OCR M3 (Mechanics 3) 2012 January

1 A particle \(P\) of mass 0.05 kg is moving on a smooth horizontal surface with speed \(2 \mathrm {~ms} ^ { - 1 }\), when it is struck by a horizontal blow in a direction perpendicular to its direction of motion. The magnitude of the impulse of the blow is \(I\) Ns. The speed of \(P\) after the blow is \(2.5 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of \(I\). Immediately before the blow \(P\) is moving parallel to a smooth vertical wall. After the blow \(P\) hits the wall and rebounds from the wall with speed \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the coefficient of restitution between \(P\) and the wall.

2
\includegraphics[max width=\textwidth, alt={}, center]{43ed8ec7-67f1-418a-8d4e-ee96448647fd-2_544_816_781_603} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. They are moving in opposite directions on a horizontal surface and they collide. Immediately before the collision, each sphere has speed \(u \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\alpha\) with the line of centres (see diagram). The coefficient of restitution between \(A\) and \(B\) is 0.5 .

1. Show that the speed of \(B\) is unchanged as a result of the collision.
2. Find the direction of motion of each of the spheres after the collision.

3 A particle \(P\) of mass 0.3 kg is projected horizontally with speed \(u \mathrm {~ms} ^ { - 1 }\) from a fixed point \(O\) on a smooth horizontal surface. At time \(t \mathrm {~s}\) after projection \(P\) is \(x \mathrm {~m}\) from \(O\) and is moving with speed \(v \mathrm {~ms} ^ { - 1 }\). There is a force of magnitude \(1.2 v ^ { 3 } \mathrm {~N}\) resisting the motion of \(P\).

1. Find an expression for \(\frac { \mathrm { d } v } { \mathrm {~d} x }\) in terms of \(v\) and hence show that \(v = \frac { u } { 4 u x + 1 }\).
2. Given that \(x = 2\) when \(t = 9\) find the value of \(u\).

4 One end of a light elastic string, of natural length 0.75 m and modulus of elasticity 44.1 N , is attached to a fixed point \(O\). A particle \(P\) of mass 1.8 kg is attached to the other end of the string. \(P\) is released from rest at \(O\) and falls vertically. Assuming there is no air resistance, find

1. the extension of the string when \(P\) is at its lowest position,
2. the acceleration of \(P\) at its lowest position.

5
\includegraphics[max width=\textwidth, alt={}, center]{43ed8ec7-67f1-418a-8d4e-ee96448647fd-3_441_450_213_808} Two uniform rods \(A B\) and \(B C\), each of length \(2 L \mathrm {~m}\) and of weight 84.5 N , 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 string attached at \(C\) and perpendicular to \(B C\). The rods \(A B\) and \(B C\) make angles \(\alpha\) and \(\beta\) to the horizontal, respectively (see diagram). It is given that \(\cos \beta = \frac { 12 } { 13 }\).

1. Find the tension in the string.
2. Hence show that the force acting on \(B C\) at \(B\) has horizontal component of magnitude 15 N and vertical component of magnitude 48.5 N , and state the direction of the component in each case.
3. Find \(\alpha\).

6 A particle \(P\) starts from rest at a point \(A\) and moves in a straight line with simple harmonic motion. At time \(t \mathrm {~s}\) after the motion starts, \(P\) 's displacement from a point \(O\) on the line is \(x \mathrm {~m}\) towards \(A\). The particle \(P\) returns to \(A\) for the first time when \(t = 0.4 \pi\). The maximum speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\) and occurs when \(P\) passes through \(O\).

1. Find the distance \(O A\).
2. Find the value of \(x\) and the velocity of \(P\) when \(t = 1\).
3. Find the number of occasions in the interval \(0 < t < 1\) at which \(P\) 's speed is the same as that when \(t = 1\), and find the corresponding values of \(x\) and \(t\).

7
\includegraphics[max width=\textwidth, alt={}, center]{43ed8ec7-67f1-418a-8d4e-ee96448647fd-4_351_314_255_861} One end of a light elastic string, of natural length \(\frac { 2 } { 3 } R \mathrm {~m}\) and with modulus of elasticity 1.2 mgN , is attached to the highest point \(A\) of a smooth fixed sphere with centre \(O\) and radius \(R \mathrm {~m}\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string and is in contact with the surface of the sphere, where the angle \(A O P\) is equal to \(\theta\) radians (see diagram).

1. Given that \(P\) is in equilibrium at the point where \(\theta = \alpha\), show that \(1.8 \alpha - \sin \alpha - 1.2 = 0\). Hence show that \(\alpha = 1.18\) correct to 3 significant figures.
   \(P\) is now released from rest at the point of the surface of the sphere where \(\theta = \frac { 2 } { 3 }\), and starts to move downwards on the surface. For an instant when \(\theta = \alpha\),
2. state the direction of the acceleration of \(P\),
3. find the magnitude of the acceleration of \(P\).