OCR MEI M3 (Mechanics 3) 2014 June

1

1. The speed \(v\) of sound in a solid material is given by \(v = \sqrt { \frac { E } { \rho } }\), where \(E\) is Young's modulus for the material and \(\rho\) is its density.
   1. Find the dimensions of Young's modulus. The density of steel is \(7800 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\) and the speed of sound in steel is \(6100 \mathrm {~ms} ^ { - 1 }\).
   2. Find Young's modulus for steel, stating the units in which your answer is measured. A tuning fork has cylindrical prongs of radius \(r\) and length \(l\). The frequency \(f\) at which the tuning fork vibrates is given by \(f = k c ^ { \alpha } E ^ { \beta } \rho ^ { \gamma }\), where \(c = \frac { l ^ { 2 } } { r }\) and \(k\) is a dimensionless constant.
   3. Find \(\alpha , \beta\) and \(\gamma\).
2. A particle P is performing simple harmonic motion along a straight line, and the centre of the oscillations is O . The points X and Y on the line are on the same side of O , at distances 3.9 m and 6.0 m from O respectively. The speed of P is \(1.04 \mathrm {~ms} ^ { - 1 }\) when it passes through X and \(0.5 \mathrm {~ms} ^ { - 1 }\) when it passes through Y.
   1. Find the amplitude and the period of the oscillations.
   2. Find the time taken for P to travel directly from X to Y .

2

1. The fixed point A is vertically above the fixed point B . A light inextensible string of length 5.4 m has one end attached to A and the other end attached to B. The string passes through a small smooth ring R of mass 0.24 kg , and R is moving at constant angular speed in a horizontal circle. The circle has radius 1.6 m , and \(\mathrm { AR } = 3.4 \mathrm {~m} , \mathrm { RB } = 2.0 \mathrm {~m}\), as shown in Fig. 2 . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a0df44f-f8f0-44d4-b2f6-70a5314706f9-3_565_504_447_753} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Find the tension in the string.
   2. Find the angular speed of R .
2. A particle P of mass 0.3 kg is joined to a fixed point O by a light inextensible string of length 1.8 m . The particle P moves without resistance in part of a vertical circle with centre O and radius 1.8 m . When OP makes an angle of \(25 ^ { \circ }\) with the downward vertical, the tension in the string is 15 N .
   1. Find the speed of P when OP makes an angle of \(25 ^ { \circ }\) with the downward vertical.
   2. Find the tension in the string when OP makes an angle of \(60 ^ { \circ }\) with the upward vertical.
   3. Find the speed of P at the instant when the string becomes slack.

3 The fixed points A and B lie on a line of greatest slope of a smooth inclined plane, with B higher than A . The horizontal distance from A to B is 2.4 m and the vertical distance is 0.7 m . The fixed point C is 2.5 m vertically above B . A light elastic string of natural length 2.2 m has one end attached to C and the other end attached to a small block of mass 9 kg which is in contact with the plane. The block is in equilibrium when it is at A, as shown in Fig. 3. \begin{figure}[h] \end{figure}

1. Show that the modulus of elasticity of the string is 37.73 N . The block starts at A and is at rest. A constant force of 18 N , acting in the direction AB , is then applied to the block so that it slides along the line AB .
2. Find the magnitude and direction of the acceleration of the block
   (A) when it leaves the point A ,
   (B) when it reaches the point B .
3. Find the speed of the block when it reaches the point B .

4 The region \(R\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = \mathrm { e } ^ { - x }\) and the line \(x = k\), where \(k\) is a positive constant.

1. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid, and show that it can be written in the form $$\frac { 1 } { 2 } - \frac { k } { \mathrm { e } ^ { 2 k } - 1 } .$$
2. The solid in part (i) is placed with its larger circular face in contact with a rough plane inclined at \(60 ^ { \circ }\) to the horizontal, as shown in Fig. 4, and you are given that no slipping occurs. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a0df44f-f8f0-44d4-b2f6-70a5314706f9-5_508_483_712_790} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} Show that, whatever the value of \(k\), the solid will not topple.
3. A uniform lamina occupies the region \(R\). Find, in terms of \(k\), the coordinates of the centre of mass of this lamina. \section*{END OF QUESTION PAPER}