Questions — OCR M2 (155 questions)

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OCR M2 2014 June Q4
9 marks Standard +0.3
4 A uniform rod \(P Q\) has weight 18 N and length 20 cm . The end \(P\) rests against a rough vertical wall. A particle of weight 3 N is attached to the rod at a point 6 cm from \(P\). The rod is held in a horizontal position, perpendicular to the wall, by a light inextensible string attached to the rod at \(Q\) and to a point \(R\) on the wall vertically above \(P\), as shown in the diagram. The string is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). The system is in limiting equilibrium.
  1. Find the tension in the string.
  2. Find the magnitude of the force exerted by the wall on the rod.
  3. Find the coefficient of friction between the wall and the rod.
OCR M2 2014 June Q5
9 marks Standard +0.3
5
  1. A car of mass 800 kg is moving at a constant speed of \(20 \mathrm {~ms} ^ { - 1 }\) on a straight road down a hill inclined at an angle \(\alpha\) to the horizontal. The engine of the car works at a constant rate of 10 kW and there is a resistance to motion of 1300 N . Show that \(\sin \alpha = \frac { 5 } { 49 }\).
  2. The car now travels up the same hill and its engine now works at a constant rate of 20 kW . The resistance to motion remains 1300 N . The car starts from rest and its speed is \(8 \mathrm {~ms} ^ { - 1 }\) after it has travelled a distance of 22.1 m . Calculate the time taken by the car to travel this distance.
OCR M2 2014 June Q6
13 marks Standard +0.3
6 Two small spheres \(A\) and \(B\), of masses \(2 m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively, are moving in opposite directions along the same straight line towards each other on a smooth horizontal surface. \(A\) has speed \(4 \mathrm {~ms} ^ { - 1 }\) and \(B\) has speed \(2 \mathrm {~ms} ^ { - 1 }\) before they collide. The coefficient of restitution between \(A\) and \(B\) is 0.4 .
  1. Find the speed of each sphere after the collision.
  2. Find, in terms of \(m\), the loss of kinetic energy during the collision.
  3. Given that the magnitude of the impulse exerted on \(A\) by \(B\) during the collision is 2.52 Ns , find \(m\).
OCR M2 2014 June Q7
12 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{5bfd0285-71cb-4dcb-8545-a379653f9a3e-4_529_403_264_829} A small smooth ring \(P\) of mass 0.4 kg is threaded onto a light inextensible string fixed at \(A\) and \(B\) as shown in the diagram, with \(A\) vertically above \(B\). The string is inclined to the vertical at angles of \(30 ^ { \circ }\) and \(45 ^ { \circ }\) at \(A\) and \(B\) respectively. \(P\) moves in a horizontal circle of radius 0.5 m about a point \(C\) vertically below \(B\).
  1. Calculate the tension in the string.
  2. Calculate the speed of \(P\). The end of the string at \(B\) is moved so both ends of the string are now fixed at \(A\).
  3. Show that, when the string is taut, \(A P\) is now 0.854 m correct to 3 significant figures. \(P\) moves in a horizontal circle with angular speed \(3.46 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  4. Find the tension in the string and the angle that the string now makes with the vertical.
OCR M2 2014 June Q8
12 marks Standard +0.3
8 A child is trying to throw a small stone to hit a target painted on a vertical wall. The child and the wall are on horizontal ground. The child is standing a horizontal distance of 8 m from the base of the wall. The child throws the stone from a height of 1 m with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(20 ^ { \circ }\) above the horizontal.
  1. Find the direction of motion of the stone when it hits the wall. The child now throws the stone with a speed of \(\mathrm { Vm } \mathrm { s } ^ { - 1 }\) from the same initial position and still at an angle of \(20 ^ { \circ }\) above the horizontal. This time the stone hits the target which is 2.5 m above the ground.
  2. Find \(V\).
OCR M2 2015 June Q1
7 marks Moderate -0.3
1 A cyclist travels along a straight horizontal road. The total mass of the cyclist and her bicycle is 80 kg and the resistance to motion is a constant 60 N .
  1. The cyclist travels at a constant speed working at a constant rate of 480 W . Find the speed at which she travels.
  2. The cyclist now instantaneously increases her power to 600 W . After travelling at this power for 14.2 s her speed reaches \(9.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance travelled at this power.
OCR M2 2015 June Q2
6 marks Moderate -0.8
2 A particle of mass 0.3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(A\). The particle moves in a horizontal circle of radius 0.343 m , with centre vertically below \(A\), at a constant angular speed of \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Find the tension in the string and the angle at which the string is inclined to the vertical.
OCR M2 2015 June Q3
6 marks Standard +0.3
3 A car of mass 1500 kg travels along a straight horizontal road with its engine working at a constant rate of \(P \mathrm {~W}\). There is a constant resistance to motion of \(R \mathrm {~N}\). Points \(A\) and \(B\) are on the road. At point \(A\) the car's speed is \(16 \mathrm {~ms} ^ { - 1 }\) and its acceleration is \(0.3875 \mathrm {~ms} ^ { - 2 }\). At point \(B\) the car's speed is \(25 \mathrm {~ms} ^ { - 1 }\) and its acceleration is \(0.2 \mathrm {~ms} ^ { - 2 }\). Find the values of \(P\) and \(R\).
OCR M2 2015 June Q4
10 marks Standard +0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{8492ec9b-3327-4d89-aaa4-bf98cdf0ebdc-2_721_513_1260_762} A uniform solid prism has cross-section \(A B C D E\) in the shape of a rectangle measuring 20 cm by 4 cm joined to a semicircle of radius 8 cm as shown in the diagram. The centre of mass of the solid lies in this cross-section.
  1. Find the distance of the centre of mass of the solid from \(A B\). The solid is placed with \(A E\) on rough horizontal ground (so the object does not slide) and is in equilibrium with a horizontal force of magnitude 4 N applied along \(C B\).
  2. Find the greatest and least possible values for the weight of the solid.
OCR M2 2015 June Q5
10 marks Standard +0.3
5 A small sphere of mass 0.2 kg is projected vertically downwards with a speed of \(5 \mathrm {~ms} ^ { - 1 }\) from a height of 1.6 m above horizontal ground. It hits the ground and rebounds vertically upwards coming to instantaneous rest at a height of \(h \mathrm {~m}\) above the ground. The coefficient of restitution between the sphere and the ground is 0.7 .
  1. Find \(h\).
  2. Find the magnitude and direction of the impulse exerted on the sphere by the ground.
  3. Find the loss of energy of the sphere between the instant of projection and the instant it comes to instantaneous rest at height \(h \mathrm {~m}\).
OCR M2 2015 June Q6
10 marks Standard +0.3
6 A particle is projected with speed \(v \mathrm {~ms} ^ { - 1 }\) from a point \(O\) on horizontal ground. The angle of projection is \(\theta ^ { \circ }\) above the horizontal. At time \(t\) seconds after the instant of projection the horizontal displacement of the particle from \(O\) is \(x \mathrm {~m}\) and the upward vertical displacement from \(O\) is \(y \mathrm {~m}\).
  1. Show that $$y = x \tan \theta - \frac { 4.9 x ^ { 2 } } { v ^ { 2 } \cos ^ { 2 } \theta } .$$ A stone is thrown from the top of a vertical cliff 100 m high. The initial speed of the stone is \(16 \mathrm {~ms} ^ { - 1 }\) and the angle of projection is \(\theta ^ { \circ }\) to the horizontal. The stone hits the sea 40 m from the foot of the cliff.
  2. Find the two possible values of \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{8492ec9b-3327-4d89-aaa4-bf98cdf0ebdc-3_623_995_1475_536} A uniform ladder \(A B\) of weight \(W \mathrm {~N}\) and length 4 m rests with its end \(A\) on rough horizontal ground and its end \(B\) against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal where \(\tan \theta = \frac { 1 } { 2 }\) (see diagram). A small object \(S\) of weight \(2 W \mathrm {~N}\) is placed on the ladder at a point \(C\), which is 1 m from \(A\). The coefficient of friction between the ladder and the ground is \(\mu\) and the system is in limiting equilibrium.
OCR M2 2015 June Q8
12 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{8492ec9b-3327-4d89-aaa4-bf98cdf0ebdc-4_342_981_255_525} Two small spheres, \(A\) and \(B\), are free to move on the inside of a smooth hollow cylinder, in such a way that they remain in contact with both the curved surface of the cylinder and its horizontal base. The mass of \(A\) is 0.4 kg , the mass of \(B\) is 0.5 kg and the radius of the cylinder is 0.6 m (see diagram). The coefficient of restitution between \(A\) and \(B\) is 0.35 . Initially, \(A\) and \(B\) are at opposite ends of a diameter of the base of the cylinder with \(A\) travelling at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) stationary. The magnitude of the force exerted on \(A\) by the curved surface of the cylinder is 6 N .
  1. Show that \(v = 3\).
  2. Calculate the speeds of the particles after \(A\) 's first impact with \(B\). Sphere \(B\) is removed from the cylinder and sphere \(A\) is now set in motion with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\). The magnitude of the total force exerted on \(A\) by the cylinder is 4.9 N .
  3. Find \(\omega\). \section*{END OF QUESTION PAPER}
OCR M2 Specimen Q1
5 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-2_236_949_269_603} A barge \(B\) is pulled along a canal by a horse \(H\), which is on the tow-path. The barge and the horse move in parallel straight lines and the tow-rope makes a constant angle of \(15 ^ { \circ }\) with the direction of motion (see diagram). The tow-rope remains taut and horizontal, and has a constant tension of 500 N .
  1. Find the work done on the barge by the tow-rope, as the barge travels a distance of 400 m . The barge moves at a constant speed and takes 10 minutes to travel the 400 m .
  2. Find the power applied to the barge.
OCR M2 Specimen Q2
7 marks Standard +0.3
2 A uniform circular cylinder, of radius 6 cm and height 15 cm , is in equilibrium on a fixed inclined plane with one of its ends in contact with the plane.
  1. Given that the cylinder is on the point of toppling, find the angle the plane makes with the horizontal. The cylinder is now placed on a horizontal board with one of its ends in contact with the board. The board is then tilted so that the angle it makes with the horizontal gradually increases.
  2. Given that the coefficient of friction between the cylinder and the board is \(\frac { 3 } { 4 }\), determine whether or not the cylinder will slide before it topples, justifying your answer.
OCR M2 Specimen Q3
8 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-2_389_698_1706_694} A uniform lamina \(A B C D\) has the shape of a square of side \(a\) adjoining a right-angled isosceles triangle whose equal sides are also of length \(a\). The weight of the lamina is \(W\). The lamina rests, in a vertical plane, on smooth supports at \(A\) and \(D\), with \(A D\) horizontal (see diagram).
  1. Show that the centre of mass of the lamina is at a horizontal distance of \(\frac { 11 } { 9 } a\) from \(A\).
  2. Find, in terms of \(W\), the magnitudes of the forces on the supports at \(A\) and \(D\).
OCR M2 Specimen Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-3_563_707_274_721} A rigid body \(A B C\) consists of two uniform rods \(A B\) and \(B C\), rigidly joined at \(B\). The lengths of \(A B\) and \(B C\) are 13 cm and 20 cm respectively, and their weights are 13 N and 20 N respectively. The distance of \(B\) from \(A C\) is 12 cm . The body hangs in equilibrium, with \(A C\) horizontal, from two vertical strings attached at \(A\) and \(C\). Find the tension in each string.
OCR M2 Specimen Q5
10 marks Standard +0.3
5 A cyclist and his machine have a combined mass of 80 kg . The cyclist ascends a straight hill \(A B\) of constant slope, starting from rest at \(A\) and reaching a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\). The level of \(B\) is 4 m above the level of \(A\).
  1. Find the gain in kinetic energy and the gain in gravitational potential energy of the cyclist and his machine. During the ascent the resistance to motion is constant and has magnitude 70 N .
  2. Given that the work done by the cyclist in ascending the hill is 8000 J , find the distance \(A B\). At \(B\) the cyclist is working at 720 watts and starts to move in a straight line along horizontal ground. The resistance to motion has the same magnitude of 70 N as before.
  3. Find the acceleration with which the cyclist starts to move horizontally.
OCR M2 Specimen Q6
10 marks Moderate -0.3
6 An athlete 'puts the shot' with an initial speed of \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(11 ^ { \circ }\) above the horizontal. At the instant of release the shot is 1.53 m above the horizontal ground. By treating the shot as a particle and ignoring air resistance, find
  1. the maximum height, above the ground, reached by the shot,
  2. the horizontal distance the shot has travelled when it hits the ground.
OCR M2 Specimen Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-4_314_757_285_708} A ball of mass 0.08 kg is attached by two strings to a fixed vertical post. The strings have lengths 2.5 m and 2.4 m , as shown in the diagram. The ball moves in a horizontal circle, of radius 2.4 m , with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Each string is taut and the lower string is horizontal. The modelling assumptions made are that both strings are light and inextensible, and that there is no air resistance.
  1. Find the tension in each string when \(v = 10.5\).
  2. Find the least value of \(v\) for which the lower string is taut.
OCR M2 Specimen Q8
13 marks Standard +0.3
8 Two uniform smooth spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 0.24 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Sphere \(A\) is travelling in a straight line on a horizontal table, with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it collides directly with sphere \(B\), which is at rest. As a result of the collision, sphere \(A\) continues in the same direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the magnitude of the impulse exerted by \(A\) on \(B\).
  2. Show that \(m \leqslant 0.08\). It is given that \(m = 0.06\).
  3. Find the coefficient of restitution between \(A\) and \(B\). On another occasion \(A\) and \(B\) are travelling towards each other, each with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide directly.
  4. Find the speeds of \(A\) and \(B\) immediately after the collision.
OCR M2 2008 January Q6
11 marks Standard +0.3
  1. Show that the tension in the string is 4.16 N , correct to 3 significant figures.
  2. Calculate \(\omega\).
    (ii) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_510_417_1238_904} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The lower part of the string is now attached to a point \(R\), vertically below \(P\). \(P B\) makes an angle \(30 ^ { \circ }\) with the vertical and \(R B\) makes an angle \(60 ^ { \circ }\) with the vertical. The bead \(B\) now moves in a horizontal circle of radius 1.5 m with constant speed \(v _ { \mathrm { m } } \mathrm { m } ^ { - 1 }\) (see Fig. 2).
    1. Calculate the tension in the string.
    2. Calculate \(v\).
OCR M2 2006 June Q6
11 marks Standard +0.3
  1. Calculate the tension in the string and hence find the angular speed of \(Q\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d6d87705-be4b-407d-b699-69fb441d88a7-4_489_1358_1286_392} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The particle \(Q\) on the plane is now fixed to a point 0.2 m from the hole at \(A\) and the particle \(P\) rotates in a horizontal circle of radius 0.2 m (see Fig. 2).
  2. Calculate the tension in the string.
  3. Calculate the speed of \(P\).
OCR M2 2008 June Q5
8 marks Standard +0.3
  1. Show that the distance from the ball to the centre of mass of the toy is 10.7 cm , correct to 1 decimal place.
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-3_312_1051_1509_587} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The toy lies on horizontal ground in a position such that the ball is touching the ground (see Fig. 2). Determine whether the toy is lying in equilibrium or whether it will move to a position where the rod is vertical.
OCR M2 2009 June Q5
11 marks Standard +0.3
  1. Fig. 1 Fig. 1 shows a uniform lamina \(B C D\) in the shape of a quarter circle of radius 6 cm . Show that the distance of the centre of mass of the lamina from \(B\) is 3.60 cm , correct to 3 significant figures. A uniform rectangular lamina \(A B D E\) has dimensions \(A B = 12 \mathrm {~cm}\) and \(A E = 6 \mathrm {~cm}\). A single plane object is formed by attaching the rectangular lamina to the lamina \(B C D\) along \(B D\) (see Fig. 2). The mass of \(A B D E\) is 3 kg and the mass of \(B C D\) is 2 kg . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-3_959_447_1123_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
  2. Taking \(x\) - and \(y\)-axes along \(A E\) and \(A B\) respectively, find the coordinates of the centre of mass of the object. The object is freely suspended at \(C\) and rests in equilibrium.
  3. Calculate the angle that \(A C\) makes with the vertical.
OCR M2 2015 June Q7
11 marks Standard +0.8
  1. Show that \(\mu = \frac { 2 } { 3 }\). A small object of weight \(a W \mathrm {~N}\) is placed on the ladder at its mid-point and the object \(S\) of weight \(2 W \mathrm {~N}\) is placed on the ladder at its lowest point \(A\).
  2. Given that the system is in equilibrium, find the set of possible values of \(a\).