OCR MEI M3 (Mechanics 3) 2012 June

1 The fixed point A is at a height \(4 b\) above a smooth horizontal surface, and C is the point on the surface which is vertically below A. A light elastic string, of natural length \(3 b\) and modulus of elasticity \(\lambda\), has one end attached to A and the other end attached to a block of mass \(m\). The block is released from rest at a point B on the surface where \(\mathrm { BC } = 3 b\), as shown in Fig. 1. You are given that the block remains on the surface and moves along the line BC . \begin{figure}[h] \end{figure}

1. Show that immediately after release the acceleration of the block is \(\frac { 2 \lambda } { 5 m }\).
2. Show that, when the block reaches C , its speed \(v\) is given by \(v ^ { 2 } = \frac { \lambda b } { m }\).
3. Show that the equation \(v ^ { 2 } = \frac { \lambda b } { m }\) is dimensionally consistent. The time taken for the block to move from B to C is given by \(k m ^ { \alpha } b ^ { \beta } \lambda ^ { \gamma }\), where \(k\) is a dimensionless constant.
4. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\). When the string has natural length 1.2 m and modulus of elasticity 125 N , and the block has mass 8 kg , the time taken for the block to move from B to C is 0.718 s .
5. Find the time taken for the block to move from B to C when the string has natural length 9 m and modulus of elasticity 20 N , and the block has mass 75 kg .

2

1. Fig. 2 shows a car of mass 800 kg moving at constant speed in a horizontal circle with centre C and radius 45 m , on a road which is banked at an angle of \(18 ^ { \circ }\) to the horizontal. The forces shown are the weight \(W\) of the car, the normal reaction, \(R\), of the road on the car and the frictional force \(F\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-3_286_970_402_561} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Given that the frictional force is zero, find the speed of the car.
   2. Given instead that the speed of the car is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the frictional force and the normal reaction.
2. One end of a light inextensible string is attached to a fixed point O , and the other end is attached to a particle P of mass \(m \mathrm {~kg}\). Starting with the string taut and P vertically below \(\mathrm { O } , \mathrm { P }\) is set in motion with a horizontal velocity of \(7 \mathrm {~ms} ^ { - 1 }\). It then moves in part of a vertical circle with centre O . The string becomes slack when the speed of P is \(2.8 \mathrm {~ms} ^ { - 1 }\). Find the length of the string. Find also the angle that OP makes with the upward vertical at the instant when the string becomes slack.

3 A particle Q is performing simple harmonic motion in a vertical line. Its height, \(x\) metres, above a fixed level at time \(t\) seconds is given by $$x = c + A \cos ( \omega t - \phi )$$ where \(c , A , \omega\) and \(\phi\) are constants.

1. Show that \(\ddot { x } = - \omega ^ { 2 } ( x - c )\). Fig. 3 shows the displacement-time graph of Q for \(0 \leqslant t \leqslant 14\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-4_547_1079_703_495} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Find exact values for \(c , A , \omega\) and \(\phi\).
3. Find the maximum speed of Q .
4. Find the height and the velocity of Q when \(t = 0\).
5. Find the distance travelled by Q between \(t = 0\) and \(t = 14\).

4

1. A uniform lamina occupies the region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(y = 3 - \sqrt { x }\) for \(0 \leqslant x \leqslant 9\). Find the coordinates of the centre of mass of this lamina.
2. Fig. 4.1 shows the region bounded by the line \(x = 2\) and the part of the circle \(y ^ { 2 } = 25 - x ^ { 2 }\) for which \(2 \leqslant x \leqslant 5\). This region is rotated about the \(x\)-axis to form a uniform solid of revolution \(S\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-5_675_659_479_705} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
   1. Find the \(x\)-coordinate of the centre of mass of \(S\). The solid \(S\) rests in equilibrium with its curved surface in contact with a rough plane inclined at \(25 ^ { \circ }\) to the horizontal. Fig. 4.2 shows a vertical section containing AB , which is a diameter and also a line of greatest slope of the flat surface of \(S\). This section also contains XY, which is a line of greatest slope of the plane. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-5_494_560_1615_749} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
   2. Find the angle \(\theta\) that AB makes with the horizontal.