OCR M3 (Mechanics 3) 2015 June

1 A particle \(P\) of mass 0.2 kg is moving on a smooth horizontal surface with speed \(3 \mathrm {~ms} ^ { - 1 }\), when it is struck by an impulse of magnitude \(I\) Ns. The impulse acts horizontally in a direction perpendicular to the original direction of motion of \(P\), and causes the direction of motion of \(P\) to change by an angle \(\alpha\), where \(\tan \alpha = \frac { 5 } { 12 }\).

1. Show that \(I = 0.25\).
2. Find the speed of \(P\) after the impulse acts.

2
\includegraphics[max width=\textwidth, alt={}, center]{2734e846-f640-4203-ac11-6b2180a21950-2_556_736_671_667} Two uniform rods \(A B\) and \(B C\), each of length \(2 L\), are freely jointed at \(B\), and \(A B\) is freely jointed to a fixed point at \(A\). The rods are held in equilibrium in a vertical plane by a light horizontal string attached at \(C\). The rods \(A B\) and \(B C\) make angles \(\alpha\) and \(\beta\) to the horizontal respectively. The weight of \(\operatorname { rod } B C\) is 75 N , and the tension in the string is 50 N (see diagram).

1. Show that \(\tan \beta = \frac { 3 } { 4 }\).
2. Given that \(\tan \alpha = \frac { 12 } { 5 }\), find the weight of \(A B\).

4 A particle of mass 0.4 kg , moving on a smooth horizontal surface, passes through a point \(O\) with velocity \(10 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) after the particle passes through \(O\), the particle has a displacement \(x \mathrm {~m}\) from \(O\), has a velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) away from \(O\), and is acted on by a force of magnitude \(\frac { 1 } { 8 } v \mathrm {~N}\) acting towards \(O\). Find

1. the time taken for the velocity of the particle to reduce from \(10 \mathrm {~ms} ^ { - 1 }\) to \(5 \mathrm {~ms} ^ { - 1 }\),
2. the average velocity of the particle over this time.

5
\includegraphics[max width=\textwidth, alt={}, center]{2734e846-f640-4203-ac11-6b2180a21950-4_337_944_255_557} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres are moving on a horizontal surface when they collide. Before the collision, \(A\) is moving with speed \(a \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\alpha\) with the line of centres and \(B\) is moving towards \(A\) with speed \(b \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\beta\) with the line of centres (see diagram). After the collision, \(A\) moves with velocity \(2 \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to the line of centres and \(B\) moves with velocity \(2 \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(45 ^ { \circ }\) with the line of centres. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).

1. Show that \(a \cos \alpha = \frac { 5 } { 6 } \sqrt { 2 }\) and find \(b \cos \beta\).
2. Find the values of \(a\) and \(\alpha\).

6 A particle \(P\) starts from rest from a point \(A\) and moves in a straight line with simple harmonic motion about a point \(O\). At time \(t\) seconds after the motion starts the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) towards \(A\). The particle \(P\) is next at rest when \(t = 0.25 \pi\) having travelled a distance of 1.2 m .

1. Find the maximum velocity of \(P\).
2. Find the value of \(x\) and the velocity of \(P\) when \(t = 0.7\).
3. Find the other values of \(t\), for \(0 < t < 1\), at which \(P\) 's speed is the same as when \(t = 0.7\). Find also the corresponding values of \(x\).

7
\includegraphics[max width=\textwidth, alt={}, center]{2734e846-f640-4203-ac11-6b2180a21950-4_282_474_1809_794} One end of a light inextensible string of length 0.5 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) is projected horizontally from the point 0.5 m below \(O\) with speed \(u \mathrm {~ms} ^ { - 1 }\). When the string makes an angle of \(\theta\) with the downward vertical the particle has speed \(v \mathrm {~ms} ^ { - 1 }\) (see diagram).

1. Show that, while the string is taut, the tension, \(T \mathrm {~N}\), in the string is given by $$T = 5.88 \cos \theta + 0.4 u ^ { 2 } - 3.92 .$$
2. Find the least value of \(u\) for which the particle will move in a complete circle.
3. If in fact \(u = 3.5 \mathrm {~ms} ^ { - 1 }\), find the speed of the particle at the point where the string first becomes slack.