Vehicle on slope with resistance

3 A car of mass \(m \mathrm {~kg}\) is towing a trailer of mass 300 kg down a straight hill inclined at \(3 ^ { \circ }\) to the horizontal at a constant speed. There are resistance forces on the car and on the trailer, and the total work done against the resistance forces in a distance of 50 m is 40000 J . The engine of the car is doing no work and the tow-bar is light and rigid.

1. Find the value of \(m\).
   The resistance force on the trailer is 200 N .
2. Find the tension in the tow-bar between the car and the trailer.

3 A cyclist exerts a constant driving force of magnitude \(F \mathrm {~N}\) while moving up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 36 } { 325 }\). A constant resistance to motion of 32 N acts on the cyclist. The total weight of the cyclist and his bicycle is 780 N . The cyclist's acceleration is \(- 0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(F\). The cyclist's speed is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill.
2. Find how far up the hill the cyclist travels before coming to rest.

7. A sledge of mass 78 kg is pulled up a slope by means of a rope. The slope is modelled as a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\). The rope is modelled as light and inextensible and is in a line of greatest slope of the plane. The coefficient of friction between the sledge and the slope is 0.25 . Given that the sledge is accelerating up the slope with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),

1. find the tension in the rope. The rope suddenly breaks. Subsequently the sledge comes to instantaneous rest and then starts sliding down the slope.
2. Find the acceleration of the sledge down the slope after it has come to instantaneous rest.
   (6 marks)
   END

7 \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-4_129_798_756_676} A winch drags a \(\log\) of mass 600 kg up a slope inclined at \(10 ^ { \circ }\) to the horizontal by means of an inextensible cable of negligible mass parallel to the slope (see diagram). The coefficient of friction between the \(\log\) and the slope is 0.15 , and the \(\log\) is initially at rest at the foot of the slope. The acceleration of the \(\log\) is \(0.11 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Calculate the tension in the cable. The cable suddenly breaks after dragging the log a distance of 10 m .
2. (a) Show that the deceleration of the log while continuing to move up the slope is \(3.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to 3 significant figures.
   (b) Calculate the time taken, after the cable breaks, for the log to return to its original position at the foot of the slope. www.ocr.org.uk after the live examination series.
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6 A trolley, of mass 100 kg , rolls at a constant speed along a straight line down a slope inclined at an angle of \(4 ^ { \circ }\) to the horizontal. Assume that a constant resistance force, of magnitude \(P\) newtons, acts on the trolley as it moves. Model the trolley as a particle.

1. Draw a diagram to show the forces acting on the trolley.
2. Show that \(P = 68.4 \mathrm {~N}\), correct to three significant figures.
3. 1. Find the acceleration of the trolley if it rolls down a slope inclined at \(5 ^ { \circ }\) to the horizontal and experiences the same constant force of magnitude \(P\) that you found in part (b).
   2. Make one criticism of the assumption that the resistance force on the trolley is constant.

8 A van, of mass 2000 kg , is towed up a slope inclined at \(5 ^ { \circ }\) to the horizontal. The tow rope is at an angle of \(12 ^ { \circ }\) to the slope. The motion of the van is opposed by a resistance force of magnitude 500 newtons. The van is accelerating up the slope at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-22_269_991_513_529} Model the van as a particle.

1. Draw a diagram to show the forces acting on the van.
2. Show that the tension in the tow rope is 3480 newtons, correct to three significant figures.

2 The battery on Carol and Martin's car is flat so the car will not start. They hope to be able to "bump start" the car by letting it run down a hill and engaging the engine when the car is going fast enough. Fig. 6.1 shows the road leading away from their house, which is at A . The road is straight, and at all times the car is steered directly along it.

• From A to B the road is horizontal.
• Between B and C , it goes up a hill with a uniform slope of \(1.5 ^ { \circ }\) to the horizontal.
• Between C and D the road goes down a hill with a uniform slope of \(3 ^ { \circ }\) to the horizontal. CD is 100 m . (This is the part of the road where they hope to get the car started.)
• From D to E the road is again horizontal.

\begin{figure}[h] \end{figure} The mass of the car is 750 kg , Carol's mass is 50 kg and Martin's mass is 80 kg .
Throughout the rest of this question, whenever Martin pushes the car, he exerts a force of 300 N along the line of the car.

1. Between A and B, Martin pushes the car and Carol sits inside to steer it. The car has an acceleration of \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Show that the resistance to the car's motion is 100 N . Throughout the rest of this question you should assume that the resistance to motion is constant at 100 N .
2. They stop at B and then Martin tries to push the car up the hill BC. Show that Martin cannot push the car up the hill with Carol inside it but can if she gets out.
   Find the acceleration of the car when Martin is pushing it and Carol is standing outside.
3. While between B and C, Carol opens the window of the car and pushes it from outside while steering with one hand. Carol is able to exert a force of 150 N parallel to the surface of the road but at an angle of \(30 ^ { \circ }\) to the line of the car. This is illustrated in Fig. 6.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9bff41e0-7be0-4932-ae50-a612abb3fe19-2_216_425_1964_870} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure} Find the acceleration of the car.
4. At C, both Martin and Carol get in the car and, starting from rest, let it run down the hill under gravity. If the car reaches a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) they can get the engine to start.