OCR MEI M2 (Mechanics 2) 2016 June

1

1. Two model railway trucks are moving freely on a straight horizontal track when they are in a direct collision. The trucks are P of mass 0.5 kg and Q of mass 0.75 kg . They are initially travelling in the same direction. Just before they collide P has a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and Q has a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-2_263_640_484_715} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure}
   1. Suppose that the speed of P is halved in the collision and that its direction of motion is not changed. Find the speed of Q immediately after the collision and find the coefficient of restitution.
   2. Show that it is not possible for both the speed of P to be halved in the collision and its direction of motion to be reversed. Both of the model trucks have flat horizontal tops. They are each travelling at the speeds they had immediately after the collision. Part of the mass of Q is a small object of mass 0.1 kg at rest at the edge of the top of Q nearest P . The object falls off, initially with negligible velocity relative to Q .
   3. Determine the speed of Q immediately after the object falls off it, making your reasoning clear. Part of the mass of P is an object of mass 0.05 kg that is fired horizontally from the top of P , parallel to and in the opposite direction to the motion of P . Immediately after the object is fired, it has a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to P .
   4. Determine the speed of P immediately after the object has been fired from it.
2. The velocities of a small object immediately before and after an elastic collision with a horizontal plane are shown in Fig. 1.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-2_172_741_1987_644} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} Show that the plane cannot be smooth.

2

1. A bullet of mass 0.04 kg is fired into a fixed uniform rectangular block along a line through the centres of opposite parallel faces, as shown in Fig. 2.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-3_209_1287_342_388} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
   \end{figure} The bullet enters the block at \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and comes to rest after travelling 0.2 m into the block.
   1. Calculate the resistive force on the bullet, assuming that this force is constant. Another bullet of the same mass is fired, as before, with the same speed into a similar block of mass 3.96 kg . The block is initially at rest and is free to slide on a smooth horizontal plane.
   2. By considering linear momentum, find the speed of the block with the bullet embedded in it and at rest relative to the block.
   3. By considering mechanical energy, find the distance the bullet penetrates the block, given the resistance of the block to the motion of the bullet is the same as in part (i).
2. Fig. 2.2 shows a block of mass 6 kg on a uniformly rough plane that is inclined at \(30 ^ { \circ }\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-3_348_636_1382_712} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure} A string with a constant tension of 91.5 N parallel to the plane pulls the block up a line of greatest slope. The speed of the block increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a distance of 8 m .

1. Use an energy method to find the magnitude of the frictional force acting on the block. Calculate the coefficient of friction between the block and the plane.
2. Calculate the power of the tension in the string when the block has a speed of \(7 \mathrm {~ms} ^ { - 1 }\). Fig. 3.1 shows a thin planar uniform rigid rectangular sheet of metal, OPQR, of width 1.65 m and height 1.2 m . The mass of the sheet is \(M \mathrm {~kg}\). The sides OP and PQ have thin rigid uniform reinforcements attached with masses \(0.6 M \mathrm {~kg}\) and \(0.4 M \mathrm {~kg}\), respectively. Fig. 3.1 also shows coordinate axes with origin at O . The sheet with its reinforcements is to be used as an inn sign.
3. Calculate the coordinates of the centre of mass of the inn sign. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_421_492_210_1334} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure} The inn sign has a weight of 300 N . It hangs in equilibrium with QR horizontal when vertical forces \(Y _ { \mathrm { Q } } \mathrm { N }\) and \(Y _ { \mathrm { R } } \mathrm { N }\) act at Q and R respectively.
4. Calculate the value of \(Y _ { \mathrm { Q } }\) and show that \(Y _ { \mathrm { R } } = 120\). The inn sign is hung from a framework, ABCD , by means of two light vertical inextensible wires attached to the sign at Q and R and the framework at B and C , as shown in Fig. 3.2. QR and BC are horizontal. The framework is made from light rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { CA }\) and CD freely pin-jointed together at \(\mathrm { A } , \mathrm { B }\) and C and to a vertical wall at A and D . Fig. 3.3 shows the dimensions of the framework in metres as well as the external forces \(X _ { \mathrm { A } } \mathrm { N } , Y _ { \mathrm { A } } \mathrm { N }\) acting at A and \(X _ { \mathrm { D } } \mathrm { N } , Y _ { \mathrm { D } } \mathrm { N }\) acting at D . You are given that \(\sin \alpha = \frac { 5 } { 13 } , \cos \alpha = \frac { 12 } { 13 } , \sin \beta = \frac { 4 } { 5 }\) and \(\cos \beta = \frac { 3 } { 5 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_543_526_1420_253} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_629_793_1343_964} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
   \end{figure}
5. Mark on the diagram in your Printed Answer Book all the forces acting on the pin-joints at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , including those internal to the rods, when the inn sign is hanging from the framework.
6. Show that \(X _ { \mathrm { D } } = 261\).
7. Calculate the forces internal to the rods \(\mathrm { AB } , \mathrm { BC }\) and CD , stating whether each rod is in tension or thrust (compression). Calculate also the values of \(Y _ { \mathrm { D } }\) and \(Y _ { \mathrm { A } }\). [Your working in this part should correspond to your diagram in part (iii).]

4 Fig. 4.1 shows a hollow circular cylinder open at one end and closed at the other. The radius of the cylinder is 0.1 m and its height is \(h \mathrm {~m} . \mathrm { O }\) and C are points on the axis of symmetry at the centres of the open and closed ends, respectively. The thin material used for the closed end has four times the density of the thin material used for the curved surface. \begin{figure}[h] \end{figure} Cylinders of this type are made with different values of \(h\).

1. Show that the centres of mass of these cylinders are on the line OC at a distance \(\frac { 5 h ^ { 2 } + 2 h } { 2 + 10 h } \mathrm {~m}\) from O . Fig. 4.2 shows one of the cylinders placed with its open end on a slope inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 2 } { 3 }\). The cylinder does not slip but is on the point of tipping.
2. Show that \(50 h ^ { 2 } + 5 h - 3 = 0\) and hence that \(h = 0.2\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-5_383_497_1178_1402} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure} Fig. 4.3 shows another of the cylinders that has weight 42 N and \(h = 0.5\). This cylinder has its open end on a rough horizontal plane. A force of magnitude \(T \mathrm {~N}\) is applied to a point P on the circumference of the closed end. This force is at an angle \(\beta\) with the horizontal such that \(\tan \beta = \frac { 3 } { 4 }\) and the force is in the vertical plane containing \(\mathrm { O } , \mathrm { C }\) and P . The cylinder does not slip but is on the point of tipping. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-5_451_679_1955_685} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{figure}
3. Calculate \(T\).