Questions — OCR (4619 questions)

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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 2016 June Q1
6 marks Standard +0.3
1 A car of mass 1400 kg is travelling on a straight horizontal road against a constant resistance to motion of 600 N . At a certain instant the car is accelerating at \(0.3 \mathrm {~ms} ^ { - 2 }\) and the engine of the car is working at a rate of 23 kW .
  1. Find the speed of the car at this instant. Subsequently the car moves up a hill inclined at \(10 ^ { \circ }\) to the horizontal at a steady speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is still a constant 600 N .
  2. Calculate the power of the car's engine as it moves up the hill.
    \(2 A\) and \(B\) are two points on a line of greatest slope of a plane inclined at \(55 ^ { \circ }\) to the horizontal. \(A\) is below the level of \(B\) and \(A B = 4 \mathrm {~m}\). A particle \(P\) of mass 2.5 kg is projected up the plane from \(A\) towards \(B\) and the speed of \(P\) at \(B\) is \(6.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of friction between the plane and \(P\) is 0.15 . Find
  3. the work done against the frictional force as \(P\) moves from \(A\) to \(B\),
  4. the initial speed of \(P\) at \(A\).
OCR M2 2016 June Q3
12 marks Standard +0.3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-2_645_1024_1290_516} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A uniform lamina \(A B D C\) is bounded by two semicircular arcs \(A B\) and \(C D\), each with centre \(O\) and of radii \(3 a\) and \(a\) respectively, and two straight edges, \(A C\) and \(D B\), which lie on the line \(A O B\) (see Fig. 1).
  1. Show that the distance of the centre of mass of the lamina from \(O\) is \(\frac { 13 a } { 3 \pi }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-3_1306_572_207_751} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The lamina has mass 3 kg and is freely pivoted to a fixed point at \(A\). The lamina is held in equilibrium with \(A B\) vertical by means of a light string attached to \(B\). The string lies in the same plane as the lamina and is at an angle of \(40 ^ { \circ }\) below the horizontal (see Fig. 2).
  2. Calculate the tension in the string.
  3. Find the direction of the force acting on the lamina at \(A\).
    \includegraphics[max width=\textwidth, alt={}, center]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-4_848_1491_251_287} A smooth solid cone of semi-vertical angle \(60 ^ { \circ }\) is fixed to the ground with its axis vertical. A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point vertically above the vertex of the cone. \(P\) rotates in a horizontal circle on the surface of the cone with constant angular velocity \(\omega\). The string is inclined to the downward vertical at an angle of \(30 ^ { \circ }\) (see diagram).
  4. Show that the magnitude of the contact force between the cone and the particle is \(\frac { 1 } { 6 } m \left( 2 \sqrt { 3 } g - 3 a \omega ^ { 2 } \right)\).
  5. Given that \(a = 0.5 \mathrm {~m}\) and \(m = 3.5 \mathrm {~kg}\), find, in either order, the greatest speed for which the particle remains in contact with the cone and the corresponding tension in the string.
OCR M2 2016 June Q5
11 marks Standard +0.3
5 A uniform ladder \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 12 } { 13 }\). A man of weight 6 W is standing on the ladder at a distance \(x\) from \(A\) and the system is in equilibrium.
  1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac { 5 W } { 24 } \left( 1 + \frac { 6 x } { a } \right)\). The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 3 }\).
  2. Find, in terms of \(a\), the greatest value of \(x\) for which the system is in equilibrium. The bottom of the ladder \(A\) is moved closer to the wall so that the ladder is now inclined at an angle \(\alpha\) to the horizontal. The man of weight 6 W can now stand at the top of the ladder \(B\) without the ladder slipping.
  3. Find the least possible value of \(\tan \alpha\).
OCR M2 2016 June Q6
10 marks Challenging +1.2
6 The masses of two particles \(A\) and \(B\) are 4 kg and 3 kg respectively. The particles are moving towards each other along a straight line on a smooth horizontal surface. \(A\) has speed \(8 \mathrm {~ms} ^ { - 1 }\) and \(B\) has speed \(10 \mathrm {~ms} ^ { - 1 }\) before they collide. The kinetic energy lost due to the collision is 121.5 J .
  1. Find the speed and direction of motion of each particle after the collision.
  2. Find the coefficient of restitution between \(A\) and \(B\).
    \includegraphics[max width=\textwidth, alt={}, center]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-5_510_1504_653_271} A particle \(P\) is projected with speed \(32 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\), from a point \(A\) on horizontal ground. At the same instant a particle \(Q\) is projected with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\beta\), where \(\sin \beta = \frac { 24 } { 25 }\), from a point \(B\) on the same horizontal ground. The particles move freely under gravity in the same vertical plane and collide with each other at the point \(C\) at the instant when they are travelling horizontally (see diagram).
  3. Calculate the height of \(C\) above the ground and the distance \(A B\). Immediately after the collision \(P\) falls vertically. \(P\) hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height 5 m above the ground.
  4. Given that the mass of \(P\) is 3 kg , find the magnitude and direction of the impulse exerted on \(P\) by the ground. The coefficient of restitution between the two particles is \(\frac { 1 } { 2 }\).
  5. Find the distance of \(Q\) from \(C\) at the instant when \(Q\) is travelling in a direction of \(25 ^ { \circ }\) below the horizontal.
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 M3 Q4
11 marks Standard +0.8
4
\includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-02_283_711_1754_722} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 5 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres, and \(B\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution is 0.75 . Find the speed and direction of motion of each sphere immediately after the collision.
OCR M3 Q5
10 marks Challenging +1.2
5
\includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-03_462_1109_283_569} Two uniform rods \(A B\) and \(B C\) have weights 64 N and 40 N respectively. The rods are freely jointed to each other at \(B\). The rod \(A B\) is freely jointed to a fixed point on horizontal ground at \(A\) and the rod \(B C\) rests against a vertical wall at \(C\). The rod \(B C\) is 1.8 m long and is horizontal. A particle of weight 9 N is attached to the rod \(B C\) at the point 0.4 m from \(C\). The point \(A\) is 1.2 m below the level of \(B C\) and 3.8 m from the wall (see diagram). The system is in equilibrium.
  1. Show that the magnitude of the frictional force at \(C\) is 27 N .
  2. Calculate the horizontal and vertical components of the force exerted on \(A B\) at \(B\).
  3. Given that friction is limiting at \(C\), find the coefficient of friction between the \(\operatorname { rod } B C\) and the wall.
OCR M3 Q6
14 marks Standard +0.8
6
\includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-03_598_839_1480_706} One end of a light inextensible string of length 0.5 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.3 kg is attached to the other end of the string. With the string taut and at an angle of \(60 ^ { \circ }\) to the upward vertical, \(P\) is projected with speed \(2 \mathrm {~ms} ^ { - 1 }\) (see diagram). \(P\) begins to move without air resistance in a vertical circle with centre \(O\). When the string makes an angle \(\theta\) with the upward vertical, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(v ^ { 2 } = 8.9 - 9.8 \cos \theta\).
  2. Find the tension in the string in terms of \(\theta\).
  3. \(P\) does not move in a complete circle. Calculate the angle through which \(O P\) turns before \(P\) leaves the circular path.
OCR M3 2006 January Q1
7 marks Standard +0.3
1
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_246_693_278_731} A particle \(P\) of mass 0.4 kg moving in a straight line has speed \(8.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). An impulse applied to \(P\) deflects it through \(45 ^ { \circ }\) and reduces its speed to \(5.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Calculate the magnitude and direction of the impulse exerted on \(P\).
\(2 \quad O\) is a fixed point on a horizontal straight line. A particle \(P\) of mass 0.5 kg is released from rest at \(O\). At time \(t\) seconds after release the only force acting on \(P\) has magnitude \(\left( 1 + k t ^ { 2 } \right) \mathrm { N }\) and acts horizontally and away from \(O\) along the line, where \(k\) is a positive constant.
  1. Find the speed of \(P\) in terms of \(k\) and \(t\).
  2. Given that \(P\) is 2 m from \(O\) when \(t = 1\), find the value of \(k\) and the time taken by \(P\) to travel 20 m from \(O\).
OCR M3 2006 January Q3
8 marks Standard +0.3
3 A light elastic string has natural length 3 m . One end is attached to a fixed point \(O\) and the other end is attached to a particle of mass 1.6 kg . The particle is released from rest in a position 5 m vertically below \(O\). Air resistance may be neglected.
  1. Given that in the subsequent motion the particle just reaches \(O\), show that the modulus of elasticity of the string is 117.6 N .
  2. Calculate the speed of the particle when it is 4.5 m below \(O\).
OCR M3 2006 January Q4
10 marks Challenging +1.2
4
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_283_711_1754_722} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 5 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres, and \(B\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution is 0.75 . Find the speed and direction of motion of each sphere immediately after the collision.
OCR M3 2006 January Q5
11 marks Challenging +1.2
5
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-3_462_1109_283_569} Two uniform rods \(A B\) and \(B C\) have weights 64 N and 40 N respectively. The rods are freely jointed to each other at \(B\). The rod \(A B\) is freely jointed to a fixed point on horizontal ground at \(A\) and the rod \(B C\) rests against a vertical wall at \(C\). The rod \(B C\) is 1.8 m long and is horizontal. A particle of weight 9 N is attached to the rod \(B C\) at the point 0.4 m from \(C\). The point \(A\) is 1.2 m below the level of \(B C\) and 3.8 m from the wall (see diagram). The system is in equilibrium.
  1. Show that the magnitude of the frictional force at \(C\) is 27 N .
  2. Calculate the horizontal and vertical components of the force exerted on \(A B\) at \(B\).
  3. Given that friction is limiting at \(C\), find the coefficient of friction between the \(\operatorname { rod } B C\) and the wall.