OCR MEI Further Mechanics Minor (Further Mechanics Minor) 2021 November

1

1. State the dimensions of force. The force \(F\) required to keep a car moving at constant speed on a circular track is given by the formula $$\mathrm { F } = \frac { \mathrm { mv } ^ { 2 } } { \mathrm { r } }$$ where
   • \(m\) is the constant mass of the car,
   • \(v\) is the speed of the car,
   • \(r\) is the radius of the circular track.
   • Verify that the formula is dimensionally consistent.
   • Determine the percentage increase in force required to keep a car moving on a circular track if the speed of the car were to increase by \(10 \%\) and if the track radius were to decrease by \(10 \%\).
   It is proposed that a new unit of force, the trackforce (Tr), should be adopted in motor-racing. 1 Tr is defined as the amount of force required to accelerate a mass of 1 ton at a rate of 1 mile per hour per second. It is given that 1 ton \(= 1016 \mathrm {~kg}\) and 1 mile \(= 1609 \mathrm {~m}\).
2. Determine the number of newtons that are equivalent to 1 Tr .

2 The diagram shows a uniform beam AB that rests with its end A on rough horizontal ground and its end B against a smooth vertical wall. The beam makes an angle of \(\theta ^ { \circ }\) with the ground.
\includegraphics[max width=\textwidth, alt={}, center]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-3_812_588_347_246} The weight of the beam is \(W N\). The beam is in limiting equilibrium and the coefficient of friction between the beam and the ground is \(\mu\). It is given that the magnitude of the contact force at A is 70 N and the magnitude of the contact force at B is 20 N . Determine, in any order,

• the value of \(W\),
• the value of \(\mu\),
• the value of \(\theta\).

3 The diagram shows an electric winch raising two crates A and B , with masses 40 kg and 25 kg , respectively. The cable connecting the winch to A , and the cable connecting A to B may both be modelled as light and inextensible. Furthermore, it can be assumed that there are no resistances to motion.
\includegraphics[max width=\textwidth, alt={}, center]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-4_499_300_447_246} Throughout the entire motion, the power \(P \mathrm {~W}\) developed by the winch is constant.
Crates A and B are both being raised at a constant speed \(\nu \mathrm { m } \mathrm { s } ^ { - 1 }\) when the cable connecting A and B breaks. After the cable between A and B breaks, crate A continues to be raised by the winch. Crate A now accelerates until it reaches a new constant speed of \(( v + 3 ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Determine

• the value of \(v\),
• the value of \(P\).

4 A child throws a ball of mass \(m \mathrm {~kg}\) vertically upwards with a speed of \(7.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball leaves the child's hand at a height of 1.6 m above horizontal ground.

1. Ignoring any possible air resistance, use an energy method to determine the maximum height reached by the ball above the ground. In fact, the ball only reaches a height of 4.1 m above the ground. For the rest of this question you should assume that the air resistance may be modelled as a constant force acting in the opposite direction to the ball's motion.
2. Show that the ball does 0.568 mJ of work against air resistance per metre travelled.
3. Calculate the speed of the ball just before it hits the ground. The ball bounces off the ground and first comes instantaneously to rest 2.8 m above the ground.
4. Determine the coefficient of restitution between the ball and the ground. In the first impact between the ball and the ground, the magnitude of the impulse exerted on the ball by the ground is 12 Ns .
5. Determine the value of \(m\).

5 Fig. 5.1 shows a solid L-shaped ornament, of uniform density. The ornament is 3 cm thick. The \(x , y\) and \(z\) axes are shown, along with the dimensions of the ornament. The measurements are in centimetres. \begin{figure}[h] \end{figure}

1. Determine, with reference to the axes shown, the coordinates of the ornament's centre of mass. Fig. 5.2 shows the ornament placed so that the shaded face (indicated in Fig. 5.1) is in contact with a plane inclined at \(\theta ^ { \circ }\) to the horizontal, with the 4 cm edge parallel to a line of greatest slope. The surface of the plane is sufficiently rough so that the ornament will not slip down the plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-6_646_844_1452_242} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}
2. Determine the minimum and maximum possible values of \(\theta\) for which the ornament does not topple. The ornament is now placed with its shaded face in contact with a rough horizontal surface. A force of magnitude \(P\) N, acting parallel to the planes of the L -shaped faces, is applied to one of the edges of the ornament, as shown in Fig. 5.3. The force is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the ornament and the surface is \(\mu\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-7_524_680_452_246} \captionsetup{labelformat=empty} \caption{Fig. 5.3}
   \end{figure} The value of \(P\) is gradually increased until the ornament is on the point of toppling but does not slide.
3. Determine the minimum value of \(\mu\).
4. Explain how your answer to part (c) would change if the angle between \(P\) and the horizontal was less than \(30 ^ { \circ }\).

6 A block rests on a horizontal surface. The coefficient of friction between the block and the surface is \(\mu\).

1. Show that if the block is given an initial speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it will move a distance of \(\frac { \mathrm { v } ^ { 2 } } { 2 \mu \mathrm {~g} }\) before coming to rest. Block B rests on the same horizontal surface as a sphere S . On the other side of S is a vertical wall, as shown below. The mass of \(B\) is 8 times the mass of \(S\).
   \includegraphics[max width=\textwidth, alt={}, center]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-8_211_1013_662_244} S is projected directly towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and hits B . It is given that
   • the coefficient of restitution between S and B is 0.8 ,
   • collisions between S and the wall are perfectly elastic,
   • the wall is perpendicular to the direction of motion of S and B .
   Furthermore, you should model the contact between B and the surface as rough and model the contact between S and the surface as smooth.
2. Determine, in terms of \(u\), expressions for
   • the speed of S
   • the speed of B
     immediately after the first collision between S and B . In each case stating the corresponding direction of motion.
   It is given that B has sufficient time to come to rest before each subsequent collision with S .
   Let \(\mathrm { X } _ { \mathrm { n } }\) be the distance B moves after the \(n\)th impact between S and B .
3. Explain why \(\mathrm { x } _ { \mathrm { n } + 1 } = \frac { 9 } { 25 } \mathrm { x } _ { \mathrm { n } }\).
4. Given that \(u = 11.2\) and the coefficient of friction between B and the surface is \(\frac { 1 } { 7 }\), show that B will travel a total distance that cannot exceed 2.8 m . \section*{END OF QUESTION PAPER} \section*{OCR
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