Questions — OCR MEI (4301 questions)

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OCR MEI M1 Q3
18 marks Standard +0.3
3 Fig. 7 illustrates a train with a locomotive, L, pulling two trucks, A and B. The locomotive has mass 90 tonnes and is subject to a resistance force of 2000 N .
Each of the trucks \(A\) and \(B\) has mass 30 tonnes and is subject to a resistance force of \(500 N\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9bff41e0-7be0-4932-ae50-a612abb3fe19-3_153_1256_457_470} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Initially the train is travelling along a straight horizontal track. The locomotive is exerting a driving force of 12000 N .
  1. Find the acceleration of the train.
  2. Find the tension in the coupling between trucks A and B . When the train is travelling at \(10 \mathrm {~ms} ^ { - 1 }\), a fault occurs with truck A and the resistance to its motion changes from 500 N to 5000 N . The driver reduces the driving force to zero and allows the train to slow down under the resistance forces and come to a stop.
  3. Find the distance the train travels while slowing down and coming to a stop. Find also the force in the coupling between trucks A and B while the train is slowing down, and state whether it is a tension or a thrust. The fault in truck A is repaired so that the resistance to its motion is again 500 N . The train continues and comes to a place where the track goes up a uniform slope at an angle of \(\alpha ^ { \circ }\) to the horizontal.
  4. When the train is on the slope, it travels at uniform speed. The driving force remains at 12000 N . Find the value of \(\alpha\).
  5. Show that the force in the coupling between trucks A and B has the same value that it had in part (ii).
OCR MEI M1 Q4
7 marks Standard +0.3
4 Fig. 5 shows blocks of mass 4 kg and 6 kg on a smooth horizontal table. They are connected by a light, inextensible string. As shown, a horizontal force \(F \mathrm {~N}\) acts on the 4 kg block and a horizontal force of 30 N acts on the 6 kg block. The magnitude of the acceleration of the system is \(2 \mathrm {~ms} ^ { - 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9bff41e0-7be0-4932-ae50-a612abb3fe19-4_119_1108_588_513} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Find the two possible values of \(F\).
  2. Find the tension in the string in each case.
OCR MEI M1 Q5
18 marks Moderate -0.3
5 Fig. 8.1 shows a sledge of mass 40 kg . It is being pulled across a horizontal surface of deep snow by a light horizontal rope. There is a constant resistance to its motion. The tension in the rope is 120 N . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9bff41e0-7be0-4932-ae50-a612abb3fe19-5_125_852_391_638} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
\end{figure} The sledge is initially at rest. After 10 seconds its speed is \(5 \mathrm {~ms} ^ { - 1 }\).
  1. Show that the resistance to motion is 100 N . When the speed of the sledge is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope breaks.
    The resistance to motion remains 100 N .
  2. Find the speed of the sledge
    (A) 1.6 seconds after the rope breaks,
    (B) 6 seconds after the rope breaks. The sledge is then pushed to the bottom of a ski slope. This is a plane at an angle of \(15 ^ { \circ }\) to the horizontal. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9bff41e0-7be0-4932-ae50-a612abb3fe19-5_263_854_1391_637} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
    \end{figure} The sledge is attached by a light rope to a winch at the top of the slope. The rope is parallel to the slope and has a constant tension of 200 N . Fig. 8.2 shows the situation when the sledge is part of the way up the slope. The ski slope is smooth.
  3. Show that when the sledge has moved from being at rest at the bottom of the slope to the point when its speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it has travelled a distance of 13.0 m (to 3 significant figures). When the speed of the sledge is \(8 \mathrm {~ms} ^ { - 1 }\), this rope also breaks.
  4. Find the time between the rope breaking and the sledge reaching the bottom of the slope.
OCR MEI M1 Q1
8 marks Moderate -0.8
1 Fig. 5 shows a block of mass 10 kg at rest on a rough horizontal floor. A light string, at an angle of \(30 ^ { \circ }\) to the vertical, is attached to the block. The tension in the string is 50 N . The block is in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-1_409_585_472_768} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Show all the forces acting on the block.
  2. Show that the frictional force acting on the block is 25 N .
  3. Calculate the normal reaction of the floor on the block.
  4. Calculate the magnitude of the total force the floor is exerting on the block.
OCR MEI M1 Q2
18 marks Moderate -0.3
2 In this question, positions are given relative to a fixed origin, O. The \(x\)-direction is east and the \(y\)-direction north; distances are measured in kilometres. Two boats, the Rosemary and the Sage, are having a race between two points A and B.
The position vector of the Rosemary at time \(t\) hours after the start is given by $$\mathbf { r } = \binom { 3 } { 2 } + \binom { 6 } { 8 } t , \text { where } 0 \leqslant t \leqslant 2 .$$ The Rosemary is at point A when \(t = 0\), and at point B when \(t = 2\).
  1. Find the distance AB .
  2. Show that the Rosemary travels at constant velocity. The position vector of the Sage is given by $$\mathbf { r } = \binom { 3 ( 2 t + 1 ) } { 2 \left( 2 t ^ { 2 } + 1 \right) }$$
  3. Plot the points A and B . Draw the paths of the two boats for \(0 \leqslant t \leqslant 2\).
  4. What can you say about the result of the race?
  5. Find the speed of the Sage when \(t = 2\). Find also the direction in which it is travelling, giving your answer as a compass bearing, to the nearest degree.
  6. Find the displacement of the Rosemary from the Sage at time \(t\) and hence calculate the greatest distance between the boats during the race.
OCR MEI M1 Q3
4 marks Moderate -0.8
3 Fig. 2 shows a sack of rice of weight 250 N hanging in equilibrium supported by a light rope AB . End A of the rope is attached to the sack. The rope passes over a small smooth fixed pulley. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-3_451_475_426_870} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Initially, end B of the rope is attached to a vertical wall as shown in Fig. 2.
  1. Calculate the horizontal and the vertical forces acting on the wall due to the rope. End B of the rope is now detached from the wall and attached instead to the top of the sack. The sack is in equilibrium with both sections of the rope vertical.
  2. Calculate the tension in the rope.
OCR MEI M1 Q4
7 marks Moderate -0.3
4 As shown in Fig. 4, boxes P and Q are descending vertically supported by a parachute. Box P has mass 75 kg . Box Q has mass 25 kg and hangs from box P by means of a light vertical wire. Air resistance on the boxes should be neglected. At one stage the boxes are slowing in their descent with the parachute exerting an upward vertical force of 1030 N on box P . The acceleration of the boxes is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) upwards and the tension in the wire is \(T \mathrm {~N}\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-3_332_358_1504_1526} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Draw a labelled diagram showing all the forces acting on box P and another diagram showing all the forces acting on box Q .
  2. Write down separate equations of motion for box P and for box Q .
  3. Calculate the tension in the wire.
OCR MEI M1 Q5
20 marks Standard +0.3
5 A cylindrical tub of mass 250 kg is on a horizontal floor. Resistance to its motion other than that due to friction is negligible. The first attempt to move the tub is by pulling it with a force of 150 N in the \(\mathbf { i }\) direction, as shown in Fig. 8.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-4_310_1349_451_435} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
\end{figure}
  1. Calculate the acceleration of the tub if friction is ignored. In fact, there is friction and the tub does not move.
  2. Write down the magnitude and direction of the frictional force opposing the pull. Two more forces are now added to the 150 N force in a second attempt to move the tub, as shown in Fig. 8.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-4_497_927_1350_646} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
    \end{figure} Angle \(\theta\) is acute and chosen so that the resultant of the three forces is in the \(\mathbf { i }\) direction.
  3. Determine the value of \(\theta\) and the resultant of the three forces. With this resultant force, the tub moves with constant acceleration and travels 1 metre from rest in 2 seconds.
  4. Show that the magnitude of the friction acting on the tub is 661 N , correct to 3 significant figures. When the speed of the tub is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it comes to a part of the floor where the friction on the tub is 200 N greater. The pulling forces stay the same.
  5. Find the velocity of the tub when it has moved a further 1.65 m .
OCR MEI M1 Q6
16 marks Standard +0.3
6 An empty open box of mass 4 kg is on a plane that is inclined at \(25 ^ { \circ }\) to the horizontal.
In one model the plane is taken to be smooth. The box is held in equilibrium by a string with tension \(T \mathrm {~N}\) parallel to the plane, as shown in Fig. 6.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-5_308_561_559_828} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
\end{figure}
  1. Calculate \(T\). A rock of mass \(m \mathrm {~kg}\) is now put in the box. The system is in equilibrium when the tension in the string, still parallel to the plane, is 50 N .
  2. Find \(m\). In a refined model the plane is rough. The empty box, of mass 4 kg , is in equilibrium when a frictional force of 20 N acts down the plane and the string has a tension of \(P \mathrm {~N}\) inclined at \(15 ^ { \circ }\) to the plane, as shown in Fig. 6.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-5_359_559_1599_830} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{figure}
  3. Draw a diagram showing all the forces acting on the box.
  4. Calculate \(P\).
  5. Calculate the normal reaction of the plane on the box.
OCR MEI M1 Q1
8 marks Standard +0.3
1 Fig. 1.1 shows a circular cylinder of mass 100 kg being raised by a light, inextensible vertical wire AB . There is negligible air resistance. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-1_306_256_368_965} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
\end{figure}
  1. Calculate the acceleration of the cylinder when the tension in the wire is 1000 N .
  2. Calculate the tension in the wire when the cylinder has an upward acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The cylinder is now raised inside a fixed smooth vertical tube that prevents horizontal motion but provides negligible resistance to the upward motion of the cylinder. When the wire is inclined at \(30 ^ { \circ }\) to the vertical, as shown in Fig. 1.2, the cylinder again has an upward acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-1_308_490_1230_849} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
    \end{figure}
  3. Calculate the new tension in the wire.
OCR MEI M1 Q2
4 marks Moderate -0.8
2 Boxes A and B slide on a smooth, horizontal plane. Box A has a mass of 4 kg and box B a mass of 5 kg . They are connected by a light, inextensible, horizontal wire. Horizontal forces of 9 N and 135 N act on A and B in the directions shown in Fig. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-1_95_915_2042_650} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Calculate the tension in the wire joining the boxes.
OCR MEI M1 Q3
4 marks Standard +0.3
3 Fig. 3 shows a system in equilibrium. The rod is firmly attached to the floor and also to an object, P. The light string is attached to P and passes over a smooth pulley with an object Q hanging freely from its other end. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-2_519_629_370_745} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Why is the tension the same throughout the string?
  2. Calculate the force in the rod, stating whether it is a tension or a thrust.
OCR MEI M1 Q4
7 marks Moderate -0.3
4 Two trucks, A and B, each of mass 10000 kg , are pulled along a straight, horizontal track by a constant, horizontal force of \(P \mathrm {~N}\). The coupling between the trucks is light and horizontal. This situation and the resistances to motion of the trucks are shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-2_198_948_1454_592} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} The acceleration of the system is \(0.2 \mathrm {~ms} ^ { 2 }\) in the direction of the pulling force of magnitude \(P\).
  1. Calculate the value of \(P\). Truck A is now subjected to an extra resistive force of 2000 N while \(P\) does not change.
  2. Calculate the new acceleration of the trucks.
  3. Calculate the force in the coupling between the trucks.
OCR MEI M1 Q5
8 marks Moderate -0.8
5 A train consists of an engine of mass 10000 kg pulling one truck of mass 4000 kg . The coupling between the engine and the truck is light and parallel to the track. The train is accelerating at \(0.25 \mathrm {~m} \mathrm {~s} ^ { 2 }\) along a straight, level track.
  1. What is the resultant force on the train in the direction of its motion? The driving force of the engine is 4000 N .
  2. What is the resistance to the motion of the train?
  3. If the tension in the coupling is 1150 N , what is the resistance to the motion of the truck? With the same overall resistance to motion, the train now climbs a uniform slope inclined at \(3 ^ { \circ }\) to the horizontal with the same acceleration of \(0.25 \mathrm {~m} \mathrm {~s} ^ { 2 }\).
  4. What extra driving force is being applied?
OCR MEI M1 Q6
14 marks Standard +0.3
6 A box of weight 147 N is held by light strings AB and BC . As shown in Fig. 7.1, AB is inclined at \(\alpha\) to the horizontal and is fixed at \(\mathrm { A } ; \mathrm { BC }\) is held at C . The box is in equilibrium with BC horizontal and \(\alpha\) such that \(\sin \alpha = 0.6\) and \(\cos \alpha = 0.8\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-4_380_542_377_791} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
\end{figure}
  1. Calculate the tension in string AB .
  2. Show that the tension in string BC is 196 N . As shown in Fig. 7.2, a box of weight 90 N is now attached at C and another light string CD is held at D so that the system is in equilibrium with BC still horizontal. CD is inclined at \(\beta\) to the horizontal. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-4_378_695_1282_687} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
    \end{figure}
  3. Explain why the tension in the string BC is still 196 N .
  4. Draw a diagram showing the forces acting on the box at C . Find the angle \(\beta\) and show that the tension in CD is 216 N , correct to three significant figures. The string section CD is now taken over a smooth pulley and attached to a block of mass \(M \mathrm {~kg}\) on a rough slope inclined at \(40 ^ { \circ }\) to the horizontal. As shown in Fig. 7.3, the part of the string attached to the box is still at \(\beta\) to the horizontal and the part attached to the block is parallel to the slope. The system is in equilibrium with a frictional force of 20 N acting on the block up the slope. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-5_436_1049_524_536} \captionsetup{labelformat=empty} \caption{Fig. 7.3}
    \end{figure}
  5. Calculate the value of \(M\).
OCR MEI M1 Q7
7 marks Moderate -0.3
7 A block of mass 4 kg is in equilibrium on a rough plane inclined at \(60 ^ { \circ }\) to the horizontal, as shown in Fig. 4. A frictional force of 10 N acts up the plane and a vertical string AB attached to the block is in tension. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-5_492_347_1545_870} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Draw a diagram showing the four forces acting on the block.
  2. By considering the components of the forces parallel to the slope, calculate the tension in the string.
  3. Calculate the normal reaction of the plane on the block.
OCR MEI M1 Q1
8 marks
1 A golf ball is hit at an angle of \(60 ^ { \circ }\) to the horizontal from a point, O, on level horizontal ground. Its initial speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The standard projectile model, in which air resistance is neglected, is used to describe the subsequent motion of the golf ball. At time \(t \mathrm {~s}\) the horizontal and vertical components of its displacement from O are denoted by \(x \mathrm {~m}\) and \(y \mathrm {~m}\).
  1. Write down equations for \(x\) and \(y\) in terms of \(t\).
  2. Hence show that the equation of the trajectory is $$y = \sqrt { 3 } x - 0.049 x ^ { 2 }$$
  3. Find the range of the golf ball.
  4. A bird is hovering at position \(( 20,16 )\). Find whether the golf ball passes above it, passes below it or hits it.
OCR MEI M1 Q2
7 marks Moderate -0.3
2 A football is kicked with speed \(31 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(20 ^ { \circ }\) to the horizontal. It travels towards the goal which is 50 m away. The height of the crossbar of the goal is 2.44 m .
  1. Does the ball go over the top of the crossbar? Justify your answer.
  2. State one assumption that you made in answering part (i).
OCR MEI M1 Q3
6 marks Moderate -0.8
3 Fig. 1 shows the speed-time graph of a runner during part of his training. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb65e726-a5e0-4060-81a6-6837dea82e64-2_1070_1588_319_273} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} For each of the following statements, say whether it is true or false. If it is false give a brief explanation.
(A) The graph shows that the runner finishes where he started.
(B) The runner's maximum speed is \(8 \mathrm {~ms} ^ { - 1 }\).
(C) At time 58 seconds, the runner is slowing down at a rate of \(1.6 \mathrm {~ms} ^ { - 2 }\).
(D) The runner travels 400 m altogether.
OCR MEI M1 Q4
3 marks Easy -1.2
4 A pellet is fired vertically upwards at a speed of \(11 \mathrm {~ms} ^ { - 1 }\). Assuming that air resistance may be neglected, calculate the speed at which the pellet hits a ceiling 2.4 m above its point of projection.
OCR MEI M1 Q5
8 marks Moderate -0.3
5 Fig. 5 shows a block of mass 10 kg at rest on a rough horizontal floor. A light string, at an angle of \(30 ^ { \circ }\) to the vertical, is attached to the block. The tension in the string is 50 N . The block is in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb65e726-a5e0-4060-81a6-6837dea82e64-3_397_577_567_795} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Show all the forces acting on the block.
  2. Show that the frictional force acting on the block is 25 N .
  3. Calculate the normal reaction of the floor on the block.
  4. Calculate the magnitude of the total force the floor is exerting on the block.
OCR MEI M1 Q6
8 marks Moderate -0.8
6 A small ball is kicked off the edge of a jetty over a calm sea. Air resistance is negligible. Fig. 6 shows
  • the point of projection, O,
  • the initial horizontal and vertical components of velocity,
  • the point A on the jetty vertically below O and at sea level,
  • the height, OA , of the jetty above the sea.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb65e726-a5e0-4060-81a6-6837dea82e64-4_451_1000_596_600} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} The time elapsed after the ball is kicked is \(t\) seconds.
  1. Find an expression in terms of \(t\) for the height of the ball above O at time \(t\). Find also an expression for the horizontal distance of the ball from O at this time.
  2. Determine how far the ball lands from A .
OCR MEI M1 Q7
7 marks Standard +0.3
7 Fig. 4 shows a particle projected over horizontal ground from a point O at ground level. The particle initially has a speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal. The particle is a horizontal distance of 44.8 m from O after 5 seconds. Air resistance should be neglected. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb65e726-a5e0-4060-81a6-6837dea82e64-5_562_757_389_729} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Write down an expression, in terms of \(\alpha\) and \(t\), for the horizontal distance of the particle from O at time \(t\) seconds after it is projected.
  2. Show that \(\cos \alpha = 0.28\).
  3. Calculate the greatest height reached by the particle.
OCR MEI M1 Q2
19 marks Moderate -0.3
2 A ball is kicked from ground level over horizontal ground. It leaves the ground at a speed of 25 ms 1 and at an angle \(\theta\) to the horizontal such that \(\cos \theta = 0.96\) and \(\sin \theta = 0.28\).
  1. Show that the height, \(y \mathrm {~m}\), of the ball above the ground \(t\) seconds after projection is given by \(y = 7 t - 4.9 t ^ { 2 }\). Show also that the horizontal distance, \(x \mathrm {~m}\), travelled by this time is given by \(x = 24 t\).
  2. Calculate the maximum height reached by the ball.
  3. Calculate the times at which the ball is at half its maximum height. Find the horizontal distance travelled by the ball between these times.
  4. Determine the following when \(t = 1.25\).
    (A) The vertical component of the velocity of the ball.
    (B) Whether the ball is rising or falling. (You should give a reason for your answer.)
    (C) The speed of the ball.
  5. Show that the equation of the trajectory of the ball is $$y = \frac { 0.7 x } { 576 } ( 240 - 7 x )$$ Hence, or otherwise, find the range of the ball.
OCR MEI M1 Q3
4 marks Moderate -0.8
3 A particle is thrown vertically upwards and returns to its point of projection after 6 seconds. Air resistance is negligible. Calculate the speed of projection of the particle and also the maximum height it reaches.