OCR Further Mechanics AS (Further Mechanics AS) 2021 November

1 One end of a light inextensible string of length 2.8 m is attached to a fixed point \(O\) on a smooth horizontal table. The other end of the string is attached to a particle \(P\) which moves on the table, with the string taut, in a circular path around \(O\). The speed of \(P\) is constant and \(P\) completes each circle in 0.84 seconds.

1. Find the magnitude of the angular velocity of \(P\).
2. Find the speed of \(P\).
3. Find the magnitude of the acceleration of \(P\).
4. State the direction of the acceleration of \(P\).

2 A car has a mass of 800 kg . The engine of the car is working at a constant power of 15 kW . In an initial model of the motion of the car it is assumed that the car is subject to a constant resistive force of magnitude \(R N\). The car is initially driven on a straight horizontal road. At the instant that its speed is \(20 \mathrm {~ms} ^ { - 1 }\) its acceleration is \(0.4 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(R = 430\).
2. Hence find the maximum constant speed at which the car can be driven along this road, according to the initial model. In a revised model the resistance to the motion of the car at any instant is assumed to be 60 v where \(v\) is the speed of the car at that instant. The car is now driven up a straight road which is inclined at an angle \(\alpha\) above the horizontal where \(\sin \alpha = 0.2\).
3. Determine the speed of the car at the instant that its acceleration is \(0.15 \mathrm {~ms} ^ { - 2 }\) up the slope, according to the revised model.

3 A particle \(A\) of mass 0.5 kg is moving with a speed of \(3.15 \mathrm {~ms} ^ { - 1 }\) on a smooth horizontal surface when it collides directly with a particle \(B\) of mass 0.8 kg which is at rest on the surface. The velocities of \(A\) and \(B\) immediately after the collision are denoted by \(\mathrm { v } _ { \mathrm { A } } \mathrm { ms } ^ { - 1 }\) and \(\mathrm { v } _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }\) respectively. You are given that \(\mathrm { v } _ { \mathrm { B } } = 2 \mathrm { v } _ { \mathrm { A } }\).

1. Find the values of \(\mathrm { V } _ { \mathrm { A } }\) and \(\mathrm { V } _ { \mathrm { B } }\).
2. Find the coefficient of restitution between \(A\) and \(B\).
3. Explain why the coefficient of restitution is a dimensionless quantity.
4. Calculate the total loss of kinetic energy as a result of the collision.
5. State, giving a reason, whether or not the collision is perfectly elastic.
6. Calculate the impulse that \(B\) exerts on \(A\) in the collision.

4 A small box \(B\) of mass 4.2 kg is initially at rest at a point \(O\) on rough horizontal ground. A horizontal force of magnitude 35 N is applied to \(B\).
\(B\) moves in a straight line until it reaches the point \(S\) which is 2.4 m from \(O\). At the instant that \(B\) reaches \(S\) its speed is \(4.5 \mathrm {~ms} ^ { - 1 }\).

1. 1. Find the energy lost due to the resistive forces acting on \(B\) as it moves from \(O\) to \(S\).
   2. Deduce the magnitude of the average resistive force acting on \(B\) as it moves from \(O\) to \(S\). When \(B\) reaches \(S\), the force is no longer applied. \(B\) continues to move directly up a smooth slope which is inclined at \(20 ^ { \circ }\) above the horizontal (see diagram).
      \includegraphics[max width=\textwidth, alt={}, center]{a65c4b75-b8b4-4a51-8abb-f857dc278271-3_275_1027_1866_244}
2. 1. State an assumption required to model the motion of \(B\) up the slope with only the information given.
   2. Using the assumption made in part (b)(i), determine the distance travelled by \(B\) up the slope until the instant when it comes to rest.

5 The escape speed of an unpowered object is the minimum speed at which it must be projected to escape the gravitational influence of the Earth if it is projected vertically upwards from the Earth's surface. A formula for the escape speed \(U\) of an unpowered object of mass \(m\) is \(U = \sqrt { \frac { 2 G m } { r } }\) where \(r\) is the radius of the Earth and \(G\) is a constant.

1. Show that the dimensions of \(G\) are \(\mathrm { M } ^ { - 1 } \mathrm {~L} ^ { 3 } \mathrm {~T} ^ { - 2 }\). A rocket is a powered object. A rocket is launched with a given launch speed and is then powered by engines which apply a constant force for a period of time after the launch. A student wishes to apply the formula given above to a rocket launch. They wish to model the minimum launch speed required for a rocket to escape the Earth’s gravitational influence. They realise that the given formula is for unpowered objects and so they include an extra term in the formula to obtain \(V = \sqrt { \frac { 2 G m } { r } } - \mathrm { kP } ^ { \alpha } \mathrm { W } ^ { \beta } \mathrm { t } ^ { \gamma }\). In their modified formula, \(G\) and \(r\) are the same as before. The other variables are defined as follows.
   • \(V\) is the required minimum launch speed of the rocket
   • \(k , \alpha , \beta\) and \(\gamma\) are dimensionless constants
   • \(P\) is the power developed by the engines of the rocket
   • \(m\) is the initial mass of the rocket
   • \(W\) is the initial weight of the rocket
   • \(t\) is the total time for which the engines of the rocket operate
   • Use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\).
   • By considering the value of \(\gamma\) found in part (b) explain the relationship between \(t\) and \(V\).

6 A smooth hemispherical shell of radius \(r \mathrm {~m}\) is held with its circular rim horizontal and uppermost. The centre of the rim is at the point \(O\) and the point on the inner surface directly below \(O\) is \(A\). A small object \(P\) of mass \(m \mathrm {~kg}\) is held at rest on the inner surface of the shell so that \(\angle \mathrm { POA } = \frac { 1 } { 3 } \pi\) radians. At the instant that \(P\) is released, an impulse is applied to \(P\) in the direction of the tangent to the surface at \(P\) in the vertical plane containing \(O , A\) and \(P\). The magnitude of the impulse is denoted by \(I\) Ns.
\(P\) immediately starts to move along the surface towards \(A\) (see diagram).
\(X\) is a point on the circular rim. \(P\) leaves the shell at \(X\).
\includegraphics[max width=\textwidth, alt={}, center]{a65c4b75-b8b4-4a51-8abb-f857dc278271-5_512_860_829_242} In an initial model of the motion of \(P\) it is assumed that \(P\) experiences no resistance to its motion.

1. Find in terms of \(r , g , m\) and \(I\) an expression for the speed of \(P\) at the instant that it leaves the shell at \(X\).
2. Find in terms of \(r , g , m\) and \(I\) an expression for the maximum height attained by \(P\) above \(X\) after it has left the shell.
3. Find an expression for the maximum mass of \(P\) for which \(P\) still leaves the shell. In a revised model it is assumed that \(P\) experiences a resistive force of constant magnitude \(R\) while it is moving.
4. Show that, in order for \(P\) to still leave the shell at \(X\) under the revised model, $$I > \sqrt { m ^ { 2 } g r + \frac { 5 \pi m r R } { 3 } } .$$
5. Show that the inequality from part (d) is dimensionally consistent.