OCR MEI M1 (Mechanics 1)

1 A train consists of a locomotive pulling 17 identical trucks. The mass of the locomotive is 120 tonnes and the mass of each truck is 40 tonnes. The locomotive gives a driving force of 121000 N . The resistance to motion on each truck is \(R \mathrm {~N}\) and the resistance on the locomotive is \(5 R \mathrm {~N}\).
Initially the train is travelling on a straight horizontal track and its acceleration is \(0.11 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(R = 1500\).
2. Find the tensions in the couplings between
   (A) the last two trucks,
   (B) the locomotive and the first truck. The train now comes to a place where the track goes up a straight, uniform slope at an angle \(\alpha\) with the horizontal, where \(\sin \alpha = \frac { 1 } { 80 }\). The driving force and the resistance forces remain the same as before.
3. Find the magnitude and direction of the acceleration of the train. The train then comes to a straight uniform downward slope at an angle \(\beta\) to the horizontal.
   The driver of the train reduces the driving force to zero and the resistance forces remain the same as before. The train then travels at a constant speed down the slope.
4. Find the value of \(\beta\).

2 A box of mass 8 kg slides on a horizontal table against a constant resistance of 11.2 N .

1. What horizontal force is applied to the box if it is sliding with acceleration of magnitude \(2 \mathrm {~ms} ^ { - 2 }\) ? Fig. 7 shows the box of mass 8 kg on a long, rough, horizontal table. A sphere of mass 6 kg is attached to the box by means of a light inextensible string that passes over a smooth pulley. The section of the string between the pulley and the box is parallel to the table. The constant frictional force of 11.2 N opposes the motion of the box. A force of 105 N parallel to the table acts on the box in the direction shown, and the acceleration of the system is in that direction. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0fbef619-ad15-4e46-be35-e17fed9952c0-2_372_878_870_683} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure}
2. What information in the question indicates that while the string is taut the box and sphere have the same acceleration?
3. Draw two separate diagrams, one showing all the horizontal forces acting on the box and the other showing all the forces acting on the sphere.
4. Show that the magnitude of the acceleration of the system is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string. The system is stationary when the sphere is at point P . When the sphere is 1.8 m above P the string breaks, leaving the sphere moving upwards at a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
5. (A) Write down the value of the acceleration of the sphere after the string breaks.
   (B) The sphere passes through P again at time \(T\) seconds after the string breaks. Show that \(T\) is the positive root of the equation \(4.9 T ^ { 2 } - 3 T - 1.8 = 0\).
   ( \(C\) ) Using part ( \(B\) ), or otherwise, calculate the total time that elapses after the sphere moves from P before the sphere again passes through P .