Questions — OCR M2 (149 questions)

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OCR M2 2007 January Q1
1 A uniform solid cylinder has height 20 cm and diameter 12 cm . It is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted until the cylinder topples when the angle of inclination is \(\alpha\). Find \(\alpha\).
OCR M2 2007 January Q2
4 marks
2 Two smooth spheres \(A\) and \(B\), of equal radius and of masses 0.2 kg and 0.1 kg respectively, are free to move on a smooth horizontal table. \(A\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with \(B\), which is stationary. The collision is perfectly elastic. Calculate the speed of \(A\) after the impact. [4]
OCR M2 2007 January Q3
3 A small sphere of mass 0.2 kg is projected vertically downwards with speed \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 40 m above horizontal ground. It hits the ground and rebounds vertically upwards, coming to instantaneous rest at its initial point of projection. Ignoring air resistance, calculate
  1. the coefficient of restitution between the sphere and the ground,
  2. the magnitude of the impulse which the ground exerts on the sphere.
OCR M2 2007 January Q4
4 A skier of mass 80 kg is pulled up a slope which makes an angle of \(20 ^ { \circ }\) with the horizontal. The skier is subject to a constant frictional force of magnitude 70 N . The speed of the skier increases from \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(A\) to \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(B\), and the distance \(A B\) is 25 m .
  1. By modelling the skier as a small object, calculate the work done by the pulling force as the skier moves from \(A\) to \(B\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-2_451_1019_1425_603} It is given that the pulling force has constant magnitude \(P \mathrm {~N}\), and that it acts at a constant angle of \(30 ^ { \circ }\) above the slope (see diagram). Calculate \(P\).
OCR M2 2007 January Q5
5 A model train has mass 100 kg . When the train is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the resistance to its motion is \(3 v ^ { 2 } \mathrm {~N}\) and the power output of the train is \(\frac { 3000 } { v } \mathrm {~W}\).
  1. Show that the driving force acting on the train is 120 N at an instant when the train is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the acceleration of the train at an instant when it is moving horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train moves with constant speed up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 98 }\).
  3. Calculate the speed of the train.
OCR M2 2007 January Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-3_540_878_989_632} A uniform lamina \(A B C D E\) of weight 30 N consists of a rectangle and a right-angled triangle. The dimensions are as shown in the 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. The lamina is freely suspended from a hinge at \(B\).
  2. Calculate the angle that \(A B\) makes with the vertical. The lamina is now held in a position such that \(B D\) is horizontal. This is achieved by means of a string attached to \(D\) and to a fixed point 15 cm directly above the hinge at \(B\).
  3. Calculate the tension in the string.
OCR M2 2007 January Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-4_782_1006_274_571} One end of a light inextensible string of length 0.8 m is attached to a fixed point \(A\) which lies above a smooth horizontal table. The other end of the string is attached to a particle \(P\), of mass 0.3 kg , which moves in a horizontal circle on the table with constant angular speed \(2 \mathrm { rad } \mathrm { s } ^ { - 1 } . A P\) makes an angle of \(30 ^ { \circ }\) with the vertical (see diagram).
  1. Calculate the tension in the string.
  2. Calculate the normal contact force between the particle and the table. The particle now moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is on the point of leaving the surface of the table.
  3. Calculate \(v\).
OCR M2 2007 January Q8
8 A missile is projected with initial speed \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. Ignoring air resistance, calculate
  1. the maximum height of the missile above the level of the point of projection,
  2. the distance of the missile from the point of projection at the instant when it is moving downwards at an angle of \(10 ^ { \circ }\) to the horizontal. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
OCR M2 2008 January Q1
1 A ball is projected with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(55 ^ { \circ }\) above the horizontal. At the instant when the ball reaches its greatest height, it hits a vertical wall, which is perpendicular to the ball's path. The coefficient of restitution between the ball and the wall is 0.65 . Calculate the speed of the ball
  1. immediately before its impact with the wall,
  2. immediately after its impact with the wall.
OCR M2 2008 January Q2
2 A particle of mass \(m \mathrm {~kg}\) is projected directly up a rough plane with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The plane makes an angle of \(30 ^ { \circ }\) with the horizontal and the coefficient of friction is 0.2 . Calculate the distance the particle travels up the plane before coming instantaneously to rest.
OCR M2 2008 January Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{982647bd-8514-40cf-b4ee-674f51df32c5-2_412_380_909_884} A uniform rod \(A B\), of weight 25 N and length 1.6 m , rests in equilibrium in a vertical plane with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth wall which is inclined at \(80 ^ { \circ }\) to the horizontal. The rod is inclined at \(60 ^ { \circ }\) to the horizontal (see diagram). Calculate the magnitude of the force acting on the rod at \(B\).
OCR M2 2008 January Q4
4 A car of mass 1200 kg has a maximum speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when travelling on a horizontal road. The car experiences a resistance of \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. The maximum power of the car's engine is 45000 W .
  1. Show that \(k = 50\).
  2. Find the maximum possible acceleration of the car when it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal road.
  3. The car climbs a hill, which is inclined at an angle of \(10 ^ { \circ }\) to the horizontal, at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the power of the car's engine.
OCR M2 2008 January Q5
5 A particle \(P\) of mass \(2 m\) is moving on a smooth horizontal surface with speed \(u\) when it collides directly with a particle \(Q\) of mass \(k m\) whose speed is \(3 u\) in the opposite direction. As a result of the collision, the directions of motion of both particles are reversed and the speed of \(P\) is halved.
  1. Find, in terms of \(u\) and \(k\), the speed of \(Q\) after the collision. Hence write down the range of possible values of \(k\).
  2. Calculate the magnitude of the impulse which \(Q\) exerts on \(P\).
  3. Given that \(k = \frac { 1 } { 2 }\), calculate the coefficient of restitution between \(P\) and \(Q\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_472_1143_221_242} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} One end of a light inextensible string is attached to a point \(P\). The other end is attached to a point \(Q , 1.96 \mathrm {~m}\) vertically below \(P\). A small smooth bead \(B\), of mass 0.3 kg , is threaded on the string and moves in a horizontal circle with centre \(Q\) and radius \(1.96 \mathrm {~m} . B\) rotates about \(Q\) with constant angular speed \(\omega\) rad s \(^ { - 1 }\) (see Fig. 1).
OCR M2 2008 January Q7
7 A missile is projected from a point \(O\) on horizontal ground with speed \(175 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\theta\). The horizontal lower surface of a cloud is 650 m above the ground.
  1. Find the value of \(\theta\) for which the missile just reaches the cloud. It is given that \(\theta = 55 ^ { \circ }\).
  2. Find the length of time for which the missile is above the lower surface of the cloud.
  3. Find the speed of the missile at the instant it enters the cloud.
OCR M2 2008 January Q8
8
  1. A uniform semicircular lamina has radius 4 cm . Show that the distance from its centre to its centre of mass is 1.70 cm , correct to 3 significant figures.
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-4_429_947_405_640} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A model bridge is made from a uniform rectangular board, \(A B C D\), with a semicircular section, \(E F G\), removed. \(O\) is the mid-point of \(E G\). \(A B = 8 \mathrm {~cm} , B C = 20 \mathrm {~cm} , A O = 12 \mathrm {~cm}\) and the radius of the semicircle is 4 cm (see Fig. 1).
    (a) Show that the distance from \(A B\) to the centre of mass of the model is 9.63 cm , correct to 3 significant figures.
    (b) Calculate the distance from \(A D\) to the centre of mass of the model.
  3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-4_572_945_1416_641} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The model bridge is smoothly pivoted at \(A\) and is supported in equilibrium by a vertical wire attached to \(D\). The weight of the model is 15 N and \(A D\) makes an angle of \(10 ^ { \circ }\) with the horizontal (see Fig. 2). Calculate the tension in the wire.
OCR M2 2009 January Q1
1 A stone is projected from a point on level ground with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(\theta ^ { \circ }\) above the horizontal. When the stone is at its greatest height it just passes over the top of a tree that is 17 m high. Calculate \(\theta\).
OCR M2 2009 January Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-2_465_643_495_749} A uniform right-angled triangular lamina \(A B C\) with sides \(A B = 12 \mathrm {~cm} , B C = 9 \mathrm {~cm}\) and \(A C = 15 \mathrm {~cm}\) is freely suspended from a hinge at its vertex \(A\). The lamina has mass 2 kg and is held in equilibrium with \(A B\) horizontal by means of a string attached to \(B\). The string is at an angle of \(30 ^ { \circ }\) to the horizontal (see diagram). Calculate the tension in the string.
OCR M2 2009 January Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-2_828_476_1338_836} A door is modelled as a lamina \(A B C D E\) consisting of a uniform rectangular section \(A B D E\) of weight 60 N and a uniform semicircular section \(B C D\) of weight 10 N and radius \(40 \mathrm {~cm} . A B\) is 200 cm and \(A E\) is 80 cm . The door is freely hinged at \(F\) and \(G\), where \(G\) is 30 cm below \(B\) and \(F\) is 30 cm above \(A\) (see diagram).
  1. Find the magnitudes and directions of the horizontal components of the forces on the door at each of \(F\) and \(G\).
  2. Calculate the distance from \(A E\) to the centre of mass of the door.
OCR M2 2009 January Q4
4 A car of mass 800 kg experiences a resistance of magnitude \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed. The car's engine is working at a constant rate of \(P \mathrm {~W}\). At an instant when the car is travelling on a horizontal road with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At an instant when the car is ascending a hill of constant slope \(12 ^ { \circ }\) to the horizontal with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that \(k = 0.900\), correct to 3 decimal places, and find \(P\). The power is increased to \(1.5 P \mathrm {~W}\).
  2. Calculate the maximum steady speed of the car on a horizontal road.
OCR M2 2009 January Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-3_729_739_868_703} A particle \(P\) of mass 0.2 kg is attached to one end of each of two light inextensible strings, one of length 0.4 m and one of length 0.3 m . 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 particle moves in a horizontal circle of radius 0.24 m at a constant angular speed of \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). Both strings are taut, the tension in \(A P\) is \(S \mathrm {~N}\) and the tension in \(B P\) is \(T \mathrm {~N}\).
  1. By resolving vertically, show that \(4 S = 3 T + 9.8\).
  2. Find another equation connecting \(S\) and \(T\) and hence calculate the tensions, correct to 1 decimal place. \section*{[Questions 6 and 7 are printed overleaf.]}
OCR M2 2009 January Q6
6 A particle is projected from a point \(O\) with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta\) above the horizontal and it moves freely under gravity. The horizontal and upward vertical displacements of the particle from \(O\) at any subsequent time, \(t\) seconds, are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(\theta\) and \(t\), and hence show that $$y = x \tan \theta - \frac { 4.9 x ^ { 2 } } { v ^ { 2 } \cos ^ { 2 } \theta } .$$
    \includegraphics[max width=\textwidth, alt={}]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-4_551_575_667_826}
    The particle subsequently passes through the point \(A\) with coordinates \(( h , - h )\) as shown in the diagram. It is given that \(v = 14\) and \(\theta = 30 ^ { \circ }\).
  2. Calculate \(h\).
  3. Calculate the direction of motion of the particle at \(A\).
  4. Calculate the speed of the particle at \(A\).
    \includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-4_278_1061_1749_543} Two small spheres, \(P\) and \(Q\), 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 and the base of the cylinder. The mass of \(P\) is 0.2 kg , the mass of \(Q\) is 0.3 kg and the radius of the cylinder is \(0.4 \mathrm {~m} . P\) and \(Q\) are stationary at opposite ends of a diameter of the base of the cylinder (see diagram). The coefficient of restitution between \(P\) and \(Q\) is \(0.5 . P\) is given an impulse of magnitude 0.8 Ns in a tangential direction.
  5. Calculate the speeds of the particles after \(P\) 's first impact with \(Q\).
    \(Q\) subsequently catches up with \(P\) and there is a second impact.
  6. Calculate the speeds of the particles after this second impact.
  7. Calculate the magnitude of the force exerted on \(Q\) by the curved surface of the cylinder after the second impact.
OCR M2 2010 January Q1
1 Find the average power exerted by a climber of mass 75 kg when climbing a vertical distance of 40 m in 2 minutes.
OCR M2 2010 January Q2
2 A small sphere of mass 0.2 kg is dropped from rest at a height of 3 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height of 1.8 m above the ground.
  1. Calculate the magnitude of the impulse which the ground exerts on the sphere.
  2. Calculate the coefficient of restitution between the sphere and the ground.
OCR M2 2010 January Q3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-2_528_688_845_731} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A uniform conical shell has mass 0.2 kg , height 0.3 m and base diameter 0.8 m . A uniform hollow cylinder has mass 0.3 kg , length 0.7 m and diameter 0.8 m . The conical shell is attached to the cylinder, with the circumference of its base coinciding with one end of the cylinder (see Fig. 1).
  1. Show that the distance of the centre of mass of the combined object from the vertex of the conical shell is 0.47 m . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-2_497_572_1836_788} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The combined object is freely suspended from its vertex and is held with its axis horizontal. This is achieved by means of a wire attached to a point on the circumference of the base of the conical shell. The wire makes an angle of \(80 ^ { \circ }\) with the slant edge of the conical shell (see Fig. 2).
  2. Calculate the tension in the wire.
OCR M2 2010 January Q4
4 A car of mass 700 kg is moving along a horizontal road against a constant resistance to motion of 400 N . At an instant when the car is travelling at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the driving force of the car at this instant.
  2. Find the power at this instant. The maximum steady speed of the car on a horizontal road is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the maximum power of the car. The car now moves at maximum power against the same resistance up a slope of constant angle \(\theta ^ { \circ }\) to the horizontal. The maximum steady speed up the slope is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  4. Find \(\theta\).