OCR MEI M3 (Mechanics 3) 2006 January

1

1. 1. Write down the dimensions of force. The period, \(t\), of a vibrating wire depends on its tension, \(F\), its length, \(l\), and its mass per unit length, \(\sigma\).
   2. Assuming that the relationship is of the form \(t = k F ^ { \alpha } l ^ { \beta } \sigma ^ { \gamma }\), where \(k\) is a dimensionless constant, use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\). Two lengths are cut from a reel of uniform wire. The first has length 1.2 m , and it vibrates under a tension of 90 N . The second has length 2.0 m , and it vibrates with the same period as the first wire.
   3. Find the tension in the second wire. (You may assume that changing the tension does not significantly change the mass per unit length.)
2. The midpoint M of a vibrating wire is moving in simple harmonic motion in a straight line, with amplitude 0.018 m and period 0.01 s .
   1. Find the maximum speed of M .
   2. Find the distance of M from the centre of the motion when its speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2

1. A moon of mass \(7.5 \times 10 ^ { 22 } \mathrm {~kg}\) moves round a planet in a circular path of radius \(3.8 \times 10 ^ { 8 } \mathrm {~m}\), completing one orbit in a time of \(2.4 \times 10 ^ { 6 } \mathrm {~s}\). Find the force acting on the moon.
2. Fig. 2 shows a fixed solid sphere with centre O and radius 4 m . Its surface is smooth. The point A on the surface of the sphere is 3.5 m vertically above the level of O . A particle P of mass 0.2 kg is placed on the surface at A and is released from rest. In the subsequent motion, when OP makes an angle \(\theta\) with the horizontal and P is still on the surface of the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b7f8bdfd-33dc-4453-8f3a-ddd24be17372-3_746_734_705_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Express \(v ^ { 2 }\) in terms of \(\theta\).
   2. Show that \(R = 5.88 \sin \theta - 3.43\).
   3. Find the radial and tangential components of the acceleration of P when \(\theta = 40 ^ { \circ }\).
   4. Find the value of \(\theta\) at the instant when P leaves the surface of the sphere.

3 A light elastic rope has natural length 15 m . One end of the rope is attached to a fixed point O and the other end is attached to a small rock of mass 12 kg . When the rock is hanging in equilibrium vertically below O , the length of the rope is 15.8 m .

1. Show that the modulus of elasticity of the rope is 2205 N . The rock is pulled down to the point 20 m vertically below O , and is released from rest in this position. It moves upwards, and comes to rest instantaneously, with the rope slack, at the point A .
2. Find the acceleration of the rock immediately after it is released.
3. Use an energy method to find the distance OA. At time \(t\) seconds after release, the rope is still taut and the displacement of the rock below the equilibrium position is \(x\) metres.
4. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 12.25 x\).
5. Write down an expression for \(x\) in terms of \(t\), and hence find the time between releasing the rock and the rope becoming slack.

4 The region between the curve \(y = 4 - x ^ { 2 }\) and the \(x\)-axis, from \(x = 0\) to \(x = 2\), is occupied by a uniform lamina. The units of the axes are metres.

1. Show that the coordinates of the centre of mass of this lamina are \(( 0.75,1.6 )\). This lamina and another exactly like it are attached to a uniform rod PQ , of mass 12 kg and length 8 m , to form a rigid body as shown in Fig. 4. Each lamina has mass 6.5 kg . The ends of the rod are at \(\mathrm { P } ( - 4,0 )\) and \(\mathrm { Q } ( 4,0 )\). The rigid body lies entirely in the \(( x , y )\) plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b7f8bdfd-33dc-4453-8f3a-ddd24be17372-4_511_956_1836_557} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
2. Find the coordinates of the centre of mass of the rigid body. The rigid body is freely suspended from the point \(\mathrm { A } ( 2,4 )\) and hangs in equilibrium.
3. Find the angle that PQ makes with the horizontal.