OCR MEI Further Mechanics A AS (Further Mechanics A AS) 2023 June

1 Throughout all parts of this question, the resistance to the motion of a car has magnitude \(\mathrm { kv } ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. At first, the car travels along a straight horizontal road with constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power developed by the car at this speed is 5000 W .

1. Show that \(k = \frac { 5 } { 8 }\).
2. Find the power the car must develop in order to maintain a constant speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when travelling along the same horizontal road. The car climbs a hill which is inclined at an angle of \(2 ^ { \circ }\) to the horizontal. The power developed by the car is 13000 W , and the car has a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Determine the mass of the car.

2 A ball P of mass \(m \mathrm {~kg}\) is held at a height of 12.8 m above a horizontal floor. P is released from rest and rebounds from the floor. After the first bounce, P reaches a maximum height of 5 m above the floor. Two models, A and B , are suggested for the motion of P .
Model A assumes that air resistance may be neglected.

1. Determine, according to model A , the coefficient of restitution between P and the floor. Model B assumes that the collision between P and the floor is perfectly elastic, but that work is done against air resistance at a constant rate of \(E\) joules per metre.
2. Show that, according to model \(\mathrm { B } , \mathrm { E } = \frac { 39 } { 89 } \mathrm { mg }\).
3. Show that both models predict that P will attain the same maximum height after the second bounce.

3 The time period \(T\) of a satellite in circular orbit around a planet satisfies the equation
\(G M T ^ { 2 } = 4 \pi ^ { 2 } R ^ { 3 }\),
where

• \(G\) is the universal gravitational constant,
• \(M\) is the mass of the planet,
• \(\quad R\) is the radius of the orbital circle.
  1. Find the dimensions of \(G\).

A student suggests the following formula to model the approach speed between two orbiting bodies.
\(v = k G { } ^ { \alpha } { } ^ { \beta } { } _ { r } \gamma _ { m _ { 1 } } m _ { 2 } \left( m _ { 1 } + m _ { 2 } \right)\),
where

• \(\quad v\) is the approach speed of the two bodies,
• \(k\) is a dimensionless constant,
• \(\quad c\) is the speed of light,
• \(\quad r\) is the distance between the two bodies,
• \(\quad m _ { 1 }\) and \(m _ { 2 }\) are the masses of the bodies.
• Use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\).
• Calculate, according to the student's model, how many times greater the approach speed is between a pair of stars which are 6.13 light-years apart and the same pair of stars if they were 8.64 light-years apart. (A light-year is a unit of distance.)

4 The diagram shows three beads, A, B and C, of masses \(0.3 \mathrm {~kg} , 0.5 \mathrm {~kg}\) and 0.7 kg respectively, threaded onto a smooth wire circuit consisting of two straight and two semi-circular sections. The circuit occupies a vertical plane, with the two straight sections horizontal and the upper section 0.45 m directly above the lower section.
\includegraphics[max width=\textwidth, alt={}, center]{a87d62b8-406d-44cd-9ffa-384005329566-5_361_961_450_248} Initially, the beads are at rest. A and B are each given an impulse so that they move towards each other, A with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B with a speed of \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the subsequent collision between A and \(\mathrm { B } , \mathrm { A }\) is brought to rest.

1. Show that the coefficient of restitution between A and B is \(\frac { 1 } { 3 }\). Bead B next collides with C.
2. Show that the speed of B before this collision is \(4.37 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures. In this collision between B and C , B is brought to rest.
3. Determine whether C next collides with A or with B .
4. Explain why, if B has a greater mass than C , B could not be brought to rest in their collision.

5 Fig. 5.1 shows the uniform cross-section of a solid S which is formed from a cylinder by boring two cylindrical tunnels the entire way through the cylinder. The radius of S is 50 cm , and the two tunnels have radii 10 cm and 30 cm . The material making up \(S\) has uniform density.
Coordinates refer to the axes shown in Fig. 5.1 and the units are centimetres. \begin{figure}[h] \end{figure} The centre of mass of \(S\) is ( \(\mathrm { x } , \mathrm { y }\) ).

1. Show that \(\bar { x } = 12\) and find the value of \(\bar { y }\). Solid \(S\) is placed onto two rails, \(A\) and \(B\), whose point of contacts with \(S\) are at ( \(- 30 , - 40\) ) and \(( 30 , - 40 )\) as shown in Fig. 5.2. Two points, \(\mathrm { P } ( 0,50 )\) and \(\mathrm { Q } ( 0 , - 50 )\), are marked on Fig. 5.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 5.2} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-6_654_640_1875_251}
   \end{figure} At first, you should assume that the contact between S and the two rails is smooth.
2. Determine the angle PQ makes with the vertical, after S settles into equilibrium. For the remainder of the question, you should assume that the contact between S and A is rough, that the contact between \(S\) and \(B\) is smooth, and that \(S\) does not move when placed on the rails. Fig. 5.3 shows only the forces exerted on S by the rails. The normal contact forces exerted by A and B on S have magnitude \(R _ { \mathrm { A } } \mathrm { N }\) and \(R _ { \mathrm { B } } \mathrm { N }\) respectively. The frictional force exerted by A on S has magnitude \(F\) N. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 5.3} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-7_652_641_593_248}
   \end{figure} The weight of S is \(W \mathrm {~N}\).
3. By taking moments about the origin, express \(F\) in the form \(\lambda W\), where \(\lambda\) is a constant to be determined.
4. Given that S is in limiting equilibrium, find the coefficient of friction between A and S .

6 A uniform beam of length 6 m and mass 10 kg rests horizontally on two supports A and B , which are 3.8 m apart. A particle \(P\) of mass 4 kg is attached 1.95 m from one end of the beam (see Fig. 6.1). \begin{figure}[h] \end{figure} When A is \(x \mathrm {~m}\) from the end of the beam, the supports exert forces of equal magnitude on the beam.

1. Determine the value of \(x\). P is now removed. The same beam is placed on the supports so that B is 0.7 m from the end of the beam. The supports remain 3.8 m apart (see Fig. 6.2). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-8_296_1082_1162_246}
   \end{figure} The contact between A and the beam is smooth. The contact between B and the beam is rough, with coefficient of friction 0.4. A small force of magnitude \(T \mathrm {~N}\) is applied to one end of the beam. The force acts in the same vertical plane as the beam and the angle the force makes with the beam is \(60 ^ { \circ }\). As \(T\) is increased, forces \(\mathrm { T } _ { \mathrm { L } }\) and \(\mathrm { T } _ { \mathrm { S } }\) are defined in the following way.
   • \(\quad \mathrm { T } _ { \mathrm { L } }\) is the value of \(T\) at which the beam would start lifting, assuming that is not already sliding.
   • \(\quad \mathrm { T } _ { \mathrm { S } }\) is the value of \(T\) at which the beam would start sliding, assuming that it has not already lifted.
   • Show that \(\mathrm { T } _ { \mathrm { L } } = 49.1\), correct to 3 significant figures.
   • Determine whether the beam will first slide or lift.
   \section*{END OF QUESTION PAPER}