OCR Further Mechanics (Further Mechanics) 2024 June

1 A particle \(P\) of mass 12.5 kg is moving on a smooth horizontal plane when it collides obliquely with a fixed vertical wall. At the instant before the collision, the velocity of \(P\) is \(- 5 \mathbf { i } + 12 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).
At the instant after the collision, the velocity of \(P\) is \(\mathbf { i } + 4 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).

1. Find the magnitude of the momentum of \(P\) before the collision.
2. Find, in vector form, the impulse that the wall exerts on \(P\).
3. State, in vector form, the impulse that \(P\) exerts on the wall.
4. Find in either order.
   • The magnitude of the impulse that the wall exerts on \(P\).
   • The angle between \(\mathbf { i }\) and the impulse that the wall exerts on \(P\).

2 One end of a light elastic string of natural length 1.4 m and modulus of elasticity 20 N is attached to a small object \(B\) of mass 2.5 kg . The other end of the string is attached to a fixed point \(O\). Object \(B\) is projected vertically upwards from \(O\) with a speed of \(u \mathrm {~ms} ^ { - 1 }\).

1. State one assumption required to model the motion of \(B\). The greatest height above \(O\) achieved by \(B\) is 8.1 m .
2. Determine the value of \(u\).

3 The mass of a truck is 6000 kg and the maximum power that its engine can generate is 90 kW . In a model of the motion of the truck it is assumed that while it is moving the total resistance to its motion is constant. At first the truck is driven along a straight horizontal road. The greatest constant speed that it can be driven at when it is using maximum power is \(25 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of the resistance to motion. The truck is being driven along the horizontal road with the engine working at 60 kW .
2. Find the acceleration of the truck at the instant when its speed is \(10 \mathrm {~ms} ^ { - 1 }\). The truck is now driven down a straight road which is inclined at an angle \(\theta\) below the horizontal. The greatest constant speed that the truck can be driven at maximum power is \(40 \mathrm {~ms} ^ { - 1 }\).
3. Determine the value of \(\theta\).

4 A particle, \(P\), of mass 6 kg is attached to one end of a light inextensible rod of length 2.4 m . The other end of the rod is smoothly hinged at a fixed point \(O\) and the rod is free to rotate in any direction. Initially, \(P\) is at rest, vertically below \(O\), when it is projected horizontally with a speed of \(12 \mathrm {~ms} ^ { - 1 }\). It subsequently describes complete vertical circles with \(O\) as the centre.
\includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-3_611_517_536_246} The angle that the rod makes with the downward vertical through \(O\) at each instant is denoted by \(\theta\) and \(A\) is the point which \(P\) passes through where \(\theta = 40 ^ { \circ }\) (see diagram).

1. Find the tangential acceleration of \(P\) at \(A\), stating its direction.
2. Determine the radial acceleration of \(P\) at \(A\), stating its direction.
3. Find the magnitude of the force in the rod when \(P\) is at \(A\), stating whether the rod is in tension or compression. The motion is now stopped when \(P\) is at \(A\), and \(P\) is then projected in such a way that it now describes horizontal circles at a constant speed with \(\theta = 40 ^ { \circ }\) (see diagram).
   \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-3_403_524_1877_242}
4. Find the speed of \(P\).
5. Explain why, wherever \(P\) 's motion is initiated from and whatever its initial velocity, it is not possible for \(P\) to describe horizontal circles at constant speed with \(\theta = 90 ^ { \circ }\).

5 In this question you may assume that if \(x\) and \(y\) are any physical quantities then \(\left[ \frac { \mathrm { dy } } { \mathrm { dx } } \right] = \left[ \frac { \mathrm { y } } { \mathrm { x } } \right]\). A machine drives a piston of mass \(m\) into a vertical cylinder. The equation below is suggested to model the power developed by the machine, \(P\), while it is not doing any other external work. $$\mathrm { P } = \mathrm { k } _ { 1 } \mathrm { mv } \frac { \mathrm { dv } } { \mathrm { dt } } + \mathrm { k } _ { 2 } \mathrm { mgv } + \mathrm { k } _ { 3 } \mathrm { E }$$ in which

• \(v\) is the velocity of the piston at a given time,
• \(g\) is the acceleration due to gravity,
• \(E\) is the rate at which heat energy is lost to the surroundings,
• \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are dimensionless constants.

Determine whether the equation is dimensionally consistent. Show all the steps in your argument.

6 Two identical spheres, \(A\) and \(B\), each of mass \(m \mathrm {~kg}\), are moving directly towards each other along the same straight line on a smooth horizontal surface until they collide. Just before they collide, the speeds of \(A\) and \(B\) are \(20 \mathrm {~ms} ^ { - 1 }\) and \(10 \mathrm {~ms} ^ { - 1 }\) respectively. The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. By finding, in terms of \(e\), an expression for the velocity of \(B\) after the collision, show that the direction of motion of \(B\) is reversed by the collision. After the collision between \(A\) and \(B\), which is not perfectly elastic, \(B\) goes on to collide directly with a fixed, vertical wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 5 } e\). After the collision between \(B\) and the wall, there are no further collisions between \(A\) and \(B\).
2. Determine the range of possible values of \(e\).
   \(7 \quad\) A body \(B\) of mass 1.5 kg is moving along the \(x\)-axis. At the instant that it is at the origin, \(O\), its velocity is \(u \mathrm {~ms} ^ { - 1 }\) in the positive \(x\)-direction. At any instant, the resistance to the motion of \(B\) is modelled as being directly proportional to \(v ^ { 2 }\) where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of \(B\) at that instant. The resistance to motion is the only horizontal force acting on \(B\). At an instant when \(B\) 's velocity is \(2 \mathrm {~ms} ^ { - 1 }\), the resistance to its motion is 24 N .

1. Show that \(B\) 's motion can be modelled by the differential equation \(\frac { 1 } { \mathrm { v } } \frac { \mathrm { dv } } { \mathrm { dx } } = - 4\).
2. 1. Solve the differential equation in part (a) to find the particular solution for \(v\) in terms of \(x\) and \(u\).
   2. By considering the behaviour of \(v\) as \(x \longrightarrow \infty\) describe one feature of the model that is not realistic. At the instant when \(B\) reaches the point \(A\), where \(\mathrm { x } = \mathrm { X }\), its speed is \(V \mathrm {~ms} ^ { - 1 }\). The work done by the resistance as \(B\) moves from \(O\) to \(A\) is denoted by \(W \mathrm {~J}\).
3. 1. Use the formula \(\mathrm { W } = \int \mathrm { F } \mathrm { dx }\) to determine an expression for \(W\) in terms of \(X\) and \(u\).
   2. Explain the relevance of the sign of your answer in part (c)(i).
   3. By writing your answer to part (c)(i) in terms of \(V\) and \(u\) show how the quantity \(W\) relates to the energy of \(B\).

8 A shape, \(S\), is formed by attaching a particle of mass \(2 m \mathrm {~kg}\) to the vertex of a uniform solid cone of mass \(8 m \mathrm {~kg}\). The height of the cone is \(h \mathrm {~m}\) and the radius of the base of the cone is 1.1 m .

1. Explain why the centre of mass of \(S\) must lie on the central axis of the cone. Two strings are attached to \(S\), one at the vertex of the cone and one at \(A\) which is a point on the edge of the base of \(S\). The other ends of the strings are attached to a horizontal ceiling in such a way that the strings are both vertical. The string attached to \(S\) at \(A\) is inextensible and has length 1.6 m . The string attached to \(S\) at the vertex is elastic with modulus of elasticity 8 mgN . Shape \(S\) is in equilibrium with its axis horizontal (see diagram).
   \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-6_654_1541_879_244}
2. Determine the natural length of the elastic string.