OCR M3 (Mechanics 3) 2010 June

1 A small ball of mass 0.8 kg is moving with speed \(10.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude 4 Ns . The speed of the ball immediately afterwards is \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The angle between the directions of motion before and after the impulse acts is \(\alpha\). Using an impulse-momentum triangle, or otherwise, find \(\alpha\).

2
\includegraphics[max width=\textwidth, alt={}, center]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-2_691_767_529_689} Two uniform rods \(A B\) and \(B C\) are of equal length and each has weight 100 N . The rods are freely jointed to each other at \(B\), and \(A\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(A B\) horizontal and \(C\) resting on a rough horizontal surface. \(C\) is vertically below the mid-point of \(A B\) (see diagram).

1. By taking moments about \(A\) for \(A B\), find the vertical component of the force on \(A B\) at \(B\). Hence find the vertical component of the contact force on \(B C\) at \(C\).
2. Calculate the magnitude of the frictional force on \(B C\) at \(C\) and state its direction. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-3_452_345_264_900} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A uniform smooth sphere \(A\) moves on a smooth horizontal surface towards a smooth vertical wall. Immediately before the sphere hits the wall it has components of velocity parallel and perpendicular to the wall each of magnitude \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after hitting the wall the components have magnitudes \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), respectively (see Fig. 1).

1. Given that the coefficient of restitution between the sphere and the wall is \(\frac { 1 } { 2 }\), state the values of \(u\) and \(v\). Shortly after hitting the wall the sphere \(A\) comes into contact with another uniform smooth sphere \(B\), which has the same mass and radius as \(A\). The sphere \(B\) is stationary and at the instant of contact the line of centres of the spheres is parallel to the wall (see Fig. 2). The contact between the spheres is perfectly elastic. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-3_524_371_1503_888} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
2. Find, for each sphere, its speed and its direction of motion immediately after the contact.
   \(4 ~ O\) is a fixed point on a horizontal plane. A particle \(P\) of mass 0.25 kg is released from rest at \(O\) and moves in a straight line on the plane. At time \(t \mathrm {~s}\) after release the only horizontal force acting on \(P\) has magnitude $$\frac { 1 } { 2400 } \left( 144 - t ^ { 2 } \right) \mathrm { N } \quad \text { for } 0 \leqslant t \leqslant 12$$ and $$\frac { 1 } { 2400 } \left( t ^ { 2 } - 144 \right) \mathrm { N } \text { for } t \geqslant 12 .$$ The force acts in the direction of \(P\) 's motion. \(P\) 's velocity at time \(t \mathrm {~s}\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find an expression for \(v\) in terms of \(t\), valid for \(t \geqslant 12\), and hence show that \(v\) is three times greater when \(t = 24\) than it is when \(t = 12\).
4. Sketch the \(( t , v )\) graph for \(0 \leqslant t \leqslant 24\).

5
\includegraphics[max width=\textwidth, alt={}, center]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-4_234_1003_1007_571} Particles \(P _ { 1 }\) and \(P _ { 2 }\) are each moving with simple harmonic motion along the same straight line. \(P _ { 1 }\) 's motion has centre \(C _ { 1 }\), period \(2 \pi \mathrm {~s}\) and amplitude \(3 \mathrm {~m} ; P _ { 2 }\) 's motion has centre \(C _ { 2 }\), period \(\frac { 4 } { 3 } \pi \mathrm {~s}\) and amplitude 4 m . The points \(C _ { 1 }\) and \(C _ { 2 }\) are 6.5 m apart. The displacements of \(P _ { 1 }\) and \(P _ { 2 }\) from their centres of oscillation at time \(t \mathrm {~s}\) are denoted by \(x _ { 1 } \mathrm {~m}\) and \(x _ { 2 } \mathrm {~m}\) respectively. The diagram shows the positions of the particles at time \(t = 0\), when \(x _ { 1 } = 3\) and \(x _ { 2 } = 4\).

1. State expressions for \(x _ { 1 }\) and \(x _ { 2 }\) in terms of \(t\), which are valid until the particles collide. The particles collide when \(t = 5.99\), correct to 3 significant figures.
2. Find the distance travelled by \(P _ { 2 }\) before the collision takes place.
3. Find the velocities of \(P _ { 1 }\) and \(P _ { 2 }\) immediately before the collision, and state whether the particles are travelling in the same direction or in opposite directions.

6 A bungee jumper of weight \(W \mathrm {~N}\) is joined to a fixed point \(O\) by a light elastic rope of natural length 20 m and modulus of elasticity 32000 N . The jumper starts from rest at \(O\) and falls vertically. The jumper is modelled as a particle and air resistance is ignored.

1. Given that the jumper just reaches a point 25 m below \(O\), find the value of \(W\).
2. Find the maximum speed reached by the jumper.
3. Find the maximum value of the deceleration of the jumper during the downward motion.

7
\includegraphics[max width=\textwidth, alt={}, center]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-5_447_693_255_726} A particle \(P\) is attached to a fixed point \(O\) by a light inextensible string of length 0.7 m . A particle \(Q\) is in equilibrium suspended from \(O\) by an identical string. With the string \(O P\) taut and horizontal, \(P\) is projected vertically downwards with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) so that it strikes \(Q\) directly (see diagram). \(P\) is brought to rest by the collision and \(Q\) starts to move with speed \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the speed of \(P\) immediately before the collision. Hence find the coefficient of restitution between \(P\) and \(Q\).
2. Given that the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(O Q\) makes an angle \(\theta\) with the downward vertical, find an expression for \(v ^ { 2 }\) in terms of \(\theta\), and show that the tension in the string \(O Q\) is \(14.7 m ( 1 + 2 \cos \theta ) \mathrm { N }\), where \(m \mathrm {~kg}\) is the mass of \(Q\).
3. Find the radial and transverse components of the acceleration of \(Q\) at the instant that the string \(O Q\) becomes slack.
4. Show that \(V ^ { 2 } = 0.8575\), where \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(Q\) when it reaches its greatest height (after the string \(O Q\) becomes slack). Hence find the greatest height reached by \(Q\) above its initial position.