Questions — OCR MEI M1 (268 questions)

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OCR MEI M1 Q3
3 Abi and Bob are standing on the ground and are trying to raise a small object of weight 250 N to the top of a building. They are using long light ropes. Fig. 7.1 shows the initial situation. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82f933a6-c17e-4b41-ae2b-3cc9d0ba975c-2_770_1068_368_530} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
\end{figure} Abi pulls vertically downwards on the rope A with a force \(F\) N. This rope passes over a small smooth pulley and is then connected to the object. Bob pulls on another rope, B, in order to keep the object away from the side of the building. In this situation, the object is stationary and in equilibrium. The tension in rope B, which is horizontal, is 25 N . The pulley is 30 m above the object. The part of rope A between the pulley and the object makes an angle \(\theta\) with the vertical.
  1. Represent the forces acting on the object as a fully labelled triangle of forces.
  2. Find \(F\) and \(\theta\). Show that the distance between the object and the vertical section of rope A is 3 m . Abi then pulls harder and the object moves upwards. Bob adjusts the tension in rope B so that the object moves along a vertical line. Fig. 7.2 shows the situation when the object is part of the way up. The tension in rope A is \(S \mathrm {~N}\) and the tension in rope B is \(T \mathrm {~N}\). The ropes make angles \(\alpha\) and \(\beta\) with the vertical as shown in the diagram. Abi and Bob are taking a rest and holding the object stationary and in equilibrium. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{82f933a6-c17e-4b41-ae2b-3cc9d0ba975c-3_384_357_520_851} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
    \end{figure}
  3. Give the equations, involving \(S , T , \alpha\) and \(\beta\), for equilibrium in the vertical and horizontal directions.
  4. Find the values of \(S\) and \(T\) when \(\alpha = 8.5 ^ { \circ }\) and \(\beta = 35 ^ { \circ }\).
  5. Abi's mass is 40 kg . Explain why it is not possible for her to raise the object to a position in which \(\alpha = 60 ^ { \circ }\).
OCR MEI M1 Q4
4 Fig. 4 illustrates points \(A , B\) and \(C\) on a straight race track. The distance \(A B\) is 300 m and \(A C\) is 500 m .
A car is travelling along the track with uniform acceleration. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82f933a6-c17e-4b41-ae2b-3cc9d0ba975c-4_70_1329_397_352} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Initially the car is at A and travelling in the direction AB with speed \(5 \mathrm {~ms} ^ { - 1 }\). After 20 s it is at C .
  1. Find the acceleration of the car.
  2. Find the speed of the car at B and how long it takes to travel from A to B .
OCR MEI M1 Q5
5
An egg falls from rest a distance of 75 cm to the floor.
Neglecting air resistance, at what speed does it hit the floor?
OCR MEI M1 Q6
6 Fig. 1 shows four forces in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82f933a6-c17e-4b41-ae2b-3cc9d0ba975c-4_364_328_1748_901} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Find the value of \(P\).
  2. Hence find the value of \(Q\).
OCR MEI M1 Q7
7 A block of weight 100 N is on a rough plane that is inclined at \(35 ^ { \circ }\) to the horizontal. The block is in equilibrium with a horizontal force of 40 N acting on it, as shown in Fig. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82f933a6-c17e-4b41-ae2b-3cc9d0ba975c-5_490_880_316_623} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Calculate the frictional force acting on the block.
OCR MEI M1 Q1
1 Fig. 2 shows two forces acting at A . The figure also shows the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) which are respectively horizontal and vertically upwards. The resultant of the two forces is \(\mathbf { F } \mathbf { N }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94f23528-931c-47b6-89aa-4b6edd25cc30-1_264_918_584_663} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Find \(\mathbf { F }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\), giving your answer correct to three significant figures.
  2. Calculate the magnitude of \(\mathbf { F }\) and the angle that \(\mathbf { F }\) makes with the upward vertical.
OCR MEI M1 Q2
2 Force \(\mathbf { F }\) is \(\left( \begin{array} { l } 4
1
2 \end{array} \right) \mathrm { N }\) and force \(\mathbf { G }\) is \(\left( \begin{array} { r } - 6
2
4 \end{array} \right) \mathrm { N }\).
  1. Find the resultant of \(\mathbf { F }\) and \(\mathbf { G }\) and calculate its magnitude.
  2. Forces \(\mathbf { F }\), \(2 \mathbf { G }\) and \(\mathbf { H }\) act on a particle which is in equilibrium. Find \(\mathbf { H }\).
OCR MEI M1 Q3
3 A box of mass 5 kg is at rest on a rough horizontal floor.
  1. Find the value of the normal reaction of the floor on the box. The box remains at rest on the floor when a force of 10 N is applied to it at an angle of \(40 ^ { \circ }\) to the upward vertical, as shown in Fig. 3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{94f23528-931c-47b6-89aa-4b6edd25cc30-2_286_470_1067_803} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure}
  2. Draw a diagram showing all the forces acting on the box.
  3. Calculate the new value of the normal reaction of the floor on the box and also the frictional force.
OCR MEI M1 Q4
4 Fig. 4 shows forces of magnitudes 20 N and 16 N inclined at \(60 ^ { \circ }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94f23528-931c-47b6-89aa-4b6edd25cc30-3_193_351_261_895} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Calculate the component of the resultant of these two forces in the direction of the 20 N force.
  2. Calculate the magnitude of the resultant of these two forces. These are the only forces acting on a particle of mass 2 kg .
  3. Find the magnitude of the acceleration of the particle and the angle the acceleration makes with the 20 N force.
OCR MEI M1 Q5
5 A particle is in equilibrium when acted on by the forces \(\left( \begin{array} { r } x
- 7
z \end{array} \right) , \left( \begin{array} { r } 4
y
- 5 \end{array} \right)\) and \(\left( \begin{array} { r } 5
4
- 7 \end{array} \right)\), where the units are newtons.
  1. Find the values of \(x , y\) and \(z\).
  2. Calculate the magnitude of \(\left( \begin{array} { r } 5
    4
    - 7 \end{array} \right)\).
OCR MEI M1 Q6
6 A small box B of weight 400 N is held in equilibrium by two light strings AB and BC . The string \(B C\) is fixed at \(C\). The end \(A\) of string \(A B\) is fixed so that \(A B\) is at an angle \(\alpha\) to the vertical where \(\alpha < 60 ^ { \circ }\). String BC is at \(60 ^ { \circ }\) to the vertical. This information is shown in Fig. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94f23528-931c-47b6-89aa-4b6edd25cc30-4_404_437_434_810} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Draw a labelled diagram showing all the forces acting on the box.
  2. In one situation string AB is fixed so that \(\alpha = 30 ^ { \circ }\). By drawing a triangle of forces, or otherwise, calculate the tension in the string BC and the tension in the string AB .
  3. Show carefully, but briefly, that the box cannot be in equilibrium if \(\alpha = 60 ^ { \circ }\) and BC remains at \(60 ^ { \circ }\) to the vertical.
OCR MEI M1 2009 January Q1
1 A particle is travelling in a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds is given by $$v = 6 + 4 t \quad \text { for } 0 \leqslant t \leqslant 5$$
  1. Write down the initial velocity of the particle and find the acceleration for \(0 \leqslant t \leqslant 5\).
  2. Write down the velocity of the particle when \(t = 5\). Find the distance travelled in the first 5 seconds. For \(5 \leqslant t \leqslant 15\), the acceleration of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. Find the total distance travelled by the particle during the 15 seconds.
OCR MEI M1 2009 January Q2
2 Fig. 2 shows an acceleration-time graph modelling the motion of a particle. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{93a5d409-ade4-418b-9c09-620d97df97de-2_684_1070_1064_536} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} At \(t = 0\) the particle has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive direction.
  1. Find the velocity of the particle when \(t = 2\).
  2. At what time is the particle travelling in the negative direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ?
OCR MEI M1 2009 January Q3
3 The resultant of the force \(\binom { - 4 } { 8 } \mathrm {~N}\) and the force \(\mathbf { F }\) gives an object of mass 6 kg an acceleration of \(\binom { 2 } { 3 } \mathrm {~ms} ^ { - 2 }\).
  1. Calculate \(\mathbf { F }\).
  2. Calculate the angle between \(\mathbf { F }\) and the vector \(\binom { 0 } { 1 }\).
OCR MEI M1 2009 January Q4
4 Sandy is throwing a stone at a plum tree. The stone is thrown from a point O at a speed of \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\alpha\) to the horizontal, where \(\cos \alpha = 0.96\). You are given that, \(t\) seconds after being thrown, the stone is \(\left( 9.8 t - 4.9 t ^ { 2 } \right) \mathrm { m }\) higher than O . When descending, the stone hits a plum which is 3.675 m higher than O . Air resistance should be neglected. Calculate the horizontal distance of the plum from O .
OCR MEI M1 2009 January Q5
5 A man of mass 75 kg is standing in a lift. He is holding a parcel of mass 5 kg by means of a light inextensible string, as shown in Fig. 5. The tension in the string is 55 N . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{93a5d409-ade4-418b-9c09-620d97df97de-3_456_476_833_833} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Find the upward acceleration.
  2. Find the reaction on the man of the lift floor.
OCR MEI M1 2009 January Q6
6 Small stones A and B are initially in the positions shown in Fig. 6 with B a height \(H \mathrm {~m}\) directly above A. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{93a5d409-ade4-418b-9c09-620d97df97de-3_318_271_1800_938} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} At the instant when B is released from rest, A is projected vertically upwards with a speed of \(29.4 \mathrm {~m} \mathrm {~s} \mathrm {~s} ^ { - 1 }\). Air resistance may be neglected. The stones collide \(T\) seconds after they begin to move. At this instant they have the same speed, \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and A is still rising. By considering when the speed of A upwards is the same as the speed of B downwards, or otherwise, show that \(T = 1.5\) and find the values of \(V\) and \(H\). Section B (36 marks)
OCR MEI M1 2009 January Q7
7 An explorer is trying to pull a loaded sledge of total mass 100 kg along horizontal ground using a light rope. The only resistance to motion of the sledge is from friction between it and the ground. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{93a5d409-ade4-418b-9c09-620d97df97de-4_327_1013_482_566} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Initially she pulls with a force of 121 N on the rope inclined at \(34 ^ { \circ }\) to the horizontal, as shown in Fig. 7, but the sledge does not move.
  1. Draw a diagram showing all the forces acting on the sledge. Show that the frictional force between the ground and the sledge is 100 N , correct to 3 significant figures. Calculate the normal reaction of the ground on the sledge. The sledge is given a small push to set it moving at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The explorer continues to pull on the rope with the same force and the same angle as before. The frictional force is also unchanged.
  2. Describe the subsequent motion of the sledge. The explorer now pulls the rope, still at an angle of \(34 ^ { \circ }\) to the horizontal, so that the tension in it is 155 N . The frictional force is now 95 N .
  3. Calculate the acceleration of the sledge. In a new situation, there is no rope and the sledge slides down a uniformly rough slope inclined at \(26 ^ { \circ }\) to the horizontal. The sledge starts from rest and reaches a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 2 seconds.
  4. Calculate the frictional force between the slope and the sledge.
OCR MEI M1 2009 January Q8
8 A toy boat moves in a horizontal plane with position vector \(\mathbf { r } = x \mathbf { i } + y \mathbf { j }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors east and north respectively. The origin of the position vectors is at O . The displacements \(x\) and \(y\) are in metres. First consider only the motion of the boat parallel to the \(x\)-axis. For this motion $$x = 8 t - 2 t ^ { 2 }$$ The velocity of the boat in the \(x\)-direction is \(v _ { x } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find an expression in terms of \(t\) for \(v _ { x }\) and determine when the boat instantaneously has zero speed in the \(x\)-direction. Now consider only the motion of the boat parallel to the \(y\)-axis. For this motion $$v _ { y } = ( t - 2 ) ( 3 t - 2 )$$ where \(v _ { y } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the boat in the \(y\)-direction at time \(t\) seconds.
  2. Given that \(y = 3\) when \(t = 1\), use integration to show that \(y = t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2\). The position vector of the boat is given in terms of \(t\) by \(\mathbf { r } = \left( 8 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2 \right) \mathbf { j }\).
  3. Find the time(s) when the boat is due north of O and also the distance of the boat from O at any such times.
  4. Find the time(s) when the boat is instantaneously at rest. Find the distance of the boat from O at any such times.
  5. Plot a graph of the path of the boat for \(0 \leqslant t \leqslant 2\).
OCR MEI M1 2010 June Q1
1 An egg falls from rest a distance of 75 cm to the floor.
Neglecting air resistance, at what speed does it hit the floor?
OCR MEI M1 2010 June Q2
2 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]{6cca1e5e-82b0-487d-8048-b9db7745dea6-2_458_479_705_833} \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 2010 June Q3
3 The three forces \(\left( \begin{array} { r } - 1
14
- 8 \end{array} \right) \mathrm { N } , \left( \begin{array} { r } 3
- 9
10 \end{array} \right) \mathrm { N }\) and \(\mathbf { F } \mathrm { N }\) act on a body of mass 4 kg in deep space and give it an acceleration of \(\left( \begin{array} { r } - 1
2
4 \end{array} \right) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  1. Calculate \(\mathbf { F }\). At one instant the velocity of the body is \(\left( \begin{array} { r } - 3
    3
    6 \end{array} \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  2. Calculate the velocity and also the speed of the body 3 seconds later.
OCR MEI M1 2010 June Q4
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]{6cca1e5e-82b0-487d-8048-b9db7745dea6-3_341_364_210_1489} \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 2010 June Q5
5 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are pointing east and north respectively.
  1. Calculate the bearing of the vector \(- 4 \mathbf { i } - 6 \mathbf { j }\). The vector \(- 4 \mathbf { i } - 6 \mathbf { j } + k ( 3 \mathbf { i } - 2 \mathbf { j } )\) is in the direction \(7 \mathbf { i } - 9 \mathbf { j }\).
  2. Find \(k\).
OCR MEI M1 2010 June Q6
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]{6cca1e5e-82b0-487d-8048-b9db7745dea6-3_458_1008_1786_571} \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 .