Questions — OCR M2 (149 questions)

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OCR M2 2012 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{d1eb99a1-04e5-43bc-87b4-d0f7c962135c-2_599_677_1151_696} A uniform beam \(A B\) of mass 15 kg and length 4 m is freely hinged to a vertical wall at \(A\). The beam is held in equilibrium in a horizontal position by a light rod \(P Q\) of length \(1.5 \mathrm {~m} . P\) is fixed to the wall vertically below \(A\) and \(P Q\) makes an angle of \(30 ^ { \circ }\) with the vertical (see diagram). The force exerted on the beam at \(Q\) by the rod is in the direction \(P Q\). Find
  1. the magnitude of the force exerted on the beam at \(Q\),
  2. the magnitude and direction of the force exerted on the beam at \(A\).
OCR M2 2012 June Q4
4 A boy throws a small ball at a vertical wall. The ball is thrown horizontally, from a point \(O\), at a speed of \(14.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it hits the wall at a point which is 0.2 m below the level of \(O\).
  1. Find the horizontal distance from \(O\) to the wall. The boy now moves so that he is 6 m from the wall. He throws the ball at an angle of \(15 ^ { \circ }\) above the horizontal. The ball again hits the wall at a point which is 0.2 m below the level from which it was thrown.
  2. Find the speed at which the ball was thrown.
OCR M2 2012 June Q5
5 A particle \(P\), of mass 2 kg , is attached to fixed points \(A\) and \(B\) by light inextensible strings, each of length 2 m . \(A\) and \(B\) are 3.2 m apart with \(A\) vertically above \(B\). The particle \(P\) moves in a horizontal circle with centre at the mid-point of \(A B\).
  1. Find the tension in each string when the angular speed of \(P\) is \(4 \mathrm { rads } ^ { - 1 }\).
  2. Find the least possible speed of \(P\).
OCR M2 2012 June Q6
6 Three particles \(A , B\) and \(C\) are in a straight line on a smooth horizontal surface. The particles have masses \(0.2 \mathrm {~kg} , 0.4 \mathrm {~kg}\) and 0.6 kg respectively. \(B\) is at rest. \(A\) is projected towards \(B\) with a speed of \(1.8 \mathrm {~ms} ^ { - 1 }\) and collides with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 3 }\).
  1. Show that the speed of \(B\) after the collision is \(0.8 \mathrm {~ms} ^ { - 1 }\) and find the speed of \(A\) after the collision.
    \(C\) is moving with speed \(0.2 \mathrm {~ms} ^ { - 1 }\) in the same direction as \(B\). Particle \(B\) subsequently collides with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e\).
  2. Find the set of values for \(e\) such that \(B\) does not collide again with \(A\).
OCR M2 2012 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{d1eb99a1-04e5-43bc-87b4-d0f7c962135c-4_353_579_248_744} The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a trapezium \(A B C D\) with \(A B\) and \(C D\) perpendicular to \(A D\). The lengths of \(A B\) and \(A D\) are each 5 cm and the length of \(C D\) is \(( a + 5 ) \mathrm { cm }\).
  1. Show the distance of the centre of mass of the prism from \(A D\) is $$\frac { a ^ { 2 } + 15 a + 75 } { 3 ( a + 10 ) } \mathrm { cm } .$$ The prism is placed with the face containing \(A B\) in contact with a horizontal surface.
  2. Find the greatest value of \(a\) for which the prism does not topple. The prism is now placed on an inclined plane which makes an angle \(\theta ^ { \circ }\) with the horizontal. \(A B\) lies along a line of greatest slope with \(B\) higher than \(A\).
  3. Using the value for \(a\) found in part (ii), and assuming the prism does not slip down the plane, find the greatest value of \(\theta\) for which the prism does not topple.
OCR M2 2013 June Q2
2 The power developed by the engine of a car as it travels at a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal road is 20 kW .
  1. Calculate the resistance to the motion of the car. The car, of mass 1500 kg , now travels down a straight road inclined at \(2 ^ { \circ }\) to the horizontal. The resistance to the motion of the car is unchanged.
  2. Find the power produced by the engine of the car when the car has speed \(32 \mathrm {~ms} ^ { - 1 }\) and is accelerating at \(0.1 \mathrm {~ms} ^ { - 2 }\).
OCR M2 2013 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-2_542_638_1208_717} A uniform semicircular arc \(A C B\) is freely pivoted at \(A\). The arc has mass 0.3 kg and is held in equilibrium by a force of magnitude \(P\) N applied at \(B\). The line of action of this force lies in the same plane as the arc, and is perpendicular to \(A B\). The diameter \(A B\) has length 4 cm and makes an angle of \(\theta ^ { \circ }\) with the downward vertical (see diagram).
  1. Given that \(\theta = 0\), find the magnitude of the force acting on the arc at \(A\).
  2. Given instead that \(\theta = 30\), find the value of \(P\).
OCR M2 2013 June Q4
4 A solid uniform cone has height 8 cm , base radius 5 cm and mass 4 kg . A uniform conical shell has height 10 cm , base radius 5 cm and mass 0.4 kg . The two shapes are joined together so that the circumferences of their circular bases coincide.
  1. Find the distance of the centre of mass of the shape from the common circular base.
    \includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-3_974_1141_484_463} The object is suspended with a string attached to the vertex of the cone and another string attached to the vertex of the conical shell. The object is in equilibrium with the strings vertical and the axis of symmetry of the object horizontal (see diagram).
  2. Find the tension in each string.
OCR M2 2013 June Q5
5 A vertical hollow cylinder of radius 0.4 m is rotating about its axis. A particle \(P\) is in contact with the rough inner surface of the cylinder. The cylinder and \(P\) rotate with the same constant angular speed. The coefficient of friction between \(P\) and the cylinder is \(\mu\).
  1. Given that the angular speed of the cylinder is \(7 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and \(P\) is on the point of moving downwards, find the value of \(\mu\). The particle is now attached to one end of a light inextensible string of length 0.5 m . The other end is fixed to a point \(A\) on the axis of the cylinder (see diagram).
    \includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-4_681_970_660_536}
  2. Find the angular speed for which the contact force between \(P\) and the cylinder becomes zero.
OCR M2 2013 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-4_243_1179_1580_443} The masses of two particles \(A\) and \(B\) are 0.2 kg and \(m \mathrm {~kg}\) respectively. The particles are moving with constant speeds \(4 \mathrm {~ms} ^ { - 1 }\) and \(u \mathrm {~ms} ^ { - 1 }\) in the same horizontal line and in the same direction (see diagram). The two particles collide and the coefficient of restitution between the particles is \(e\). After the collision, \(A\) and \(B\) continue in the same direction with speeds \(4 \left( 1 - e + e ^ { 2 } \right) \mathrm { ms } ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively.
  1. Find \(u\) and \(m\) in terms of \(e\).
  2. Find the value of \(e\) for which the speed of \(A\) after the collision is least and find, in this case, the total loss in kinetic energy due to the collision.
  3. Find the possible values of \(e\) for which the magnitude of the impulse that \(B\) exerts on \(A\) is 0.192 Ns .
    \includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-5_744_887_264_589} The diagram shows a surface consisting of a horizontal part \(O A\) and a plane \(A B\) inclined at an angle of \(70 ^ { \circ }\) to the horizontal. A particle is projected from the point \(O\) with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal \(O A\). The particle hits the plane \(A B\) at the point \(P\), with speed \(14 \mathrm {~ms} ^ { - 1 }\) and at right angles to the plane, 1.4 s after projection.
  4. Show that the value of \(u\) is 15.9 , correct to 3 significant figures, and find the value of \(\theta\).
  5. Find the height of \(P\) above the level of \(A\). The particle rebounds with speed \(v \mathrm {~ms} ^ { - 1 }\). The particle next lands at \(A\).
  6. Find the value of \(v\).
  7. Find the coefficient of restitution between the particle and the plane at \(P\).
OCR M2 2014 June Q1
1 A football is kicked from horizontal ground with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. The greatest height the football reaches above ground level is 2.44 m . By modelling the football as a particle and ignoring air resistance, find
  1. the value of \(\theta\),
  2. the range of the football.
OCR M2 2014 June Q2
2 A uniform solid cylinder of height 12 cm and radius \(r \mathrm {~cm}\) is in equilibrium on a rough inclined plane with one of its circular faces in contact with the plane.
  1. The cylinder is on the point of toppling when the angle of inclination of the plane to the horizontal is \(21 ^ { \circ }\). Find \(r\). The cylinder is now placed on a different inclined plane with one of its circular faces in contact with the plane. This plane is also inclined at \(21 ^ { \circ }\) to the horizontal. The coefficient of friction between this plane and the cylinder is \(\mu\).
  2. The cylinder slides down this plane but does not topple. Find an inequality for \(\mu\).
OCR M2 2014 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{5bfd0285-71cb-4dcb-8545-a379653f9a3e-2_778_579_1304_744} A uniform lamina \(A B C D E\) consists of a rectangle \(A B D E\) and an isosceles triangle \(B C D\) joined along their common edge. \(A B = D E = 8 \mathrm {~cm} , A E = B D = 10 \mathrm {~cm}\) and \(B C = C D = 13 \mathrm {~cm}\) (see diagram).
  1. Find the distance of the centre of mass of the lamina from \(A E\).
  2. The lamina is freely suspended from \(B\) and hangs in equilibrium. Calculate the angle that \(B D\) makes with the vertical.
OCR M2 2014 June Q4
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
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
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
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
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
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
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
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
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
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
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
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}