OCR M3 (Mechanics 3) 2013 January

1
\includegraphics[max width=\textwidth, alt={}, center]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-2_477_534_261_770} A ball of mass 0.6 kg is moving with speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line. It is struck by an impulse \(I \mathrm { Ns }\) acting at an acute angle \(\theta\) to its direction of motion (see diagram). The impulse causes the direction of motion of the ball to change by an acute angle \(\alpha\), where \(\sin \alpha = \frac { 8 } { 17 }\). After the impulse acts the ball is moving with a speed of \(3.4 \mathrm {~ms} ^ { - 1 }\). Find \(I\) and \(\theta\).

2 Two uniform smooth spheres \(A\) and \(B\), of equal radius and equal mass, are moving towards each other on a horizontal surface. Immediately before they collide, \(A\) has speed \(0.3 \mathrm {~ms} ^ { - 1 }\) along the line of centres and \(B\) has speed \(0.6 \mathrm {~ms} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line of centres (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-2_302_1013_1247_502} After the collision, the direction of motion of \(B\) is at right angles to its original direction of motion. Find

1. the speed of \(B\) after the collision,
2. the speed and direction of motion of \(A\) after the collision,
3. the coefficient of restitution between \(A\) and \(B\).

3 At time \(t = 0 \mathrm {~s}\) a particle \(P\), of mass 0.3 kg , is 1 m away from a point \(O\) on a smooth horizontal plane and is moving away from \(O\) with speed \(\sqrt { 5 } \mathrm {~ms} ^ { - 1 }\). The only horizontal force acting on \(P\) has magnitude \(1.5 x \mathrm {~N}\), where \(x\) is the distance \(O P\), and acts away from \(O\).

1. Show that the speed of \(P , v \mathrm {~ms} ^ { - 1 }\), is given by \(v = \sqrt { 5 } x\).
2. Find an expression for \(v\) in terms of \(t\).

4 A smooth cylinder of radius \(a \mathrm {~m}\) is fixed with its axis horizontal and \(O\) is the centre of a cross-section. Particle \(P\), of mass 0.4 kg , and particle \(Q\), of mass 0.6 kg , are connected by a light inextensible string of length \(\pi a \mathrm {~m}\). The string is held at rest with \(P\) and \(Q\) at opposite ends of the horizontal diameter of the crosssection through \(O\) (see Fig. 1). The string is released and \(Q\) begins to descend. When \(O P\) has rotated through \(\theta\) radians, with \(P\) remaining in contact with the cylinder, the speed of each particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that \(v ^ { 2 } = 3.92 a ( 3 \theta - 2 \sin \theta )\) and find an expression in terms of \(\theta\) for the normal force of the cylinder on \(P\) at this time.
2. Given that \(P\) leaves the surface of the cylinder when \(\theta = \alpha\), show that \(\sin \alpha = k \alpha\) where \(k\) is a constant to be found.

5 A particle \(P\), of mass 2.5 kg , is in equilibrium suspended from a fixed point \(A\) by a light elastic string of natural length 3 m and modulus of elasticity 36.75 N . Another particle \(Q\), of mass 1 kg , is released from rest at \(A\) and falls freely until it reaches \(P\) and becomes attached to it.

1. Show that the speed of the combined particles, immediately after \(Q\) becomes attached to \(P\), is \(2 \sqrt { 2 } \mathrm {~ms} ^ { - 1 }\). The combined particles fall a further distance \(X \mathrm {~m}\) before coming to instantaneous rest.
2. Find a quadratic equation satisfied by \(X\), and show that it simplifies to \(35 X ^ { 2 } - 56 X - 80 = 0\).

6 A uniform \(\operatorname { rod } A B\), of weight \(W\) and length \(2 l\) is in equilibrium at \(60 ^ { \circ }\) to the horizontal with \(A\) resting against a smooth vertical plane and \(B\) resting on a rough section of a horizontal plane. Another uniform rod \(C D\), of length \(\sqrt { 3 } l\) and weight \(W\), is freely jointed to the mid-point of \(A B\) at \(C\); its other end \(D\) rests on a smooth section of the horizontal plane. \(C D\) is inclined at \(30 ^ { \circ }\) to the horizontal (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-4_508_1075_438_495}

1. Show that the force exerted by the horizontal plane on \(C D\) is \(\frac { 1 } { 2 } W\). Find the normal component of the force exerted by the horizontal plane on \(A B\).
2. Find the magnitude and direction of the force exerted by \(C D\) on \(A B\).
3. Given that \(A B\) is in limiting equilibrium, find the coefficient of friction between \(A B\) and the horizontal plane.

7 A simple pendulum consists of a light inextensible string of length 0.8 m and a particle \(P\) of mass \(m \mathrm {~kg}\). The pendulum is hanging vertically at rest from a fixed point \(O\) when \(P\) is given a horizontal velocity of \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, in the subsequent motion, the maximum angle between the string and the downward vertical is 0.107 radians, correct to 3 significant figures.
2. Show that the motion may be modelled as simple harmonic motion, and find the period of this motion.
3. Find the time after the start of the motion when the velocity of the particle is first \(- 0.2 \mathrm {~ms} ^ { - 1 }\) and find the angular displacement of \(O P\) from the downward vertical at this time.