OCR MEI M2 (Mechanics 2) 2008 June

Question 1

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Disc A of mass 6 kg and disc B of mass 0.5 kg are moving in the same straight line. The relative positions of the discs and the \(\mathbf { i }\) direction are shown in Fig. 1.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-2_282_1325_402_450} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
\end{figure} The discs collide directly. The impulse on A in the collision is \(- 12 \mathbf { i }\) Ns and after the collision A has velocity \(3 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B has velocity \(11 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
1. Show that the velocity of A just before the collision is \(5 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and find the velocity of B at this time.
2. Calculate the coefficient of restitution in the collision.
3. After the collision, a force of \(- 2 \mathbf { i } \mathrm {~N}\) acts on B for 7 seconds. Find the velocity of B after this time.
A ball bounces off a smooth plane. The angles its path makes with the plane before and after the impact are \(\alpha\) and \(\beta\), as shown in Fig. 1.2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-2_317_1082_1468_575} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
\end{figure} The velocity of the ball before the impact is \(u \mathbf { i } - v \mathbf { j }\) and the coefficient of restitution in the impact is \(e\). Write down an expression in terms of \(u , v , e , \mathbf { i }\) and \(\mathbf { j }\) for the velocity of the ball immediately after the impact. Hence show that \(\tan \beta = e \tan \alpha\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-3_581_486_274_383} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-3_593_392_264_1370} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
\end{figure} A uniform wire is bent to form a bracket OABCD . The sections \(\mathrm { OA } , \mathrm { AB }\) and BC lie on three sides of a square and CD is parallel to AB . This is shown in Fig. 2.1 where the dimensions, in centimetres, are also given.
1. Show that, referred to the axes shown in Fig. 2.1, the \(x\)-coordinate of the centre of mass of the bracket is 3.6 . Find also the \(y\)-coordinate of its centre of mass.
2. The bracket is now freely suspended from D and hangs in equilibrium. Draw a diagram showing the position of the centre of mass and calculate the angle of CD to the vertical.
3. The bracket is now hung by means of vertical, light strings BP and DQ attached to B and to D , as shown in Fig. 2.2. The bracket has weight 5 N and is in equilibrium with OA horizontal. Calculate the tensions in the strings BP and DQ . The original bracket shown in Fig. 2.1 is now changed by adding the section OE, where AOE is a straight line. This section is made of the same type of wire and has length \(L \mathrm {~cm}\), as shown in Fig. 2.3.
  \(\begin{array} { l l l l } \begin{array} { l } \text { not to }
  \text { scale } \end{array} & 2 & 6 &
  \mathrm {~L} \longrightarrow & \mathrm {~L} & &
  \mathrm {~L} & \mathrm { O } & 6 & \mathrm {~A} \end{array}\) Fig. 2.3 The value of \(L\) is chosen so that the centre of mass is now on the \(y\)-axis.
4. Calculate \(L\).

Question 3

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Fig. 3.1 shows a framework in a vertical plane constructed of light, rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { AD }\) and BD . The rods are freely pin-jointed to each other at \(\mathrm { A } , \mathrm { B }\) and D and to a vertical wall at C and D. There are vertical loads of \(L \mathrm {~N}\) at A and \(3 L \mathrm {~N}\) at B . Angle DAB is \(30 ^ { \circ }\), angle DBC is \(60 ^ { \circ }\) and ABC is a straight, horizontal line. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-4_538_617_497_804} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
1. Draw a diagram showing the loads and the internal forces in the four rods.
2. Find the internal forces in the rods in terms of \(L\), stating whether each rod is in tension or in thrust (compression). [You may leave answers in surd form. Note that you are not required to find the external forces acting at C and at D.]
Fig. 3.2 shows uniform beams PQ and QR , each of length 2 lm and of weight \(W \mathrm {~N}\). The beams are freely hinged at Q and are in equilibrium on a rough horizontal surface when inclined at \(60 ^ { \circ }\) to the horizontal. You are given that the total force acting at Q on QR due to the hinge is horizontal. This force, \(U \mathrm {~N}\), is shown in Fig. 3.3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-4_428_566_1699_536} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-4_296_282_1699_1407} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
\end{figure} Show that the frictional force between the floor and each beam is \(\frac { \sqrt { 3 } } { 6 } W \mathrm {~N}\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-5_641_885_269_671} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} A small sphere of mass 0.15 kg is attached to one end, B, of a light, inextensible piece of fishing line of length 2 m . The other end of the line, A , is fixed and the line can swing freely. The sphere swings with the line taut from a point where the line is at an angle of \(40 ^ { \circ }\) with the vertical, as shown in Fig. 4.
1. Explain why no work is done on the sphere by the tension in the line.
2. Show that the sphere has dropped a vertical distance of about 0.4679 m when it is at the lowest point of its swing and calculate the amount of gravitational potential energy lost when it is at this point.
3. Assuming that there is no air resistance and that the sphere swings from rest from the position shown in Fig. 4, calculate the speed of the sphere at the lowest point of its swing.
4. Now consider the case where
  - there is a force opposing the motion that results in an energy loss of 0.6 J for every metre travelled by the sphere,
the sphere is given an initial speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (and it is descending) with AB at \(40 ^ { \circ }\) to the vertical.
A block of mass 3 kg slides down a uniform, rough slope that is at an angle of \(30 ^ { \circ }\) to the horizontal. The acceleration of the block is \(\frac { 1 } { 8 } g\). Show that the coefficient of friction between the block and the slope is \(\frac { 1 } { 4 } \sqrt { 3 }\).