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OCR M3 2008 June Q1
6 marks Standard +0.3
1 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 1.8 m and modulus of elasticity 1.35 mg N . The other end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. \(P\) is held at rest at a point on the surface 3 m from \(O\). The particle is then released. Find
  1. the initial acceleration of \(P\),
  2. the speed of \(P\) at the instant the string becomes slack.
OCR M3 2008 June Q2
6 marks Standard +0.3
2 A particle \(P\) of mass 0.2 kg is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits a horizontal smooth surface. The direction of motion of \(P\) immediately before impact makes an angle of \(27 ^ { \circ }\) with the surface. Given that the coefficient of restitution between the particle and the surface is 0.6 , find
  1. the vertical component of the velocity of \(P\) immediately after impact,
  2. the magnitude of the impulse exerted on \(P\).
OCR M3 2008 June Q3
10 marks Challenging +1.2
3 \includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-2_387_561_1055_794} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 0.8 kg and 2.0 kg respectively. The spheres are on a horizontal surface. \(A\) is moving with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the line of centres when it collides with \(B\), which is stationary (see diagram). The coefficient of restitution between the spheres is 0.75 . Find the speed and direction of motion of \(A\) immediately after the collision.
OCR M3 2008 June Q4
10 marks Standard +0.3
4 A particle \(P\) of mass \(m \mathrm {~kg}\) is held at rest at a point \(O\) on a fixed plane inclined at an angle \(\sin ^ { - 1 } \left( \frac { 4 } { 7 } \right)\) to the horizontal. \(P\) is released and moves down the plane. The total resistance acting on \(P\) is \(0.2 m v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\).
  1. Show that \(5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = 28 - v\) and hence find an expression for \(v\) in terms of \(t\).
  2. Find the acceleration of \(P\) when \(t = 10\).
OCR M3 2008 June Q5
11 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-3_581_903_267_621} Two uniform rods \(X A\) and \(X B\) are freely jointed at \(X\). The lengths of the rods are 1.5 m and 1.3 m respectively, and their weights are 150 N and 130 N respectively. The rods are in equilibrium in a vertical plane with \(A\) and \(B\) in contact with a rough horizontal surface. \(A\) and \(B\) are at distances horizontally from \(X\) of 0.9 m and 0.5 m respectively, and \(X\) is 1.2 m above the surface (see diagram).
  1. The normal components of the contact forces acting on the rods at \(A\) and \(B\) are \(R _ { A } \mathrm {~N}\) and \(R _ { B } \mathrm {~N}\) respectively. Show that \(R _ { A } = 125\) and find \(R _ { B }\).
  2. Find the frictional components of the contact forces acting on the rods at \(A\) and \(B\).
  3. Find the horizontal and vertical components of the force exerted on \(X A\) at \(X\), stating their directions.
OCR M3 2008 June Q6
14 marks Standard +0.8
6 A particle \(P\) of mass 0.1 kg moves in a straight line on a smooth horizontal surface. A force of \(( 0.36 - 0.144 x ) \mathrm { N }\) acts on \(P\) in the direction from \(O\) to \(P\), where \(x \mathrm {~m}\) is the displacement of \(P\) from a point \(O\) on the surface at time \(t \mathrm {~s}\).
  1. By using the substitution \(x = y + 2.5\), or otherwise, show that \(P\) moves with simple harmonic motion of period 5.24 s , correct to 3 significant figures. The maximum value of \(x\) during the motion is 3 .
  2. Write down the amplitude of \(P\) 's motion and find the two possible values of \(x\) for which \(P\) 's speed is \(0.48 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. On each of the first two occasions when \(P\) has speed \(0.48 \mathrm {~m} \mathrm {~s} ^ { - 1 } , P\) is moving towards \(O\). Find the time interval between
    (a) these first two occasions,
    (b) the second and third occasions when \(P\) has speed \(0.48 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
OCR M3 2008 June Q7
15 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-4_517_677_267_733} A particle \(P\) of mass \(m \mathrm {~kg}\) is slightly disturbed from rest at the highest point on the surface of a smooth fixed sphere of radius \(a\) m and centre \(O\). The particle starts to move downwards on the surface. While \(P\) remains on the surface \(O P\) makes an angle of \(\theta\) radians with the upward vertical and has angular speed \(\omega\) rad s \(^ { - 1 }\) (see diagram). The sphere exerts a force of magnitude \(R \mathrm {~N}\) on \(P\).
  1. Show that \(a \omega ^ { 2 } = 2 g ( 1 - \cos \theta )\).
  2. Find an expression for \(R\) in terms of \(m , g\) and \(\theta\). At the instant that \(P\) loses contact with the surface of the sphere, find
  3. the transverse component of the acceleration of \(P\),
  4. the rate of change of \(R\) with respect to time \(t\), in terms of \(m , g\) and \(a\).
OCR M3 2009 June Q1
6 marks Moderate -0.3
1 A smooth sphere of mass 0.3 kg bounces on a fixed horizontal surface. Immediately before the sphere bounces the components of its velocity horizontally and vertically downwards are \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The speed of the sphere immediately after it bounces is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the vertical component of the velocity of the sphere immediately after impact is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and hence find the coefficient of restitution between the surface and the sphere.
  2. State the direction of the impulse on the sphere and find its magnitude.
OCR M3 2009 June Q2
8 marks Standard +0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-2_460_725_731_708} Two uniform rods, \(A B\) and \(B C\), are freely jointed to each other at \(B\), and \(C\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(A\) resting on a rough horizontal surface. This surface is 1.5 m below the level of \(B\) and the horizontal distance between \(A\) and \(B\) is 3 m (see diagram). The weight of \(A B\) is 80 N and the frictional force acting on \(A B\) at \(A\) is 14 N .
  1. Write down the horizontal component of the force acting on \(A B\) at \(B\) and show that the vertical component of this force is 33 N upwards.
  2. Given that the force acting on \(B C\) at \(C\) has magnitude 50 N , find the weight of \(B C\). \includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-2_421_949_1793_598} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 4 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision both spheres have speed \(3 \mathrm {~ms} ^ { - 1 }\). The spheres are moving in opposite directions, each at \(60 ^ { \circ }\) to the line of centres (see diagram). After the collision \(A\) moves in a direction perpendicular to the line of centres.
  3. Show that the speed of \(B\) is unchanged as a result of the collision, and find the angle that the new direction of motion of \(B\) makes with the line of centres.
  4. Find the coefficient of restitution between the spheres.
OCR M3 2009 June Q4
11 marks Challenging +1.2
4 A motor-cycle, whose mass including the rider is 120 kg , is decelerating on a horizontal straight road. The motor-cycle passes a point \(A\) with speed \(40 \mathrm {~ms} ^ { - 1 }\) and when it has travelled a distance of \(x \mathrm {~m}\) beyond \(A\) its speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engine develops a constant power of 8 kW and resistances are modelled by a force of \(0.25 v ^ { 2 } \mathrm {~N}\) opposing the motion.
  1. Show that \(\frac { 480 v ^ { 2 } } { v ^ { 3 } - 32000 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 1\).
  2. Find the speed of the motor-cycle when it has travelled 500 m beyond \(A\).
OCR M3 2009 June Q5
11 marks Challenging +1.8
5 \includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-3_591_668_776_737} Each of two identical strings has natural length 1.5 m and modulus of elasticity 18 N . One end of one of the strings is attached to \(A\) and one end of the other string is attached to \(B\), where \(A\) and \(B\) are fixed points which are 3 m apart and at the same horizontal level. \(M\) is the mid-point of \(A B\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of each of the strings. \(P\) is held at rest at the point 0.8 m vertically above \(M\), and then released. The lowest point reached by \(P\) in the subsequent motion is 2 m below \(M\) (see diagram).
  1. Find the maximum tension in each of the strings during \(P\) 's motion.
  2. By considering energy,
    (a) show that the value of \(m\) is 0.42 , correct to 2 significant figures,
    (b) find the speed of \(P\) at \(M\).
OCR M3 2009 June Q6
13 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-4_368_131_274_1005} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length \(L \mathrm {~m}\). The other end of the string is attached to a fixed point \(O\). The particle is held at rest with the string taut and then released. \(P\) starts to move and in the subsequent motion the angular displacement of \(O P\), at time \(t \mathrm {~s}\), is \(\theta\) radians from the downward vertical (see diagram). The initial value of \(\theta\) is 0.05 .
  1. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - \frac { g } { L } \sin \theta\).
  2. Hence show that the motion of \(P\) is approximately simple harmonic.
  3. Given that the period of the approximate simple harmonic motion is \(\frac { 4 } { 7 } \pi \mathrm {~s}\), find the value of \(L\).
  4. Find the value of \(\theta\) when \(t = 0.7 \mathrm {~s}\), and the value of \(t\) when \(\theta\) next takes this value.
  5. Find the speed of \(P\) when \(t = 0.7 \mathrm {~s}\). \includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-4_422_501_1500_822} A hollow cylinder has internal radius \(a\). The cylinder is fixed with its axis horizontal. A particle \(P\) of mass \(m\) is at rest in contact with the smooth inner surface of the cylinder. \(P\) is given a horizontal velocity \(u\), in a vertical plane perpendicular to the axis of the cylinder, and begins to move in a vertical circle. While \(P\) remains in contact with the surface, \(O P\) makes an angle \(\theta\) with the downward vertical, where \(O\) is the centre of the circle. The speed of \(P\) is \(v\) and the magnitude of the force exerted on \(P\) by the surface is \(R\) (see diagram).
  6. Find \(v ^ { 2 }\) in terms of \(u , a , g\) and \(\theta\) and show that \(R = \frac { m u ^ { 2 } } { a } + m g ( 3 \cos \theta - 2 )\).
  7. Given that \(P\) just reaches the highest point of the circle, find \(u ^ { 2 }\) in terms of \(a\) and \(g\), and show that in this case the least value of \(v ^ { 2 }\) is \(a g\).
  8. Given instead that \(P\) oscillates between \(\theta = \pm \frac { 1 } { 6 } \pi\) radians, find \(u ^ { 2 }\) in terms of \(a\) and \(g\).
OCR M3 2010 June Q1
6 marks Standard +0.3
1 A small ball of mass 0.8 kg is moving with speed \(10.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude 4 Ns . The speed of the ball immediately afterwards is \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The angle between the directions of motion before and after the impulse acts is \(\alpha\). Using an impulse-momentum triangle, or otherwise, find \(\alpha\).
OCR M3 2010 June Q2
7 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-2_691_767_529_689} Two uniform rods \(A B\) and \(B C\) are of equal length and each has weight 100 N . The rods are freely jointed to each other at \(B\), and \(A\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(A B\) horizontal and \(C\) resting on a rough horizontal surface. \(C\) is vertically below the mid-point of \(A B\) (see diagram).
  1. By taking moments about \(A\) for \(A B\), find the vertical component of the force on \(A B\) at \(B\). Hence find the vertical component of the contact force on \(B C\) at \(C\).
  2. Calculate the magnitude of the frictional force on \(B C\) at \(C\) and state its direction. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-3_452_345_264_900} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A uniform smooth sphere \(A\) moves on a smooth horizontal surface towards a smooth vertical wall. Immediately before the sphere hits the wall it has components of velocity parallel and perpendicular to the wall each of magnitude \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after hitting the wall the components have magnitudes \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), respectively (see Fig. 1).
OCR M3 2010 June Q5
11 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-4_234_1003_1007_571} Particles \(P _ { 1 }\) and \(P _ { 2 }\) are each moving with simple harmonic motion along the same straight line. \(P _ { 1 }\) 's motion has centre \(C _ { 1 }\), period \(2 \pi \mathrm {~s}\) and amplitude \(3 \mathrm {~m} ; P _ { 2 }\) 's motion has centre \(C _ { 2 }\), period \(\frac { 4 } { 3 } \pi \mathrm {~s}\) and amplitude 4 m . The points \(C _ { 1 }\) and \(C _ { 2 }\) are 6.5 m apart. The displacements of \(P _ { 1 }\) and \(P _ { 2 }\) from their centres of oscillation at time \(t \mathrm {~s}\) are denoted by \(x _ { 1 } \mathrm {~m}\) and \(x _ { 2 } \mathrm {~m}\) respectively. The diagram shows the positions of the particles at time \(t = 0\), when \(x _ { 1 } = 3\) and \(x _ { 2 } = 4\).
  1. State expressions for \(x _ { 1 }\) and \(x _ { 2 }\) in terms of \(t\), which are valid until the particles collide. The particles collide when \(t = 5.99\), correct to 3 significant figures.
  2. Find the distance travelled by \(P _ { 2 }\) before the collision takes place.
  3. Find the velocities of \(P _ { 1 }\) and \(P _ { 2 }\) immediately before the collision, and state whether the particles are travelling in the same direction or in opposite directions.
OCR M3 2010 June Q6
12 marks Challenging +1.2
6 A bungee jumper of weight \(W \mathrm {~N}\) is joined to a fixed point \(O\) by a light elastic rope of natural length 20 m and modulus of elasticity 32000 N . The jumper starts from rest at \(O\) and falls vertically. The jumper is modelled as a particle and air resistance is ignored.
  1. Given that the jumper just reaches a point 25 m below \(O\), find the value of \(W\).
  2. Find the maximum speed reached by the jumper.
  3. Find the maximum value of the deceleration of the jumper during the downward motion.
OCR M3 2010 June Q7
17 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-5_447_693_255_726} A particle \(P\) is attached to a fixed point \(O\) by a light inextensible string of length 0.7 m . A particle \(Q\) is in equilibrium suspended from \(O\) by an identical string. With the string \(O P\) taut and horizontal, \(P\) is projected vertically downwards with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) so that it strikes \(Q\) directly (see diagram). \(P\) is brought to rest by the collision and \(Q\) starts to move with speed \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the speed of \(P\) immediately before the collision. Hence find the coefficient of restitution between \(P\) and \(Q\).
  2. Given that the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(O Q\) makes an angle \(\theta\) with the downward vertical, find an expression for \(v ^ { 2 }\) in terms of \(\theta\), and show that the tension in the string \(O Q\) is \(14.7 m ( 1 + 2 \cos \theta ) \mathrm { N }\), where \(m \mathrm {~kg}\) is the mass of \(Q\).
  3. Find the radial and transverse components of the acceleration of \(Q\) at the instant that the string \(O Q\) becomes slack.
  4. Show that \(V ^ { 2 } = 0.8575\), where \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(Q\) when it reaches its greatest height (after the string \(O Q\) becomes slack). Hence find the greatest height reached by \(Q\) above its initial position.
OCR M3 2011 June Q1
4 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{e004bbb5-f9e1-4ea2-8357-39db9392cb8c-2_355_572_260_788} A particle \(P\) of mass 0.3 kg is moving in a straight line with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is deflected through an angle \(\theta\) by an impulse of magnitude \(I\) N s. The impulse acts at right angles to the initial direction of motion of \(P\) (see diagram). The speed of \(P\) immediately after the impulse acts is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that \(\cos \theta = 0.8\) and find the value of \(I\).
OCR M3 2011 June Q2
10 marks Challenging +1.2
2 \includegraphics[max width=\textwidth, alt={}, center]{e004bbb5-f9e1-4ea2-8357-39db9392cb8c-2_403_999_982_575} Two uniform rods \(A B\) and \(A C\), of lengths 3 m and 4 m respectively, have weights 300 N and 400 N respectively. The rods are freely jointed at \(A\). The mid-points of the rods are joined by a light inextensible string. The rods are in equilibrium in a vertical plane with the string taut and \(B\) and \(C\) in contact with a smooth horizontal surface. The point \(A\) is 2.4 m above the surface (see diagram).
  1. Show that the force exerted by the surface on \(A B\) is 374 N and find the force exerted by the surface on \(A C\).
  2. Find the tension in the string.
  3. Find the horizontal and vertical components of the force exerted on \(A B\) at \(A\) and state their directions.
OCR M3 2011 June Q3
10 marks Challenging +1.2
3 A particle \(P\) of mass 0.25 kg is projected horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) on a smooth horizontal surface and moves in a straight line on the surface. The only horizontal force acting on \(P\) has magnitude \(0.2 v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after it is projected from \(O\). This force is directed towards \(O\).
  1. Find an expression for \(v\) in terms of \(t\). The particle \(P\) passes through a point \(X\) with speed \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the average speed of \(P\) for its motion between \(O\) and \(X\).
OCR M3 2011 June Q4
11 marks Standard +0.3
4 One end of a light inextensible string of length 2 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) is held at rest with the string taut so that \(O P\) makes an angle of 0.15 radians with the downward vertical. \(P\) is released and \(t\) seconds afterwards \(O P\) makes an angle of \(\theta\) radians with the downward vertical.
  1. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 4.9 \sin \theta\) and give a reason why the motion is approximately simple harmonic. Using the simple harmonic approximation,
  2. obtain an expression for \(\theta\) in terms of \(t\) and hence find the values of \(t\) at the first and second occasions when \(\theta = - 0.1\),
  3. find the angular speed of \(O P\) and the linear speed of \(P\) when \(t = 0.5\). \includegraphics[max width=\textwidth, alt={}, center]{e004bbb5-f9e1-4ea2-8357-39db9392cb8c-3_606_1006_973_568} Two uniform smooth identical spheres \(A\) and \(B\) are moving towards each other on a horizontal surface when they collide. Immediately before the collision \(A\) and \(B\) are moving with speeds \(u _ { A } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(u _ { B } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, at acute angles \(\alpha\) and \(\beta\), respectively, to the line of centres. Immediately after the collision \(A\) and \(B\) are moving with speeds \(v _ { A } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v _ { B } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, at right angles and at acute angle \(\gamma\), respectively, to the line of centres (see diagram).
  4. Given that \(\sin \beta = 0.96\) and \(\frac { v _ { B } } { u _ { B } } = 1.2\), find the value of \(\sin \gamma\).
  5. Given also that, before the collision, the component of \(A\) 's velocity parallel to the line of centres is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the values of \(u _ { B }\) and \(v _ { B }\).
  6. Find the coefficient of restitution between the spheres.
  7. Given that the kinetic energy of \(A\) immediately before the collision is \(6.5 m \mathrm {~J}\), where \(m \mathrm {~kg}\) is the mass of \(A\), find the value of \(v _ { A }\).
OCR M3 2011 June Q6
11 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{e004bbb5-f9e1-4ea2-8357-39db9392cb8c-4_559_525_258_808} A particle \(P\) of weight 6 N is attached to the highest point \(A\) of a fixed smooth sphere by a light elastic string. The sphere has centre \(O\) and radius 0.8 m . The string has natural length \(\frac { 1 } { 10 } \pi \mathrm {~m}\) and modulus of elasticity \(9 \mathrm {~N} . P\) is released from rest at a point \(X\) on the sphere where \(O X\) makes an angle of \(\frac { 1 } { 4 } \pi\) radians with the upwards vertical. \(P\) remains in contact with the sphere as it moves upwards to \(A\). At time \(t\) seconds after the release, \(O P\) makes an angle of \(\theta\) radians with the upwards vertical (see diagram). When \(\theta = \frac { 1 } { 6 } \pi , P\) passes through the point \(Y\).
  1. Show that as \(P\) moves from \(X\) to \(Y\) its gravitational potential energy increases by \(2 \cdot 4 ( \sqrt { 3 } - \sqrt { 2 } ) \mathrm { J }\) and the elastic potential energy in the string decreases by \(0.4 \pi \mathrm {~J}\).
  2. Verify that the transverse acceleration of \(P\) is zero when \(\theta = \frac { 1 } { 6 } \pi\), and hence find the maximum speed of \(P\).
OCR M3 2011 June Q7
14 marks Standard +0.3
7 One end of a light inextensible string of length 0.8 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.3 kg is attached to the other end of the string. \(P\) is projected horizontally from the point 0.8 m vertically below \(O\) with speed \(5.6 \mathrm {~m} \mathrm {~s} ^ { - 1 } . P\) starts to move in a vertical circle with centre \(O\). The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the string makes an angle \(\theta\) with the downward vertical.
  1. While the string remains taut, show that \(v ^ { 2 } = 15.68 ( 1 + \cos \theta )\), and find the tension in the string in terms of \(\theta\).
  2. For the instant when the string becomes slack, find the value of \(\theta\) and the value of \(v\).
  3. Find, in either order, the speed of \(P\) when it is at its greatest height after the string becomes slack, and the greatest height reached by \(P\) above its point of projection. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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OCR M3 2012 June Q2
8 marks Challenging +1.2
2 \includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-3_442_636_255_715} \(B\) is a point on a smooth plane surface inclined at an angle of \(15 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 0.45 kg is released from rest at the point \(A\) which is 2.5 m vertically above \(B\). The particle \(P\) rebounds from the surface at an angle of \(60 ^ { \circ }\) to the line of greatest slope through \(B\), with a speed of \(u \mathrm {~ms} ^ { - 1 }\). The impulse exerted on \(P\) by the surface has magnitude \(I\) Ns and is in a direction making an angle of \(\theta ^ { \circ }\) with the upward vertical through \(B\) (see diagram).
  1. Explain why \(\theta = 15\).
  2. Find the values of \(u\) and \(I\).
OCR M3 2012 June Q3
10 marks Standard +0.3
3 A particle \(P\) of mass \(m \mathrm {~kg}\) is released from rest and falls vertically. When \(P\) has fallen a distance of \(x \mathrm {~m}\) it has a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only forces acting on \(P\) are its weight and air resistance of magnitude \(\frac { 1 } { 400 } m v ^ { 2 } \mathrm {~N}\).
  1. Find \(v ^ { 2 }\) in terms of \(x\) and show that \(v ^ { 2 }\) must be less than 3920 .
  2. Find the speed of \(P\) when it has fallen 100 m .