OCR MEI M3 (Mechanics 3) 2008 June

1

1. 1. Write down the dimensions of velocity, acceleration and force. A ball of mass \(m\) is thrown vertically upwards with initial velocity \(U\). When the velocity of the ball is \(v\), it experiences a force \(\lambda v ^ { 2 }\) due to air resistance where \(\lambda\) is a constant.
   2. Find the dimensions of \(\lambda\). A formula approximating the greatest height \(H\) reached by the ball is $$H \approx \frac { U ^ { 2 } } { 2 g } - \frac { \lambda U ^ { 4 } } { 4 m g ^ { 2 } }$$ where \(g\) is the acceleration due to gravity.
   3. Show that this formula is dimensionally consistent. A better approximation has the form \(H \approx \frac { U ^ { 2 } } { 2 g } - \frac { \lambda U ^ { 4 } } { 4 m g ^ { 2 } } + \frac { 1 } { 6 } \lambda ^ { 2 } U ^ { \alpha } m ^ { \beta } g ^ { \gamma }\).
   4. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
2. A girl of mass 50 kg is practising for a bungee jump. She is connected to a fixed point O by a light elastic rope with natural length 24 m and modulus of elasticity 2060 N . At one instant she is 30 m vertically below O and is moving vertically upwards with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She comes to rest instantaneously, with the rope slack, at the point A . Find the distance OA .

2 A particle P of mass 0.3 kg is connected to a fixed point O by a light inextensible string of length 4.2 m . Firstly, P is moving in a horizontal circle as a conical pendulum, with the string making a constant angle with the vertical. The tension in the string is 3.92 N .

1. Find the angle which the string makes with the vertical.
2. Find the speed of P . P now moves in part of a vertical circle with centre O and radius 4.2 m . When the string makes an angle \(\theta\) with the downward vertical, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2). You are given that \(v = 8.4\) when \(\theta = 60 ^ { \circ }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2a4afead-e772-4d86-bc8d-86ffa5bca507-2_382_648_1985_751} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
3. Find the tension in the string when \(\theta = 60 ^ { \circ }\).
4. Show that \(v ^ { 2 } = 29.4 + 82.32 \cos \theta\).
5. Find \(\theta\) at the instant when the string becomes slack.

3 A small block B has mass 2.5 kg . A light elastic string connects B to a fixed point P , and a second light elastic string connects \(B\) to a fixed point \(Q\), which is 6.5 m vertically below \(P\). The string PB has natural length 3.2 m and stiffness \(35 \mathrm { Nm } ^ { - 1 }\); the string BQ has natural length 1.8 m and stiffness \(5 \mathrm { Nm } ^ { - 1 }\). The block B is released from rest in the position 4.4 m vertically below P . You are given that B performs simple harmonic motion along part of the line PQ, and that both strings remain taut throughout the motion. Air resistance may be neglected. At time \(t\) seconds after release, the length of the string PB is \(x\) metres (see Fig. 3). \begin{figure}[h] \end{figure}

1. Find, in terms of \(x\), the tension in the string PB and the tension in the string BQ .
2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 64 - 16 x\).
3. Find the value of \(x\) when B is at the centre of oscillation.
4. Find the period of oscillation.
5. Write down the amplitude of the motion and find the maximum speed of B.
6. Find the time after release when \(B\) is first moving downwards with speed \(0.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4

1. A uniform solid of revolution is obtained by rotating through \(2 \pi\) radians about the \(y\)-axis the region bounded by the curve \(y = 8 - 2 x ^ { 2 }\) for \(0 \leqslant x \leqslant 2\), the \(x\)-axis and the \(y\)-axis.
   1. Find the \(y\)-coordinate of the centre of mass of this solid. The solid is now placed on a rough plane inclined at an angle \(\theta\) to the horizontal. It rests in equilibrium with its circular face in contact with the plane as shown in Fig. 4. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{2a4afead-e772-4d86-bc8d-86ffa5bca507-4_511_568_616_831} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure}
   2. Given that the solid is on the point of toppling, find \(\theta\).
2. Find the \(y\)-coordinate of the centre of mass of a uniform lamina in the shape of the region bounded by the curve \(y = 8 - 2 x ^ { 2 }\) for \(- 2 \leqslant x \leqslant 2\), and the \(x\)-axis.