\lambda\).
6. Find \(\frac { d ^ { 2 } V } { d \theta ^ { 2 } }\) and hence determine whether the position of equilibrium at \(\theta = \frac { 1 } { 4 } \pi\) is stable or unstable.

4 The region bounded by the curve \(\mathrm { y } = 2 \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { x } }\) for \(0 \leqslant x \leqslant 2\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\), is occupied by a uniform lamina.

1. Find the exact value of the \(y\)-coordinate of the centre of mass of the lamina. As shown in the diagram below, a uniform lamina occupies the closed region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\) where $$f ( x ) = \begin{cases} 2 \mathrm { e } ^ { \frac { 1 } { 2 } x } & 0 \leqslant x \leqslant 2
   \frac { 2 } { 3 } ( 5 - x ) \mathrm { e } & 2 \leqslant x \leqslant 5 . \end{cases}$$ \includegraphics[max width=\textwidth, alt={}, center]{27b790da-800f-4f5e-8f63-d52159efb48e-4_863_1179_762_443}
2. Find the exact value of the \(x\)-coordinate of the centre of mass of the lamina.

5 A uniform rod \(A B\) has mass \(2 m\) and length 4a.

1. Show by integration that the moment of inertia of the rod about an axis perpendicular to the rod through \(A\) is \(\frac { 32 } { 3 } \mathrm { ma } ^ { 2 }\) The rod is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). A particle of mass \(m\) is moving horizontally in the plane in which the rod is free to rotate. The particle has speed \(v\), and strikes the rod at \(B\). In the subsequent motion the particle adheres to the rod and the combined rigid body \(Q\), consisting of the rod and the particle, starts to rotate.
2. Find, in terms of \(v\) and \(a\), the initial angular speed of \(Q\). At time \(t\) seconds the angle between \(Q\) and the downward vertical is \(\theta\) radians.
3. Show that \(\dot { \theta } ^ { 2 } = \mathrm { k } \frac { \mathrm { g } } { \mathrm { a } } ( \cos \theta - 1 ) + \frac { 9 \mathrm { v } ^ { 2 } } { 400 \mathrm { a } ^ { 2 } }\), stating the value of the constant \(k\).
4. Find, in terms of \(a\) and \(g\), the set of values of \(v ^ { 2 }\) for which \(Q\) makes complete revolutions. When \(Q\) is horizontal, the force exerted by the axis on \(Q\) has vertically upwards component \(R\).
5. Find \(R\) in terms of \(m\) and \(g\).
   \includegraphics[max width=\textwidth, alt={}, center]{27b790da-800f-4f5e-8f63-d52159efb48e-6_844_509_248_778} A compound pendulum consists of a uniform rod \(A B\) of length 1 m and mass 3 kg , a particle of mass 1 kg attached to the rod at \(A\) and a circular disc of radius \(\frac { 1 } { 3 } \mathrm {~m}\), mass 6 kg and centre \(C\). The end \(B\) of the rod is rigidly attached to a point on the circumference of the disc in such a way that \(A B C\) is a straight line. The pendulum is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(P\) on the rod where \(\mathrm { AP } = \mathrm { xm }\) and \(\mathrm { x } < \frac { 1 } { 2 }\) (see diagram).
6. Show that the moment of inertia of the pendulum about the axis of rotation is \(\left( 10 x ^ { 2 } - 19 x + 12 \right) \mathrm { kg } \mathrm { m } ^ { 2 }\). The pendulum is making small oscillations about the equilibrium position, such that at time \(t\) seconds the angular displacement that the pendulum makes with the downward vertical is \(\theta\) radians.
7. Find the angular acceleration of the pendulum, in terms of \(x , g\) and \(\theta\).
8. Show that the motion is approximately simple harmonic, and show that the approximate period of oscillations, in seconds, is given by \(2 \pi \sqrt { \frac { 20 x ^ { 2 } - 38 x + 24 } { ( 19 - 20 x ) g } }\).
9. Hence find the value of \(x\) for which the approximate period of oscillations is least.