Questions — OCR M2 (149 questions)

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OCR M2 2005 June Q2
2 A particle is projected horizontally with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 10 m above horizontal ground. The particle moves freely under gravity. Calculate the speed and direction of motion of the particle at the instant it hits the ground.
OCR M2 2005 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-2_451_533_1676_808} One end of a light inextensible string of length 1.6 m is attached to a point \(P\). The other end is attached to the point \(Q\), vertically below \(P\), where \(P Q = 0.8 \mathrm {~m}\). A small smooth bead \(B\), of mass 0.01 kg , is threaded on the string and moves in a horizontal circle, with centre \(Q\) and radius \(0.6 \mathrm {~m} . Q B\) rotates with constant angular speed \(\omega\) rad s \(^ { - 1 }\) (see diagram).
  1. Show that the tension in the string is 0.1225 N .
  2. Find \(\omega\).
  3. Calculate the kinetic energy of the bead.
OCR M2 2005 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-3_168_956_246_593} Three smooth spheres \(A , B\) and \(C\), of equal radius and of masses \(m \mathrm {~kg} , 2 m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively, lie in a straight line and are free to move on a smooth horizontal table. Sphere \(A\) is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with sphere \(B\) which is stationary. As a result of the collision \(B\) starts to move with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the coefficient of restitution between \(A\) and \(B\).
  2. Find, in terms of \(m\), the magnitude of the impulse that \(A\) exerts on \(B\), and state the direction of this impulse. Sphere \(B\) subsequently collides with sphere \(C\) which is stationary. As a result of this impact \(B\) and \(C\) coalesce.
  3. Show that there will be another collision.
OCR M2 2005 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-3_319_650_1219_749} A uniform \(\operatorname { rod } A B\) of length 60 cm and weight 15 N is freely suspended from its end \(A\). The end \(B\) of the rod is attached to a light inextensible string of length 80 cm whose other end is fixed to a point \(C\) which is at the same horizontal level as \(A\). The rod is in equilibrium with the string at right angles to the rod (see diagram).
  1. Show that the tension in the string is 4.5 N .
  2. Find the magnitude and direction of the force acting on the rod at \(A\).
OCR M2 2005 June Q6
6 A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of \(5 ^ { \circ }\) to the horizontal. At a certain point \(P\) on the hill the car's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(Q\) is 400 m further up the hill from \(P\), and at \(Q\) the car's speed is \(15 \mathrm {~ms} ^ { - 1 }\).
  1. Calculate the work done by the car's engine as the car moves from \(P\) to \(Q\), assuming that any resistances to the car's motion may be neglected. Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.
  2. Given that the acceleration of the car at \(Q\) is zero, show that the power of the engine as the car passes through \(Q\) is 12.0 kW , correct to 3 significant figures.
  3. Given that the power of the car's engine at \(P\) is the same as at \(Q\), calculate the car's retardation at \(P\).
OCR M2 2005 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_76_243_269_365}
\includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_332_1427_322_360} A barrier is modelled as a uniform rectangular plank of wood, \(A B C D\), rigidly joined to a uniform square metal plate, \(D E F G\). The plank of wood has mass 50 kg and dimensions 4.0 m by 0.25 m . The metal plate has mass 80 kg and side 0.5 m . The plank and plate are joined in such a way that \(C D E\) is a straight line (see diagram). The barrier is smoothly pivoted at the point \(D\). In the closed position, the barrier rests on a thin post at \(H\). The distance \(C H\) is 0.25 m .
  1. Calculate the contact force at \(H\) when the barrier is in the closed position. In the open position, the centre of mass of the barrier is vertically above \(D\).
  2. Calculate the angle between \(A B\) and the horizontal when the barrier is in the open position.
OCR M2 2005 June Q8
8 A particle is projected with speed \(49 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta\) from a point \(O\) on a horizontal plane, and moves freely under gravity. The horizontal and upward vertical displacements of the particle from \(O\) at time \(t\) seconds after projection are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(\theta\) and \(t\), and hence show that $$y = x \tan \theta - \frac { x ^ { 2 } \left( 1 + \tan ^ { 2 } \theta \right) } { 490 } .$$
    \includegraphics[max width=\textwidth, alt={}]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_627_1249_1699_447}
    The particle passes through the point where \(x = 70\) and \(y = 30\). The two possible values of \(\theta\) are \(\theta _ { 1 }\) and \(\theta _ { 2 }\), and the corresponding points where the particle returns to the plane are \(A _ { 1 }\) and \(A _ { 2 }\) respectively (see diagram).
  2. Find \(\theta _ { 1 }\) and \(\theta _ { 2 }\).
  3. Calculate the distance between \(A _ { 1 }\) and \(A _ { 2 }\).
OCR M2 2006 June Q1
1 A child of mass 35 kg runs up a flight of stairs in 10 seconds. The vertical distance climbed is 4 m . Assuming that the child's speed is constant, calculate the power output.
OCR M2 2006 June Q2
2 A small sphere of mass 0.3 kg is dropped from rest at a height of 2 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height of 1.4 m above the ground. Ignoring air resistance, calculate the magnitude of the impulse which the ground exerts on the sphere when it rebounds.
OCR M2 2006 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{d6d87705-be4b-407d-b699-69fb441d88a7-2_710_572_721_788} A uniform solid hemisphere of weight 12 N and radius 6 cm is suspended by two vertical strings. One string is attached to the point \(O\), the centre of the plane face, and the other string is attached to the point \(A\) on the rim of the plane face. The hemisphere hangs in equilibrium and \(O A\) makes an angle of \(60 ^ { \circ }\) with the vertical (see diagram).
  1. Find the horizontal distance from the centre of mass of the hemisphere to the vertical through \(O\).
  2. Calculate the tensions in the strings.
OCR M2 2006 June Q4
4 A car of mass 900 kg is travelling at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a level road. The total resistance to motion is 450 N .
  1. Calculate the power output of the car's engine. A roof box of mass 50 kg is mounted on the roof of the car. The total resistance to motion of the vehicle increases to 500 N .
  2. The car's engine continues to work at the same rate. Calculate the maximum speed of the car on the level road. The power output of the car's engine increases to 15000 W . The resistance to motion of the car, with roof box, remains 500 N .
  3. Calculate the instantaneous acceleration of the car on the level road when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  4. The car climbs a hill which is at an angle of \(5 ^ { \circ }\) to the horizontal. Calculate the instantaneous retardation of the car when its speed is \(26 \mathrm {~ms} ^ { - 1 }\).
OCR M2 2006 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{d6d87705-be4b-407d-b699-69fb441d88a7-3_657_549_1219_799} A uniform lamina \(A B C D E\) consists of a square and an isosceles triangle. The square has sides of 18 cm and \(B C = C D = 15 \mathrm {~cm}\) (see diagram).
  1. Taking \(x\) - and \(y\)-axes along \(A E\) and \(A B\) respectively, find the coordinates of the centre of mass of the lamina.
  2. The lamina is freely suspended from \(B\). Calculate the angle that \(B D\) makes with the vertical. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d6d87705-be4b-407d-b699-69fb441d88a7-4_441_1355_265_394} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A light inextensible string of length 1 m passes through a small smooth hole \(A\) in a fixed smooth horizontal plane. One end of the string is attached to a particle \(P\), of mass 0.5 kg , which hangs in equilibrium below the plane. The other end of the string is attached to a particle \(Q\), of mass 0.3 kg , which rotates with constant angular speed in a circle of radius 0.2 m on the surface of the plane (see Fig. 1).
OCR M2 2006 June Q7
7 A small ball is projected at an angle of \(50 ^ { \circ }\) above the horizontal, from a point \(A\), which is 2 m above ground level. The highest point of the path of the ball is 15 m above the ground, which is horizontal. Air resistance may be ignored.
  1. Find the speed with which the ball is projected from \(A\). The ball hits a net at a point \(B\) when it has travelled a horizontal distance of 45 m .
  2. Find the height of \(B\) above the ground.
  3. Find the speed of the ball immediately before it hits the net.
OCR M2 2006 June Q8
8 Two uniform smooth spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 2 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Sphere \(A\) is travelling in a straight line on a smooth horizontal surface, with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), 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 \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the greatest possible value of \(m\). It is given that \(m = 1\).
  2. Find the coefficient of restitution between \(A\) and \(B\). On another occasion \(A\) and \(B\) are travelling towards each other, each with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide directly.
  3. Find the kinetic energy lost due to the collision.
OCR M2 2007 June Q1
1 A man drags a sack at constant speed in a straight line along horizontal ground by means of a rope attached to the sack. The rope makes an angle of \(35 ^ { \circ }\) with the horizontal and the tension in the rope is 40 N . Calculate the work done in moving the sack 100 m .
OCR M2 2007 June Q2
2 Calculate the range on a horizontal plane of a small stone projected from a point on the plane with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(27 ^ { \circ }\).
OCR M2 2007 June Q3
3 A rocket of mass 250 kg is moving in a straight line in space. There is no resistance to motion, and the mass of the rocket is assumed to be constant. With its motor working at a constant rate of 450 kW the rocket's speed increases from \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(150 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a time \(t\) seconds.
  1. Calculate the value of \(t\).
  2. Calculate the acceleration of the rocket at the instant when its speed is \(120 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
OCR M2 2007 June Q4
4 A ball is projected from a point \(O\) on the edge of a vertical cliff. The horizontal and vertically upward components of the initial velocity are \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. At time \(t\) seconds after projection the ball is at the point \(( x , y )\) referred to horizontal and vertically upward axes through \(O\). Air resistance may be neglected.
  1. Express \(x\) and \(y\) in terms of \(t\), and hence show that \(y = 3 x - \frac { 1 } { 10 } x ^ { 2 }\). The ball hits the sea at a point which is 25 m below the level of \(O\).
  2. Find the horizontal distance between the cliff and the point where the ball hits the sea.
OCR M2 2007 June Q5
5 A cyclist and her bicycle have a combined mass of 70 kg . The cyclist ascends a straight hill \(A B\) of constant slope, starting from rest at \(A\) and reaching a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\). The level of \(B\) is 6 m above the level of \(A\). For the cyclist's motion from \(A\) to \(B\), find
  1. the increase in kinetic energy,
  2. the increase in gravitational potential energy. During the ascent the resistance to motion is constant and has magnitude 60 N . The work done by the cyclist in moving from \(A\) to \(B\) is 8000 J .
  3. Calculate the distance \(A B\).
OCR M2 2007 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{9951c978-37e6-4d89-9fe3-c1e5e28b221e-3_670_613_274_767} A particle \(P\) of mass 0.3 kg is attached to one end of each of two light inextensible strings. The other end of the longer string is attached to a fixed point \(A\) and the other end of the shorter string is attached to a fixed point \(B\), which is vertically below \(A\). \(A P\) makes an angle of \(30 ^ { \circ }\) with the vertical and is 0.4 m long. \(P B\) makes an angle of \(60 ^ { \circ }\) with the vertical. The particle moves in a horizontal circle with constant angular speed and with both strings taut (see diagram). The tension in the string \(A P\) is 5 N . Calculate
  1. the tension in the string \(P B\),
  2. the angular speed of \(P\),
  3. the kinetic energy of \(P\).
OCR M2 2007 June Q7
7 Two small spheres \(A\) and \(B\), with masses 0.3 kg and \(m \mathrm {~kg}\) respectively, lie at rest on a smooth horizontal surface. \(A\) is projected directly towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) and hits \(B\). The direction of motion of \(A\) is reversed in the collision. The speeds of \(A\) and \(B\) after the collision are \(1 \mathrm {~ms} ^ { - 1 }\) and \(3 \mathrm {~ms} ^ { - 1 }\) respectively. The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Show that \(m = 0.7\).
  2. Find \(e\). B continues to move at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and strikes a vertical wall at right angles. The coefficient of restitution between \(B\) and the wall is \(f\).
  3. Find the range of values of \(f\) for which there will be a second collision between \(A\) and \(B\).
  4. Find, in terms of \(f\), the magnitude of the impulse that the wall exerts on \(B\).
  5. Given that \(f = \frac { 3 } { 4 }\), calculate the final speeds of \(A\) and \(B\), correct to 1 decimal place.
OCR M2 2007 June Q8
8 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9951c978-37e6-4d89-9fe3-c1e5e28b221e-4_451_481_274_833} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} An object consists of a uniform solid hemisphere of weight 40 N and a uniform solid cylinder of weight 5 N . The cylinder has height \(h \mathrm {~m}\). The solids have the same base radius 0.8 m and are joined so that the hemisphere's plane face coincides with one of the cylinder's faces. The centre of the common face is the point \(O\) (see Fig. 1). The centre of mass of the object lies inside the hemisphere and is at a distance of 0.2 m from \(O\).
  1. Show that \(h = 1.2\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9951c978-37e6-4d89-9fe3-c1e5e28b221e-4_620_1065_1297_541} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} One end of a light inextensible string is attached to a point on the circumference of the upper face of the cylinder. The string is horizontal and its other end is tied to a fixed point on a rough plane. The object rests in equilibrium on the plane with its axis of symmetry vertical. The plane makes an angle of \(30 ^ { \circ }\) with the horizontal (see Fig. 2). The tension in the string is \(T \mathrm {~N}\) and the frictional force acting on the object is \(F \mathrm {~N}\).
  2. By taking moments about \(O\), express \(F\) in terms of \(T\).
  3. Find another equation connecting \(T\) and \(F\). Hence calculate the tension and the frictional force.
OCR M2 2008 June Q1
1 A car is pulled at constant speed along a horizontal straight road by a force of 200 N inclined at \(35 ^ { \circ }\) to the horizontal. Given that the work done by the force is 5000 J , calculate the distance moved by the car.
OCR M2 2008 June Q2
2 A bullet of mass 9 grams passes horizontally through a fixed vertical board of thickness 3 cm . The speed of the bullet is reduced from \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(150 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it passes through the board. The board exerts a constant resistive force on the bullet. Calculate the magnitude of this resistive force.
OCR M2 2008 June Q3
3 The resistance to the motion of a car of mass 600 kg is \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed and \(k\) is a constant. The car ascends a hill of inclination \(\alpha\), where \(\sin \alpha = \frac { 1 } { 10 }\). The power exerted by the car's engine is 12000 W and the car has constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(k = 0.6\). The power exerted by the car's engine is increased to 16000 W .
  2. Calculate the maximum speed of the car while ascending the hill. The car now travels on horizontal ground and the power remains 16000 W .
  3. Calculate the acceleration of the car at an instant when its speed is \(32 \mathrm {~ms} ^ { - 1 }\).