Questions — OCR M2 (155 questions)

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OCR M2 2010 January Q5
12 marks Standard +0.3
5 Two spheres of the same radius with masses 2 kg and 3 kg are moving directly towards each other on a smooth horizontal plane with speeds \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The spheres collide and the kinetic energy lost is 81 J . Calculate the speed and direction of motion of each sphere after the collision.
OCR M2 2010 January Q6
17 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{8e1225a2-cb98-4b71-a4af-0150f093f852-3_698_1047_1297_550} A particle \(P\) is projected with speed \(V _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta _ { 1 }\) from a point \(O\) on horizontal ground. When \(P\) is vertically above a point \(A\) on the ground its height is 250 m and its velocity components are \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally and \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically upwards (see diagram).
  1. Show that \(V _ { 1 } = 86.0\) and \(\theta _ { 1 } = 62.3 ^ { \circ }\), correct to 3 significant figures. At the instant when \(P\) is vertically above \(A\), a second particle \(Q\) is projected from \(O\) with speed \(V _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta _ { 2 } . P\) and \(Q\) hit the ground at the same time and at the same place.
  2. Calculate the total time of flight of \(P\) and the total time of flight of \(Q\).
  3. Calculate the range of the particles and hence calculate \(V _ { 2 }\) and \(\theta _ { 2 }\).
OCR M2 2010 January Q7
15 marks Standard +0.8
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-4_444_771_258_687} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A particle \(P\) of mass 0.2 kg is moving on the smooth inner surface of a fixed hollow hemisphere which has centre \(O\) and radius \(5 \mathrm {~m} . P\) moves with constant angular speed \(\omega\) in a horizontal circle at a vertical distance of 3 m below the level of \(O\) (see Fig.1).
  1. Calculate the magnitude of the force exerted by the hemisphere on \(P\).
  2. Calculate \(\omega\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-4_592_773_1231_687} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} A light inextensible string is now attached to \(P\). The string passes through a small smooth hole at the lowest point of the hemisphere and a particle of mass 0.1 kg hangs in equilibrium at the end of the string. \(P\) moves in the same horizontal circle as before (see Fig. 2).
  3. Calculate the new angular speed of \(P\).
OCR M2 2011 January Q1
7 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-02_378_471_260_836} A uniform square frame \(A B C D\) has sides of length 0.6 m . The side \(A D\) is removed from the frame, and the open frame \(A B C D\) is attached at \(A\) to a fixed point (see diagram).
  1. Calculate the distance of the centre of mass of the open frame from \(A\). The open frame rotates about \(A\) in the plane \(A B C D\) with angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  2. Calculate the speed of the centre of mass of the open frame.
OCR M2 2011 January Q2
8 marks Standard +0.3
2 The resistance to the motion of a car is \(k v ^ { \frac { 3 } { 2 } } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed and \(k\) is a constant. The power exerted by the car's engine is 15000 W , and the car has constant speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a horizontal road.
  1. Show that \(k = 4.8\). With the engine operating at a much lower power, the car descends a hill of inclination \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\). At an instant when the speed of the car is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Given that the mass of the car is 700 kg , calculate the power of the engine. \includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-02_579_447_1658_849} A particle \(P\) of mass 0.4 kg is attached to one end of each of two light inextensible strings which are both taut. 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 \(A\). The string \(A P\) makes an angle of \(30 ^ { \circ }\) with the vertical and is 0.5 m long. The string \(B P\) makes an angle of \(60 ^ { \circ }\) with the vertical. \(P\) moves with constant angular speed in a horizontal circle with centre vertically below \(B\) (see diagram). The tension in the string \(A P\) is twice the tension in the string \(B P\). Calculate
OCR M2 2011 January Q4
8 marks Moderate -0.3
4 A block of mass 25 kg is dragged 30 m up a slope inclined at \(5 ^ { \circ }\) to the horizontal by a rope inclined at \(20 ^ { \circ }\) to the slope. The tension in the rope is 100 N and the resistance to the motion of the block is 70 N . The block is initially at rest. Calculate
  1. the work done by the tension in the rope,
  2. the change in the potential energy of the block,
  3. the speed of the block after it has moved 30 m up the slope.
OCR M2 2011 January Q5
12 marks Challenging +1.2
5 A uniform solid is made of a hemisphere with centre \(O\) and radius 0.6 m , and a cylinder of radius 0.6 m and height 0.6 m . The plane face of the hemisphere and a plane face of the cylinder coincide. (The formula for the volume of a sphere is \(\frac { 4 } { 3 } \pi r ^ { 3 }\).)
  1. Show that the distance of the centre of mass of the solid from \(O\) is 0.09 m .
  2. \includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-03_636_1036_982_593} The solid is placed with the curved surface of the hemisphere on a rough horizontal surface and the axis inclined at \(45 ^ { \circ }\) to the horizontal. The equilibrium of the solid is maintained by a horizontal force of 2 N applied to the highest point on the circumference of its plane face (see diagram). Calculate
    1. the mass of the solid,
    2. the set of possible values of the coefficient of friction between the surface and the solid.
OCR M2 2011 January Q6
14 marks Standard +0.3
6 A small ball \(B\) is projected with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(30 ^ { \circ }\) from a point \(O\) on a horizontal plane, and moves freely under gravity.
  1. Calculate the height of \(B\) above the plane when moving horizontally. \(B\) has mass 0.4 kg . At the instant when \(B\) is moving horizontally it receives an impulse of magnitude \(I \mathrm { Ns }\) in its direction of motion which immediately increases the speed of \(B\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Calculate \(I\). For the instant when \(B\) returns to the plane, calculate
  3. the speed and direction of motion of \(B\),
  4. the time of flight, and the distance of \(B\) from \(O\).
OCR M2 2011 January Q8
Standard +0.3
8
6
6
  • 6
  • {}
    6
  		• (continued)
    6
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    OCR M2 2011 January Q10
    Standard +0.3
    10
    7
    {}
    7
  		• (continued)
    7
  • 7
  • (continued)
    \section*{OCR
    RECOGNISING ACHIEVEMENT}
    •
    OCR M2 2012 January Q1
    5 marks Moderate -0.8
    1 A particle \(P\) is projected with speed \(40 \mathrm {~ms} ^ { - 1 }\) at an angle of \(35 ^ { \circ }\) above the horizontal from a point \(O\). For the instant 3 s after projection, calculate the magnitude and direction of the velocity of \(P\).
    OCR M2 2012 January Q2
    7 marks Standard +0.8
    2 \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5addd79d-d502-455c-936f-27005483164e-2_655_334_440_861} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A child's toy is a uniform solid consisting of a hemisphere of radius \(r \mathrm {~cm}\) joined to a cone of base radius \(r \mathrm {~cm}\). The curved surface of the cone makes an angle \(\alpha\) with its base. The two shapes are joined at the plane faces with their circumferences coinciding (see Fig. 1). The distance of the centre of mass of the toy above the common circular plane face is \(x \mathrm {~cm}\).
    [0pt] [The volume of a sphere is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) and the volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
    1. Show that \(x = \frac { r \left( \tan ^ { 2 } \alpha - 3 \right) } { 8 + 4 \tan \alpha }\). The toy is placed on a horizontal surface with the hemisphere in contact with the surface. The toy is released from rest from the position in which the common plane circular face is vertical (see Fig. 2). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{5addd79d-d502-455c-936f-27005483164e-2_193_670_1827_699} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure}
    2. Find the set of values of \(\alpha\) such that the toy moves to the upright position.
    OCR M2 2012 January Q3
    8 marks Standard +0.3
    3 \includegraphics[max width=\textwidth, alt={}, center]{5addd79d-d502-455c-936f-27005483164e-3_483_787_260_641} A uniform rod \(A B\) of mass 10 kg and length 2.4 m rests with \(A\) on rough horizontal ground. The rod makes an angle of \(60 ^ { \circ }\) with the horizontal and is supported by a fixed smooth peg \(P\). The distance \(A P\) is 1.6 m (see diagram).
    1. Calculate the magnitude of the force exerted by the peg on the rod.
    2. Find the least value of the coefficient of friction between the rod and the ground needed to maintain equilibrium.
    OCR M2 2012 January Q4
    12 marks Standard +0.3
    4 A particle \(P\) of mass 0.2 kg is attached to one end of a light inextensible string of length 1.2 m . The other end of the string is fixed at a point \(A\) which is 0.6 m above a smooth horizontal table. \(P\) moves on the table in a circular path whose centre \(O\) is vertically below \(A\).
    1. Given that the angular speed of \(P\) is \(2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find
      1. the tension in the string,
      2. the normal reaction between the particle and the table.
      3. Find the greatest possible speed of \(P\), given that the particle remains in contact with the table.
    OCR M2 2012 January Q5
    12 marks Standard +0.3
    5 A car of mass 1500 kg travels up a line of greatest slope of a straight road inclined at \(5 ^ { \circ }\) to the horizontal. The power of the car's engine is constant and equal to 25 kW and the resistance to the motion of the car is constant and equal to 750 N . The car passes through point \(A\) with speed \(10 \mathrm {~ms} ^ { - 1 }\).
    1. Find the acceleration of the car at \(A\). The car later passes through a point \(B\) with speed \(20 \mathrm {~ms} ^ { - 1 }\). The car takes 28s to travel from \(A\) to \(B\).
    2. Find the distance \(A B\).
    OCR M2 2012 January Q6
    13 marks Standard +0.3
    6 A small ball of mass 0.5 kg is held at a height of 3.136 m above a horizontal floor. The ball is released from rest and rebounds from the floor. The coefficient of restitution between the ball and floor is \(e\).
    1. Find in terms of \(e\) the speed of the ball immediately after the impact with the floor and the impulse that the floor exerts on the ball. The ball continues to bounce until it eventually comes to rest.
    2. Show that the time between the first bounce and the second bounce is \(1.6 e\).
    3. Write down, in terms of \(e\), the time between
      1. the second bounce and the third bounce,
      2. the third bounce and the fourth bounce.
      3. Given that the time from the ball being released until it comes to rest is 5 s , find the value of \(e\).
    OCR M2 2012 January Q7
    15 marks Challenging +1.2
    7 A particle \(P\) is projected horizontally with speed \(15 \mathrm {~ms} ^ { - 1 }\) from the top of a vertical cliff. At the same instant a particle \(Q\) is projected from the bottom of the cliff, with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. \(P\) and \(Q\) move in the same vertical plane. The height of the cliff is 60 m and the ground at the bottom of the cliff is horizontal.
    1. Given that the particles hit the ground simultaneously, find the value of \(\theta\) and find also the distance between the points of impact with the ground.
    2. Given instead that the particles collide, find the value of \(\theta\), and determine whether \(Q\) is rising or falling immediately before this collision.
    OCR M2 2005 June Q1
    5 marks Standard +0.3
    1 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-2_531_533_269_806} A uniform solid cone has vertical height 20 cm and base radius \(r \mathrm {~cm}\). It is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted until the cone topples when the angle of inclination is \(24 ^ { \circ }\) (see diagram).
    1. Find \(r\), correct to 1 decimal place. A uniform solid cone of vertical height 20 cm and base radius 2.5 cm is placed on the plane which is inclined at an angle of \(24 ^ { \circ }\).
    2. State, with justification, whether this cone will topple.
    OCR M2 2005 June Q2
    6 marks Moderate -0.8
    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
    8 marks Standard +0.3
    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
    9 marks Standard +0.3
    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
    10 marks Standard +0.3
    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
    10 marks Standard +0.3
    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
    11 marks Standard +0.3
    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
    13 marks Standard +0.3
    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 \(A _ { 2 }\).