OCR MEI M3 (Mechanics 3) 2007 June

1

1. 1. Write down the dimensions of the following quantities. \begin{displayquote} Velocity
      Acceleration
      Force
      Density (which is mass per unit volume)
      Pressure (which is force per unit area) \end{displayquote} For a fluid with constant density \(\rho\), the velocity \(v\), pressure \(P\) and height \(h\) at points on a streamline are related by Bernoulli's equation $$P + \frac { 1 } { 2 } \rho v ^ { 2 } + \rho g h = \mathrm { constant } ,$$ where \(g\) is the acceleration due to gravity.
   2. Show that the left-hand side of Bernoulli's equation is dimensionally consistent.
2. In a wave tank, a float is performing simple harmonic motion with period 3.49 s in a vertical line. The height of the float above the bottom of the tank is \(h \mathrm {~m}\) at a time \(t \mathrm {~s}\). When \(t = 0\), the height has its maximum value. The value of \(h\) varies between 1.6 and 2.2.
   1. Sketch a graph showing how \(h\) varies with \(t\).
   2. Express \(h\) in terms of \(t\).
   3. Find the magnitude and direction of the acceleration of the float when \(h = 1.7\).

2 A fixed hollow sphere with centre O has an inside radius of 2.7 m . A particle P of mass 0.4 kg moves on the smooth inside surface of the sphere. At first, P is moving in a horizontal circle with constant speed, and OP makes a constant angle of \(60 ^ { \circ }\) with the vertical (see Fig. 2.1). \begin{figure}[h] \end{figure}

1. Find the normal reaction acting on P .
2. Find the speed of P . The particle P is now placed at the lowest point of the sphere and is given an initial horizontal speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then moves in part of a vertical circle. When OP makes an angle \(\theta\) with the upward vertical and P is still in contact with the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\) (see Fig. 2.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-3_716_778_1653_696} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
3. Find \(v ^ { 2 }\) in terms of \(\theta\).
4. Show that \(R = 4.16 - 11.76 \cos \theta\).
5. Find the speed of P at the instant when it leaves the surface of the sphere.

3 A light elastic string has natural length 1.2 m and stiffness \(637 \mathrm { Nm } ^ { - 1 }\).

1. The string is stretched to a length of 1.3 m . Find the tension in the string and the elastic energy stored in the string. One end of this string is attached to a fixed point \(A\). The other end is attached to a heavy ring \(R\) which is free to move along a smooth vertical wire. The shortest distance from A to the wire is 1.2 m (see Fig. 3). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-4_357_337_669_863} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} The ring is in equilibrium when the length of the string \(A R\) is 1.3 m .
2. Show that the mass of the ring is 2.5 kg . The ring is given an initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards from its equilibrium position. It first comes to rest, instantaneously, in the position where the length of AR is 1.5 m .
3. Find \(u\).
4. Determine whether the ring will rise above the level of A .

4

1. The region bounded by the curve \(y = x ^ { 3 }\) for \(0 \leqslant x \leqslant 2\), the \(x\)-axis and the line \(x = 2\), is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina. [8]
2. The region bounded by the circular arc \(y = \sqrt { 4 - x ^ { 2 } }\) for \(1 \leqslant x \leqslant 2\), the \(x\)-axis and the line \(x = 1\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution, as shown in Fig. 4.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-5_627_499_593_785} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
   1. Show that the \(x\)-coordinate of the centre of mass of this solid of revolution is 1.35 . This solid is placed on a rough horizontal surface, with its flat face in a vertical plane. It is held in equilibrium by a light horizontal string attached to its highest point and perpendicular to its flat face, as shown in Fig. 4.2. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-5_573_613_1662_728} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
   2. Find the least possible coefficient of friction between the solid and the horizontal surface.