Questions M2 (1391 questions)

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OCR M2 2012 January Q2
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
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
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
    (a) the tension in the string,
    (b) the normal reaction between the particle and the table.
  2. Find the greatest possible speed of \(P\), given that the particle remains in contact with the table.
OCR M2 2012 January Q5
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
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
    (a) the second bounce and the third bounce,
    (b) the third bounce and the fourth bounce.
  4. 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
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 2013 January Q1
1 A block is being pushed in a straight line along horizontal ground by a force of 18 N inclined at \(15 ^ { \circ }\) below the horizontal. The block moves a distance of 6 m in 5 s with constant speed. Find
  1. the work done by the force,
  2. the power with which the force is working.
OCR M2 2013 January Q2
2 A car of mass 1500 kg travels along a straight horizontal road. The resistance to the motion of the car is \(k v ^ { \frac { 1 } { 2 } } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. At the instant when the engine produces a power of 15000 W , the car has speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is accelerating at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the value of \(k\). It is given that the greatest steady speed of the car on this road is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the greatest power that the engine can produce.
OCR M2 2013 January Q3
3 A particle \(A\) is released from rest from the top of a smooth plane, which makes an angle of \(30 ^ { \circ }\) with the horizontal. The particle \(A\) collides 2 s later with a particle \(B\), which is moving up a line of greatest slope of the plane. The coefficient of restitution between the particles is 0.4 and the speed of \(B\) immediately before the collision is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 } . B\) has velocity \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the plane immediately after the collision. Find
  1. the speed of \(A\) immediately after the collision,
  2. the distance \(A\) moves up the plane after the collision. The masses of \(A\) and \(B\) are 0.5 kg and \(m \mathrm {~kg}\), respectively.
  3. Find the value of \(m\).
OCR M2 2013 January Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{b5b63135-6d02-4c4a-835a-9834d2852d6b-2_506_561_1765_735} A uniform square lamina \(A B C D\) of side 6 cm has a semicircular piece, with \(A B\) as diameter, removed (see diagram).
  1. Find the distance of the centre of mass of the remaining shape from \(C D\). The remaining shape is suspended from a fixed point by a string attached at \(C\) and hangs in equilibrium.
  2. Find the angle between \(C D\) and the vertical.
    \includegraphics[max width=\textwidth, alt={}, center]{b5b63135-6d02-4c4a-835a-9834d2852d6b-3_691_1200_262_434} A uniform rod \(A B\), of mass 3 kg and length 4 m , is in limiting equilibrium with \(A\) on rough horizontal ground. The rod is at an angle of \(60 ^ { \circ }\) to the horizontal and is supported by a small smooth peg \(P\), such that the distance \(A P\) is 2.5 m (see diagram). Find
  3. the force acting on the rod at \(P\),
  4. the coefficient of friction between the ground and the rod.
OCR M2 2013 January Q6
6 A particle of mass 0.5 kg is held at rest at a point \(P\), which is at the bottom of an inclined plane. The particle is given an impulse of 1.8 Ns directed up a line of greatest slope of the plane.
  1. Find the speed at which the particle starts to move. The particle subsequently moves up the plane to a point \(Q\), which is 0.3 m above the level of \(P\).
  2. Given that the plane is smooth, find the speed of the particle at \(Q\). It is given instead that the plane is rough. The particle is now projected up the plane from \(P\) with initial speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and comes to rest at a point \(R\) which is 0.2 m above the level of \(P\).
  3. Given that the plane is inclined at \(30 ^ { \circ }\) to the horizontal, find the magnitude of the frictional force on the particle.
OCR M2 2013 January Q7
7 A particle is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta\) above the horizontal from a point \(O\). At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the particle from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\) and \(\theta\) and hence obtain the equation of trajectory $$y = x \tan \theta - \frac { g x ^ { 2 } \sec ^ { 2 } \theta } { 2 u ^ { 2 } } .$$ In a shot put competition, a shot is thrown from a height of 2.1 m above horizontal ground. It has initial velocity of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta\) above the horizontal. The shot travels a horizontal distance of 22 m before hitting the ground.
  2. Show that \(12.1 \tan ^ { 2 } \theta - 22 \tan \theta + 10 = 0\), and find the value of \(\theta\).
  3. Find the time of flight of the shot.
OCR M2 2013 January Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{b5b63135-6d02-4c4a-835a-9834d2852d6b-4_739_860_1114_616} A conical shell has radius 6 m and height 8 m . The shell, with its vertex \(V\) downwards, is rotating about its vertical axis. A particle, of mass 0.4 kg , is in contact with the rough inner surface of the shell. The particle is 4 m above the level of \(V\) (see diagram). The particle and shell rotate with the same constant angular speed. The coefficient of friction between the particle and the shell is \(\mu\).
  1. The frictional force on the particle is \(F \mathrm {~N}\), and the normal force of the shell on the particle is \(R \mathrm {~N}\). It is given that the speed of the particle is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), which is the smallest possible speed for the particle not to slip.
    (a) By resolving vertically, show that \(4 F + 3 R = 19.6\).
    (b) By finding another equation connecting \(F\) and \(R\), find the values of \(F\) and \(R\) and show that \(\mu = 0.336\), correct to 3 significant figures.
  2. Find the largest possible angular speed of the shell for which the particle does not slip. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
    7
OCR M2 2005 June Q1
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
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 \(26 \mathrm {~ms} ^ { - 1 }\).