OCR MEI Further Mechanics A AS (Further Mechanics A AS) 2020 November

1 Brent is riding his bicycle along a straight horizontal road.
While riding along this road Brent can attain a maximum speed of \(6.25 \mathrm {~ms} ^ { - 1 }\) and the wind resistance acting on Brent and his bicycle is constant and equal to 19.2 N . Brent and his bicycle have a combined mass of 72 kg . Brent later begins to ride up a hill which is inclined at an angle of \(3 ^ { \circ }\) to the horizontal.
Given that the wind resistance and the maximum power developed by the bicycle is unchanged, determine Brent's maximum speed up the hill.

2 George is investigating the time it takes for a ball to reach a certain height when projected vertically upwards. George believes that the time, \(t\), for the ball to reach a certain height, \(h\), depends on

• the ball's mass \(m\),
• the projection speed \(u\), and
• the height \(h\).

George suggests the following formula to model this situation
\(t = k m ^ { \alpha } u ^ { \beta } h ^ { \gamma }\),
where \(k\) is a dimensionless constant.

1. Use dimensional analysis to show that \(t = \frac { k h } { u }\).
2. Hence explain why George’s formula is unrealistic. Mandy argues that any model of this situation must consider the acceleration due to gravity, \(g\). She suggests the alternative formula
   \(t = \frac { u - \sqrt { u ^ { 2 } + g h } } { g }\).
3. Show that Mandy's formula is dimensionally consistent.
4. Explain why Mandy’s formula is incorrect.

3 Fig. 3 shows a light square lamina ABCD , of side length 0.75 m , suspended vertically by wires attached to A and B so that AB is horizontal. A particle P of mass \(m \mathrm {~kg}\) is attached to the edge DC . The lamina hangs in equilibrium. \begin{figure}[h] \end{figure} The tension in the wire attached to A is 14 N and the tension in the wire attached to B is \(T \mathrm {~N}\). The wire at A makes an angle of \(25 ^ { \circ }\) with the horizontal and the wire at B makes an angle of \(60 ^ { \circ }\) with the horizontal.

1. Determine the value of \(T\).
2. Determine
   1. the value of \(m\),
   2. the distance of P from D . P is moved to the midpoint of CD . A couple is applied to the lamina so that it remains in equilibrium with AB horizontal and the tension in both wires unchanged.
3. Determine
   • the magnitude of the couple,
   • the direction of the couple.

4 Fig. 4 shows a uniform beam of length \(2 a\) and weight \(W\) leaning against a block of weight \(2 W\) which is on a rough horizontal plane. The beam is freely hinged to the plane at O and makes an angle \(\theta\) with the horizontal. The contact between the beam and the block is smooth. The beam and block are in equilibrium, and it may be assumed that the block does not topple. \begin{figure}[h] \end{figure} Let

• \(S\) be the normal contact force between the beam and the block,
• \(R\) be the normal contact force between the plane and the block,
• \(F\) be the frictional force between the plane and the block.

Partially complete force diagrams showing the beam and the block separately are given in the Printed Answer Booklet.

1. Add the forces listed above to these diagrams. It is given that \(\theta = 30 ^ { \circ }\).
2. Determine the minimum possible value of the coefficient of friction between the block and the plane.
3. In each case explain, with justification, how your answer to part (b) would change (assuming the rest of the system remained unchanged) if
   1. \(\theta < 30 ^ { \circ }\),
   2. the contact between the beam and the block were rough.

5 Throughout this question it may be assumed that there are no resistances to motion.
Model trucks A and B, with masses 5 kg and 3 kg respectively, rest on a set of straight, horizontal rails. Truck A is given an impulse of 3.8 Ns towards B .

1. Calculate the initial speed of A. Truck A collides directly with B. After the collision, B moves with a speed of \(0.6 \mathrm {~ms} ^ { - 1 }\).
2. Determine
   1. the velocity of A after the collision,
   2. the kinetic energy lost due to the collision.
3. B continues to move with a speed of \(0.6 \mathrm {~ms} ^ { - 1 }\) and collides with a model truck C, of mass 4 kg , which is travelling at a speed of \(0.2 \mathrm {~ms} ^ { - 1 }\) towards B on the same set of rails. After the collision between B and C , the speeds of B and C are in the ratio 1 to 2 . Determine the two possible values of the coefficient of restitution between B and C .

6 Fig. 6.1 shows a solid uniform prism OABCDEFG . The \(\mathrm { Ox } , \mathrm { Oy }\) and Oz axes are also shown. The cross-section of the prism is a trapezium. Fig. 6.2 shows the face OABC . The dimensions shown in the figures are in centimetres. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The centre of mass of the prism has coordinates \(( \bar { x } , \bar { y } , \bar { z } )\).

1. Determine the values of \(\bar { x } , \bar { y }\) and \(\bar { z }\).
2. By considering triangle PBA, where P has coordinates ( \(\bar { x } , 0 , \bar { z }\) ), determine whether it is possible for the prism to rest with the face ABEF in contact with a horizontal plane without toppling.

7 Fig. 7.1 shows one end of a light inextensible string attached to a block A of mass 4.4 kg . The other end of the string is attached to a block B of mass 5.2 kg . Block A is in contact with a smooth horizontal plane. The string is taut and passes over a small smooth pulley at the end of the plane. Block B is inside a hollow vertical tube and the vertical sides of B are in contact with the tube. Initially B is 1.6 m above the horizontal base of the tube. \begin{figure}[h] \end{figure} The blocks are released from rest. It may be assumed that in the subsequent motion A does not reach the pulley and the string remains taut. Block B reaches the base of the tube with speed \(3.5 \mathrm {~ms} ^ { - 1 }\).

1. Given that the frictional force exerted by the tube on B is constant, use an energy method to show that the magnitude of this force is 14.21 N . Blocks A and B remain attached to the opposite ends of a light inextensible string, but A is now in contact with a rough plane inclined at \(\theta ^ { \circ }\) to the horizontal, as shown in Fig. 7.2. The string connecting A and B is taut and passes over a small smooth pulley at the top of the plane. Block B is inside the same hollow vertical tube as before with the vertical sides of B in contact with the tube. It may be assumed that the frictional force exerted by the tube on B remains unchanged. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-8_623_723_552_260} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{figure} The coefficient of friction between block A and the plane is \(\frac { 3 } { 11 }\).
   The blocks are released from rest, with block B 1.6 m above the base of the tube. It may be assumed that in the subsequent motion A does not reach the pulley and the string remains taut.
2. Given that block B reaches the base of the tube with speed \(0.7 \mathrm {~ms} ^ { - 1 }\), show that \(\theta\) satisfies the equation
   \(3 \cos \theta + 11 \sin \theta = k\),
   where \(k\) is a constant to be determined. \section*{END OF QUESTION PAPER} \section*{}