OCR Further Mechanics (Further Mechanics) 2021 June

1 A particle \(Q\) of mass \(m \mathrm {~kg}\) is acted on by a single force so that it moves with constant acceleration \(\mathbf { a } = \binom { 1 } { 2 } \mathrm {~ms} ^ { - 2 }\). Initially \(Q\) is at the point \(O\) and is moving with velocity \(\mathbf { u } = \binom { 2 } { - 5 } \mathrm {~ms} ^ { - 1 }\). After \(Q\) has been moving for 5 seconds it reaches the point \(A\).

1. Use the equation \(\mathbf { v } . \mathbf { v } = \mathbf { u } . \mathbf { u } + 2 \mathbf { a } . \mathbf { x }\) to show that at \(A\) the kinetic energy of \(Q\) is 37 m J .
2. 1. Show that the power initially generated by the force is - 8 mW W.
   2. The power in part (b)(i) is negative. Explain what this means about the initial motion of \(Q\).
3. 1. Find the time at which the power generated by the force is instantaneously zero.
   2. Find the minimum kinetic energy of \(Q\) in terms of \(m\).

2 A particle \(P\) of mass 4.5 kg is free to move along the \(x\)-axis. In a model of the motion it is assumed that \(P\) is acted on by two forces:

• a constant force of magnitude \(f \mathrm {~N}\) in the positive \(x\) direction;
• a resistance to motion, \(R \mathrm {~N}\), whose magnitude is proportional to the speed of \(P\).

At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). When \(t = 0 , P\) is at the origin \(O\) and is moving in the positive direction with speed \(u \mathrm {~ms} ^ { - 1 }\), and when \(v = 5 , R = 2\). \begin{enumerate}[label=(\alph*)] \item Show that, according to the model, \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 10 f - 4 v } { 45 }\). \item

1. By solving the differential equation in part (a), show that \(v = \frac { 1 } { 2 } \left( 5 f - ( 5 f - 2 u ) \mathrm { e } ^ { - \frac { 4 } { 45 } t } \right)\).
2. Describe briefly how, according to the model, the speed of \(P\) varies over time in each of the following cases.
   The flat surface of a smooth solid hemisphere of radius \(r\) is fixed to a horizontal plane on a planet where the acceleration due to gravity is denoted by \(\gamma\). \(O\) is the centre of the flat surface of the hemisphere. A particle \(P\) is held at a point on the surface of the hemisphere such that the angle between \(O P\) and the upward vertical through \(O\) is \(\alpha\), where \(\cos \alpha = \frac { 3 } { 4 }\). \(P\) is then released from rest. \(F\) is the point on the plane where \(P\) first hits the plane (see diagram).
   1. Find an exact expression for the distance \(O F\). The acceleration due to gravity on and near the surface of the planet Earth is roughly \(6 \gamma\).
   2. Explain whether \(O F\) would increase, decrease or remain unchanged if the action were repeated on the planet Earth.