Questions — OCR M2 (155 questions)

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OCR M2 2009 June Q4
11 marks Moderate -0.3
4 A light inextensible string of length 0.6 m has one end fixed to a point \(A\) on a smooth horizontal plane. The other end of the string is attached to a particle \(B\), of mass 0.4 kg , which rotates about \(A\) with constant angular speed \(2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) on the surface of the plane.
  1. Calculate the tension in the string. A particle \(P\) of mass 0.1 kg is attached to the mid-point of the string. The line \(A P B\) is straight and rotation continues at \(2 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  2. Calculate the tension in the section of the string \(A P\).
  3. Calculate the total kinetic energy of the system.
OCR M2 2009 June Q6
13 marks Standard +0.3
6 Two uniform spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 0.4 kg and the mass of \(B\) is 0.2 kg . The spheres \(A\) and \(B\) are travelling in the same direction in a straight line on a smooth horizontal surface, \(A\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and \(B\) with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v < 5\). A collides directly with \(B\) and the impulse between them has magnitude 0.9 Ns . Immediately after the collision, the speed of \(B\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate \(v\). \(B\) subsequently collides directly with a stationary sphere \(C\) of mass 0.1 kg and the same radius as \(A\) and \(B\). The coefficient of restitution between \(B\) and \(C\) is 0.6 .
  2. Determine whether there will be a further collision between \(A\) and \(B\).
OCR M2 2009 June Q7
14 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-4_440_657_906_744} A ball is projected with an initial speed of \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(25 ^ { \circ }\) below the horizontal from a point on the top of a vertical wall. The point of projection is 8 m above horizontal ground. The ball hits a vertical fence which is at a horizontal distance of 9 m from the wall (see diagram).
  1. Calculate the height above the ground of the point where the ball hits the fence.
  2. Calculate the direction of motion of the ball immediately before it hits the fence.
  3. It is given that \(30 \%\) of the kinetic energy of the ball is lost when it hits the fence. Calculate the speed of the ball immediately after it hits the fence.
OCR M2 2011 June Q1
7 marks Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-2_314_931_242_607} A sledge with its load has mass 70 kg . It moves down a slope and the resistance to the motion of the sledge is 90 N . The speed of the sledge is controlled by the constant tension in a light rope, which is attached to the sledge and parallel to the slope (see diagram). While travelling 20 m down the slope, the speed of the sledge decreases from \(2.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it descends a vertical distance of 3 m .
  1. Calculate the change in energy of the sledge and its load.
  2. Calculate the tension in the rope.
OCR M2 2011 June Q2
7 marks Moderate -0.3
2 A car of mass 1250 kg travels along a straight road inclined at \(2 ^ { \circ }\) to the horizontal. The resistance to the motion of the car is \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. The car travels at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up the slope and the engine of the car works at a constant rate of 21 kW .
  1. Calculate the value of \(k\).
  2. Calculate the constant speed of the car on a horizontal road.
OCR M2 2011 June Q3
7 marks Standard +0.3
3 A uniform lamina \(A B C D E\) consists of a square \(A C D E\) and an equilateral triangle \(A B C\) which are joined along their common edge \(A C\) to form a pentagon whose sides are each 8 cm in length.
  1. Calculate the distance of the centre of mass of the lamina from \(A C\).
  2. The lamina is freely suspended from \(A\) and hangs in equilibrium. Calculate the angle that \(A C\) makes with the vertical.
OCR M2 2011 June Q4
11 marks Moderate -0.3
4 Two small spheres \(A\) and \(B\) are moving towards each other along a straight line on a smooth horizontal surface. \(A\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before they collide directly. The direction of motion of \(B\) is reversed in the collision. The speeds of \(A\) and \(B\) after the collision are \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(2.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.
  1. (a) Show that the direction of motion of \(A\) is unchanged by the collision.
    (b) Calculate the coefficient of restitution between \(A\) and \(B\). The mass of \(B\) is 0.2 kg .
  2. Find the mass of \(A\). \(B\) continues to move at \(2.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and strikes a vertical wall at right angles. The wall exerts an impulse of magnitude 0.68 N s on \(B\).
  3. Calculate the coefficient of restitution between \(B\) and the wall.
OCR M2 2011 June Q5
12 marks Moderate -0.3
5 A particle is projected with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\) from a point \(O\) and moves freely under gravity. The horizontal and vertically upwards displacements of the particle from \(O\) at any subsequent time \(t \mathrm {~s}\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of the particle.
  2. Calculate the values of \(x\) when \(y = 0.6\).
  3. Find the direction of motion of the particle when \(y = 0.6\) and the particle is rising.
OCR M2 2011 June Q6
14 marks Standard +0.3
6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-3_538_478_758_836} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A container is constructed from a hollow cylindrical shell and a hollow cone which are joined along their circumferences. The cylindrical shell has radius 0.2 m , and the cone has semi-vertical angle \(30 ^ { \circ }\). Two identical small spheres \(P\) and \(Q\) move independently in horizontal circles on the smooth inner surface of the container (see Fig. 1). Each sphere has mass 0.3 kg .
  1. \(P\) moves in a circle of radius 0.12 m and is in contact with only the conical part of the container. Calculate the angular speed of \(P\).
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-3_278_209_1845_1009} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} \(Q\) moves with speed \(2.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is in contact with both the cylindrical and conical surfaces of the container (see Fig. 2). Calculate the magnitude of the force which the cylindrical shell exerts on the sphere.
  3. Calculate the difference between the mechanical energy of \(P\) and of \(Q\). \section*{[Question 7 is printed overleaf.]}
OCR M2 2011 June Q7
14 marks Challenging +1.2
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-4_474_912_260_493} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A uniform solid cone of height 0.8 m and semi-vertical angle \(60 ^ { \circ }\) lies with its curved surface on a horizontal plane. The point \(P\) on the circumference of the base is in contact with the plane. \(V\) is the vertex of the cone and \(P Q\) is a diameter of its base. The weight of the cone is 550 N . A force of magnitude \(F \mathrm {~N}\) and line of action \(P Q\) is applied to the base of the cone (see Fig. 1). The cone topples about \(V\) without sliding.
  1. Calculate the least possible value of \(F\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-4_528_1143_1302_500} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The force of magnitude \(F \mathrm {~N}\) is removed and an increasing force of magnitude \(T \mathrm {~N}\) acting upwards in the vertical plane of symmetry of the cone and perpendicular to \(P Q\) is applied to the cone at \(Q\) (see Fig. 2). The coefficient of friction between the cone and the horizontal plane is \(\mu\).
  2. Given that the cone slides before it topples about \(P\), calculate the greatest possible value for \(\mu\).
OCR M2 2012 June Q1
5 marks Moderate -0.8
1 A particle, of mass 0.8 kg , moves along a smooth horizontal surface. It hits a vertical wall, which is at right angles to the direction of motion of the particle, and rebounds. The speed of the particle as it hits the wall is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of restitution between the particle and the wall is 0.3 . Find
  1. the impulse that the wall exerts on the particle,
  2. the kinetic energy lost in the impact.
OCR M2 2012 June Q2
8 marks Moderate -0.3
2 A car of mass 1600 kg moves along a straight horizontal road. The resistance to the motion of the car has constant magnitude 800 N and the car's engine is working at a constant rate of 20 kW .
  1. Find the acceleration of the car at an instant when the car's speed is \(20 \mathrm {~ms} ^ { - 1 }\). The car now moves up a hill inclined at \(4 ^ { \circ }\) to the horizontal. The car's engine continues to work at 20 kW and the magnitude of the resistance to motion remains at 800 N .
  2. Find the greatest steady speed at which the car can move up the hill.
OCR M2 2012 June Q3
9 marks Standard +0.3
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
10 marks Standard +0.3
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
13 marks Standard +0.3
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
13 marks Standard +0.3
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
14 marks Standard +0.3
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
7 marks Standard +0.3
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
10 marks Standard +0.3
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
8 marks Standard +0.3
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
10 marks Standard +0.3
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
15 marks Standard +0.8
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
4 marks Moderate -0.3
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
5 marks Standard +0.3
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
8 marks Standard +0.3
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.