OCR MEI M2 (Mechanics 2) 2011 June

1

1. Sphere P , of mass 10 kg , and sphere Q , of mass 15 kg , move with their centres on a horizontal straight line and have no resistances to their motion. \(\mathrm { P } , \mathrm { Q }\) and the positive direction are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-2_332_803_434_712} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} Initially, P has a velocity of \(- 1.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is acted on by a force of magnitude 13 N acting in the direction PQ . After \(T\) seconds, P has a velocity of \(4.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and has not reached Q .
   1. Calculate \(T\). The force of magnitude 13 N is removed. P is still travelling at \(4.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with Q , which has a velocity of \(- 0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Suppose that P and Q coalesce in the collision to form a single object.
   2. Calculate their common velocity after the collision. Suppose instead that P and Q separate after the collision and that P has a velocity of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) afterwards.
   3. Calculate the velocity of Q after the collision and also the coefficient of restitution in the collision.
2. Fig. 1.2 shows a small ball projected at a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) below the horizontal over smooth horizontal ground. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-2_424_832_1918_699} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} The ball is initially 3.125 m above the ground. The coefficient of restitution between the ball and the ground is 0.6 . Calculate the angle with the horizontal of the ball's trajectory immediately after the second bounce on the ground.

2 Any non-exact answers to this question should be given correct to four significant figures.
A thin, straight beam AB is 2 m long. It has a weight of 600 N and its centre of mass G is 0.8 m from end A. The beam is freely pivoted about a horizontal axis through A. The beam is held horizontally in equilibrium.
Initially this is done by means of a support at B, as shown in Fig.2.1. \begin{figure}[h] \end{figure}

1. Calculate the force on the beam due to the support at B . The support at B is now removed and replaced by a wire attached to the beam 0.3 m from A and inclined at \(50 ^ { \circ }\) to the beam, as shown in Fig. 2.2. The beam is still horizontal and in equilibrium. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-3_275_803_1226_671} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
2. Calculate the tension in the wire. The forces acting on the beam at A due to the hinge are a horizontal force \(X \mathrm {~N}\) in the direction AB , and a downward vertical force \(Y \mathrm {~N}\), which have a resultant of magnitude \(R\) at \(\alpha\) to the horizontal.
3. Calculate \(X , Y , R\) and \(\alpha\). The dotted lines in Fig. 2.3 are the lines of action of the tension in the wire and the weight of the beam. These lines of action intersect at P . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-3_460_791_2074_678} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
   \end{figure}
4. State with a reason the size of the angle GAP.

3 A bracket is being made from a sheet of uniform thin metal. Firstly, a plate is cut from a square of the sheet metal in the shape OABCDEFHJK, shown shaded in Fig. 3.1. The dimensions shown in the figure are in centimetres; axes \(\mathrm { O } x\) and \(\mathrm { O } y\) are also shown. \begin{figure}[h] \end{figure}

1. Show that, referred to the axes given in Fig. 3.1, the centre of mass of the plate OABCDEFHJK has coordinates (0.8, 2.5). The plate is hung using light vertical strings attached to \(\mathbf { J }\) and \(\mathbf { H }\). The edge \(\mathbf { J H }\) is horizontal and the plate is in equilibrium. The weight of the plate is 3.2 N .
2. Calculate the tensions in each of the strings. The plate is now bent to form the bracket. This is shown in Fig. 3.2: the rectangle IJKO is folded along the line IA so that it is perpendicular to the plane ABCGHI ; the rectangle DEFG is folded along the line DG so it is also perpendicular to the plane ABCGHI but on the other side of it. Fig. 3.2 also shows the axes \(\mathrm { O } x , \mathrm { O } y\) and \(\mathrm { O } z\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-4_611_782_1713_678} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
3. Show that, referred to the axes given in Fig. 3.2, the centre of mass of the bracket has coordinates ( \(1,2.7,0\) ). The bracket is now hung freely in equilibrium from a string attached to O .
4. Calculate the angle between the edge OI and the vertical.

4

1. A parachutist and her equipment have a combined mass of 80 kg . During a descent where the parachutist loses 1600 m in height, her speed reduces from \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and she does \(1.3 \times 10 ^ { 6 } \mathrm {~J}\) of work against resistances. Use an energy method to calculate the value of \(V\).
2. A vehicle of mass 800 kg is climbing a hill inclined at \(\theta\) to the horizontal, where \(\sin \theta = 0.1\). At one time the vehicle has a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is accelerating up the hill at \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) against a resistance of 1150 N .
   1. Show that the driving force on the vehicle is 2134 N and calculate its power at this time. The vehicle is pulling a sledge, of mass 300 kg , which is sliding up the hill. The sledge is attached to the vehicle by a light, rigid coupling parallel to the slope. The force in the coupling is 900 N .
   2. Assuming that the only resistance to the motion of the sledge is due to friction, calculate the coefficient of friction between the sledge and the ground.