OCR M3 (Mechanics 3) 2016 June

1
\includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-2_285_1096_255_488} A particle \(P\) of mass 0.3 kg is moving with speed \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal surface when it is struck by a horizontal impulse. After the impulse acts \(P\) has speed \(0.6 \mathrm {~ms} ^ { - 1 }\) and is moving in a direction making an angle \(30 ^ { \circ }\) with its original direction of motion (see diagram).

1. Find the magnitude of the impulse and the angle its line of action makes with the original direction of motion of \(P\). Subsequently a second impulse acts on \(P\). After this second impulse acts, \(P\) again moves from left to right with speed \(0.4 \mathrm {~ms} ^ { - 1 }\) in a direction parallel to its original direction of motion.
2. State the magnitude of the second impulse, and show the direction of the second impulse on a diagram.

2 A particle \(Q\) of mass 0.2 kg is projected horizontally with velocity \(4 \mathrm {~ms} ^ { - 1 }\) from a fixed point \(A\) on a smooth horizontal surface. At time \(t \mathrm {~s}\) after projection \(Q\) is \(x \mathrm {~m}\) from \(A\) and is moving away from \(A\) with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a force of \(3 \cos 2 t \mathrm {~N}\) acting on \(Q\) in the positive \(x\)-direction.

1. Find an expression for the velocity of \(Q\) at time \(t\). State the maximum and minimum values of the velocity of \(Q\) as \(t\) varies.
2. Find the average velocity of \(Q\) between times \(t = \pi\) and \(t = \frac { 3 } { 2 } \pi\).
   \includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-2_549_1237_1724_415} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2 m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively. The spheres are approaching each other on a horizontal surface when they collide. Before the collision \(A\) is moving with speed \(5 \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\alpha\) with the line of centres, where \(\cos \alpha = \frac { 4 } { 5 }\), and \(B\) is moving with speed \(3 \frac { 1 } { 4 } \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\beta\) with the line of centres, where \(\cos \beta = \frac { 5 } { 13 }\). A straight vertical wall is situated to the right of \(B\), perpendicular to the line of centres (see diagram). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).

1. Find the speed of \(A\) after the collision. Find also the component of the velocity of \(B\) along the line of centres after the collision.
   \(B\) subsequently hits the wall.
2. Explain why \(A\) and \(B\) will have a second collision if the coefficient of restitution between \(B\) and the wall is sufficiently large. Find the set of values of the coefficient of restitution for which this second collision will occur. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{4} \includegraphics[alt={},max width=\textwidth]{c0f31235-80aa-4838-844f-b706de55e7cd-3_193_1451_705_306}
   \end{figure} \(A\) and \(C\) are two fixed points, 1.5 m apart, on a smooth horizontal plane. A light elastic string of natural length 0.4 m and modulus of elasticity 20 N has one end fixed to point \(A\) and the other end fixed to a particle \(B\). Another light elastic string of natural length 0.6 m and modulus of elasticity 15 N has one end fixed to point \(C\) and the other end fixed to the particle \(B\). The particle is released from rest when \(A B C\) forms a straight line and \(A B = 0.4 \mathrm {~m}\) (see diagram). Find the greatest kinetic energy of particle \(B\) in the subsequent motion.
   \includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-3_586_533_1409_758} One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs at rest. \(P\) is then projected horizontally from this position with speed \(2 \sqrt { a g }\). When the string makes an angle \(\theta\) with the upward vertical \(P\) has speed \(v\) (see diagram). The tension in the string is \(T\).
3. Find an expression for \(T\) in terms of \(m , g\) and \(\theta\), and hence find the height of \(P\) above its initial level when the string becomes slack.
   \(P\) is now projected horizontally from the same initial position with speed \(U\).
4. Find the set of values of \(U\) for which the string does not remain taut in the subsequent motion.
   \includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-4_566_1013_255_525} Two uniform rods \(A B\) and \(A C\) are freely jointed at \(A\). Rod \(A B\) is of length \(2 l\) and weight \(W\); \(\operatorname { rod } A C\) is of length \(4 l\) and weight \(2 W\). The rods rest in equilibrium in a vertical plane on two rough horizontal steps, so that \(A B\) makes an angle of \(\theta\) with the horizontal, where \(\sin \theta = \frac { 4 } { 5 }\), and \(A C\) makes an angle of \(\varphi\) with the horizontal, where \(\sin \varphi = \frac { 3 } { 5 }\) (see diagram). The force of the step acting on \(A B\) at \(B\) has vertical component \(R\) and horizontal component \(F\).
5. By taking moments about \(A\) for the rod \(A B\), find an equation relating \(W , R\) and \(F\).
6. Show that \(R = \frac { 73 } { 50 } W\), and find the vertical component of the force acting on \(A C\) at \(C\).
7. The coefficient of friction at \(B\) is equal to that at \(C\). Given that one of the rods is on the point of slipping, explain which rod this must be, and find the coefficient of friction.

7 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of modulus of elasticity 24 mgN and natural length 0.6 m . The other end of the string is attached to a fixed point \(O\); the particle \(P\) rests in equilibrium at a point \(A\) with the string vertical.

1. Show that the distance \(O A\) is 0.625 m . Another particle \(Q\), of mass \(3 m \mathrm {~kg}\), is released from rest from a point 0.4 m above \(P\) and falls onto \(P\). The two particles coalesce.
2. Show that the combined particle initially moves with speed \(2.1 \mathrm {~ms} ^ { - 1 }\).
3. Show that the combined particle initially performs simple harmonic motion, and find the centre of this motion and its amplitude.
4. Find the time that elapses between \(Q\) being released from rest and the combined particle first reaching the highest point of its subsequent motion. \section*{END OF QUESTION PAPER}