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OCR Further Mechanics 2022 June Q3
6 marks Challenging +1.2
3 A particle \(P\) of mass 6 kg moves in a straight line under the action of a single force of magnitude \(F N\) which acts in the direction of motion of \(P\).
At time \(t\) seconds, where \(t \geqslant 0 , F\) is given by \(\mathrm { F } = \frac { 1 } { 5 - 4 \mathrm { e } ^ { - \mathrm { t } ^ { 2 } } }\).
When \(t = 0\), the speed of \(P\) is \(1.9 \mathrm {~ms} ^ { - 1 }\).
  1. Find the impulse of the force over the period \(0 \leqslant t \leqslant 2\).
  2. Find the speed of \(P\) at the instant when \(t = 2\).
  3. Find the work done by the force on \(P\) over the period \(0 \leqslant t \leqslant 2\).
OCR Further Mechanics 2022 June Q4
8 marks Standard +0.3
4 When two objects are placed a distance apart in outer space each applies a gravitational force to the other. It is suggested that the magnitude of this force depends on the masses of both objects and the distance between them. Assuming that this suggestion is correct, it is further assumed that the magnitude of this force is given by a relationship of the form $$\mathrm { F } = \mathrm { Gm } _ { 1 } ^ { \alpha } \mathrm { m } _ { 2 } ^ { \beta } \mathrm { r } ^ { \gamma }$$ where
  • \(F\) is the magnitude of the force
  • \(m _ { 1 }\) and \(m _ { 2 }\) are the masses of the two objects
  • \(r\) is the distance between the two objects
  • \(G\) is a constant.
    1. Using a dimensional argument based on Newton's third law explain why \(\alpha = \beta\).
It is given that the magnitude of the gravitational force is given by such a relationship and that \(G = 6.67 \times 10 ^ { - 11 } \mathrm {~m} ^ { 3 } \mathrm {~kg} ^ { - 1 } \mathrm {~s} ^ { - 2 }\).
  • Write down the dimensions of \(G\).
  • By using dimensional analysis, determine the values of \(\alpha , \beta\) and \(\gamma\). You are given that the mass of the Earth is \(5.97 \times 10 ^ { 24 } \mathrm {~kg}\) and that the distance of the Moon from the Earth is \(3.84 \times 10 ^ { 8 } \mathrm {~m}\). You may assume that the only force acting on the Moon is the gravitational force due to the Earth.
  • By modelling the Earth as stationary and assuming that the Moon moves in a circular orbit around the Earth, determine the period of the motion of the Moon. Give your answer to the nearest day.
    •
    OCR Further Mechanics 2022 June Q6
    10 marks Challenging +1.2
    6 A particle \(P\) of mass 2.5 kg is free to move along the \(x\)-axis. When its displacement from the origin is \(x \mathrm {~m}\) its velocity is \(v \mathrm {~ms} ^ { - 1 }\). At time \(t = 0\) seconds, \(P\) is at the point where \(x = 1\) and is travelling in the negative \(x\)-direction with speed \(5 \mathrm {~ms} ^ { - 1 }\). At this time an impulse of \(I\) Ns is applied to \(P\) in the positive \(x\)-direction so that \(P\) moves in the positive \(x\)-direction with speed \(18 \mathrm {~ms} ^ { - 1 }\).
    1. Find the value of \(I\). Subsequently, whenever \(P\) is in motion, two forces act on it. The first force acts in the positive \(x\)-direction and has magnitude \(\frac { 5 v ^ { 2 } } { x } N\). The second force acts in the negative \(x\)-direction and has magnitude 60 vN .
    2. Show that the motion of \(P\) can be modelled by the differential equation \(\frac { \mathrm { dV } } { \mathrm { dx } } = \frac { \mathrm { aV } } { \mathrm { x } } + \mathrm { b }\) where \(a\) and \(b\) are constants whose values should be determined.
    3. By solving the differential equation derived in part (b) find an expression for \(v\) in terms of \(x\). You are given that \(\mathrm { x } = \frac { 4 } { 3 \mathrm { e } ^ { - 24 \mathrm { t } } + 1 }\) when \(t \geqslant 0\).
    4. Describe in detail the motion of \(P\) when \(t \geqslant 0\).
    OCR Further Mechanics 2022 June Q7
    15 marks Challenging +1.2
    7 The training rig for a parachutist comprises a fixed platform and a fixed hook, \(H\). The platform is 3.5 m above horizontal ground level. The hook, which is not directly above the platform, is 6.5 m above the ground. One end of a light inextensible cord of length 4.5 m is attached to \(H\) and the other is attached to a trainee parachutist of mass 90 kg standing on the edge of the platform with the cord straight and taut. The trainee is then projected off the platform with a velocity of \(7 \mathrm {~ms} ^ { - 1 }\) perpendicular to the cord in a downward direction. The motion of the trainee all takes place in a single vertical plane and while the cord is attached to \(H\) it remains straight and taut. When the speed of the trainee reaches \(5.5 \mathrm {~ms} ^ { - 1 }\) the cord is detached from \(H\) and the trainee then moves under the influence of gravity alone until landing on the ground (see diagram).
    \includegraphics[max width=\textwidth, alt={}, center]{857eca7f-c42d-49a9-ac39-a2fb5bcb9cd5-6_615_1211_934_242} The trainee is modelled as a particle and air resistance is modelled as being negligible.
    1. Show that at the instant before the cord is detached from \(H\), the tension in the cord has a magnitude of 1005.5 N . The point on the ground vertically below the edge of the platform is denoted by \(O\). The point on the ground where the trainee lands is denoted by \(T\).
    2. Determine the distance \(O T\). The ground around \(T\) is in fact an elastic mat of thickness 0.5 m which is angled so that it is perpendicular to the direction of motion of the trainee on landing. The mat, which is very rough, is modelled as an elastic spring of natural length 0.5 m . It is assumed that the trainee strikes the mat at ground level and is brought to rest once the mat has been compressed by 0.3 m .
    3. Determine the modulus of elasticity of the mat. Give your answer to the nearest integer.
    OCR Further Mechanics 2022 June Q8
    13 marks Challenging +1.8
    8 Two smooth circular discs, \(A\) and \(B\), have equal radii and are free to move on a smooth horizontal plane. The masses of \(A\) and \(B\) are 1 kg and \(m \mathrm {~kg}\) respectively. \(B\) is initially placed at rest with its centre at the origin, \(O\). \(A\) is projected towards \(B\) with a velocity of \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta\) to the negative \(y\)-axis where \(\tan \theta = \frac { 5 } { 2 }\). At the instant of collision the line joining their centres lies on the \(x\)-axis. There are two straight vertical walls on the plane. One is perpendicular to the \(x\)-axis and the other is perpendicular to the \(y\)-axis. The walls are an equal distance from \(O\) (see diagram).
    \includegraphics[max width=\textwidth, alt={}, center]{857eca7f-c42d-49a9-ac39-a2fb5bcb9cd5-7_944_1241_694_242} After \(A\) and \(B\) have collided with each other, each of them goes on to collide with a wall. Each then rebounds and they collide again at the same place as their first collision, with disc \(B\) again at \(O\). The coefficient of restitution between \(A\) and \(B\) is denoted by \(e\). The coefficient of restitution between \(A\) and the wall that it collides with is also \(e\) while the coefficient of restitution between \(B\) and the wall that it collides with is \(\frac { 5 } { 9 } e\). It is assumed that any resistance to the motion of \(A\) and \(B\) may be ignored.
    1. Explain why it must be the case that the collision between \(A\) and the wall that it collides with is not inelastic.
    2. Show that \(\mathrm { e } = \frac { 1 } { \mathrm {~m} }\).
    3. Show that \(m = \frac { 5 } { 3 }\).
    4. State one limitation of the model used.
    OCR Further Mechanics 2023 June Q1
    8 marks Standard +0.3
    1 One end of a light inextensible string of length 0.8 m is attached to a particle \(P\) of mass \(m \mathrm {~kg}\). The other end of the string is attached to a fixed point \(O\). Initially \(P\) hangs in equilibrium vertically below \(O\). It is then projected horizontally with a speed of \(5.3 \mathrm {~ms} ^ { - 1 }\) so that it moves in a vertical circular path with centre \(O\) (see diagram).
    \includegraphics[max width=\textwidth, alt={}, center]{894be707-4f7b-4647-b209-805522556196-2_686_586_450_248} At a certain instant, \(P\) first reaches the point where the string makes an angle of \(\frac { 1 } { 3 } \pi\) radians with the downward vertical through \(O\).
    1. Show that at this instant the speed of \(P\) is \(4.5 \mathrm {~ms} ^ { - 1 }\).
    2. Find the magnitude and direction of the radial acceleration of \(P\) at this instant.
    3. Find the magnitude of the tangential acceleration of \(P\) at this instant.
    OCR Further Mechanics 2023 June Q2
    11 marks Moderate -0.3
    2 Materials have a measurable property known as the Young's Modulus, E.
    If a force is applied to one face of a block of the material then the material is stretched by a distance called the extension. Young's modulus is defined as the ratio \(\frac { \text { Stress } } { \text { Strain } }\) where Stress is defined as the force per unit area and Strain is the ratio of the extension of the block to the length of the block.
    1. Show that Strain is a dimensionless quantity.
    2. By considering the dimensions of both Stress and Strain determine the dimensions of \(E\). It is suggested that the speed of sound in a material, \(c\), depends only upon the value of Young's modulus for the material, \(E\), the volume of the material, \(V\), and the density (or mass per unit volume) of the material, \(\rho\).
    3. Use dimensional analysis to suggest a formula for \(c\) in terms of \(E , V\) and \(\rho\).
    4. The speed of sound in a certain material is \(500 \mathrm {~ms} ^ { - 1 }\).
      1. Use your formula from part (c) to predict the speed of sound in the material if the value of Young’s modulus is doubled but all other conditions are unchanged.
      2. With reference to your formula from part (c), comment on the effect on the speed of sound in the material if the volume is doubled but all other conditions are unchanged.
    5. Suggest one possible limitation caused by using dimensional analysis to set up the model in part (c).
    OCR Further Mechanics 2023 June Q3
    7 marks Standard +0.8
    3 Two smooth circular discs \(A\) and \(B\) are moving on a smooth horizontal plane when they collide. The mass of \(A\) is 5 kg and the mass of \(B\) is 3 kg . At the instant before they collide,
    • the velocity of \(A\) is \(4 \mathrm {~ms} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line of centres,
    • the velocity of \(B\) is \(6 \mathrm {~ms} ^ { - 1 }\) along the line of centres
      (see diagram).
      \includegraphics[max width=\textwidth, alt={}, center]{894be707-4f7b-4647-b209-805522556196-4_531_1683_651_191}
    The coefficient of restitution for collisions between the two discs is \(\frac { 3 } { 4 }\).
    Determine the angle that the velocity of \(A\) makes with the line of centres after the collision.
    \(4 A B C D\) is a uniform lamina in the shape of a kite with \(\mathrm { BA } = \mathrm { BC } = 0.37 \mathrm {~m} , \mathrm { DA } = \mathrm { DC } = 0.91 \mathrm {~m}\) and \(\mathrm { AC } = 0.7 \mathrm {~m}\) (see diagram). The centre of mass of \(A B C D\) is \(G\).
    1. Explain why \(G\) lies on \(B D\).
    2. Show that the distance of \(G\) from \(B\) is 0.36 m . The lamina \(A B C D\) is freely suspended from the point \(A\).
    3. Determine the acute angle that \(C D\) makes with the horizontal, stating which of \(C\) or \(D\) is higher.
    OCR Further Mechanics 2023 June Q5
    13 marks Challenging +1.3
    5 A particle \(P\) of mass 2 kg moves along the \(x\)-axis.
    At time \(t = 0 , P\) passes through the origin \(O\) with speed \(3 \mathrm {~ms} ^ { - 1 }\).
    At time \(t\) seconds the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\), where \(t \geqslant 0 , x \geqslant 0\) and \(v \geqslant 0\). While \(P\) is in motion the only force acting on \(P\) is a resistive force \(F\) of magnitude \(\left( v ^ { 2 } + 1 \right) \mathrm { N }\) acting in the negative \(x\)-direction.
    1. Find an expression for \(v\) in terms of \(x\).
    2. Determine the distance travelled by \(P\) while its speed drops from \(3 \mathrm {~ms} ^ { - 1 }\) to \(2 \mathrm {~ms} ^ { - 1 }\). Particle \(Q\) is identical to particle \(P\). At a different time, \(Q\) is moving along the \(x\)-axis under the influence of a single constant resistive force of magnitude 1 N . When \(t ^ { \prime } = 0 , Q\) is at the origin and its speed is \(3 \mathrm {~ms} ^ { - 1 }\).
    3. By comparing the motion of \(P\) with the motion of \(Q\) explain why \(P\) must come to rest at some finite time when \(t < 6\) with \(x < 9\).
    4. Sketch the velocity-time graph for \(P\). You do not need to indicate any values on your sketch.
    5. Determine the maximum displacement of \(P\) from \(O\) during \(P\) 's motion.
    OCR Further Mechanics 2023 June Q6
    12 marks Challenging +1.2
    6 A particle \(P\) of mass 3 kg is moving on a smooth horizontal surface under the influence of a variable horizontal force \(\mathbf { F } \mathrm { N }\). At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P , \mathbf { v } \mathrm {~ms} ^ { - 1 }\), is given by $$\mathbf { v } = ( 32 \sinh ( 2 t ) ) \mathbf { i } + ( 32 \cosh ( 2 t ) - 257 ) \mathbf { j } .$$
      1. By considering kinetic energy, determine the work done by \(\mathbf { F }\) over the interval \(0 \leqslant t \leqslant \ln 2\).
      2. Explain the significance of the sign of the answer to part (a)(i).
    2. Determine the rate at which \(\mathbf { F }\) is working at the instant when \(P\) is moving parallel to the i-direction.
    OCR Further Mechanics 2023 June Q7
    7 marks Challenging +1.2
    7 Two particles \(A\) and \(B\) are connected by a light inextensible string of length 1.26 m . Particle \(A\) has a mass of 1.25 kg and moves on a smooth horizontal table in a circular path of radius 0.9 m and centre \(O\). The string passes through a small smooth hole at \(O\). Particle \(B\) has a mass of 2 kg and moves in a horizontal circle as shown in the diagram. The angle that the portion of string below the table makes with the downwards vertical through \(O\) is \(\theta\), where \(\cos \theta = \frac { 4 } { 5 }\) (see diagram).
    \includegraphics[max width=\textwidth, alt={}, center]{894be707-4f7b-4647-b209-805522556196-6_369_810_493_244}
    1. Determine the angular speed of \(A\) and the angular speed of \(B\). At the start of the motion, \(A , O\) and \(B\) all lie in the same vertical plane.
    2. Find the first subsequent time when \(A , O\) and \(B\) all lie in the same vertical plane.
    OCR Further Mechanics 2023 June Q8
    8 marks Challenging +1.8
    8 One end of a light elastic string of natural length 2.1 m and modulus of elasticity 4.8 N is attached to a particle, \(P\), of mass 1.75 kg . The other end of the string is attached to a fixed point, \(O\), which is on a rough inclined plane. The angle between the plane and the horizontal is \(\theta\) where \(\sin \theta = \frac { 3 } { 5 }\). The coefficient of friction between \(P\) and the plane is 0.732 . Particle \(P\) is placed on the plane at \(O\) and then projected down a line of greatest slope of the plane with an initial speed of \(2.4 \mathrm {~ms} ^ { - 1 }\). Determine the distance that \(P\) has travelled from \(O\) at the instant when it first comes to rest. You can assume that during its motion \(P\) does not reach the bottom of the inclined plane.
    OCR Further Mechanics 2024 June Q1
    8 marks Standard +0.3
    1 A particle \(P\) of mass 12.5 kg is moving on a smooth horizontal plane when it collides obliquely with a fixed vertical wall. At the instant before the collision, the velocity of \(P\) is \(- 5 \mathbf { i } + 12 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).
    At the instant after the collision, the velocity of \(P\) is \(\mathbf { i } + 4 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).
    1. Find the magnitude of the momentum of \(P\) before the collision.
    2. Find, in vector form, the impulse that the wall exerts on \(P\).
    3. State, in vector form, the impulse that \(P\) exerts on the wall.
    4. Find in either order.
      • The magnitude of the impulse that the wall exerts on \(P\).
      • The angle between \(\mathbf { i }\) and the impulse that the wall exerts on \(P\).
    OCR Further Mechanics 2024 June Q2
    5 marks Challenging +1.2
    2 One end of a light elastic string of natural length 1.4 m and modulus of elasticity 20 N is attached to a small object \(B\) of mass 2.5 kg . The other end of the string is attached to a fixed point \(O\). Object \(B\) is projected vertically upwards from \(O\) with a speed of \(u \mathrm {~ms} ^ { - 1 }\).
    1. State one assumption required to model the motion of \(B\). The greatest height above \(O\) achieved by \(B\) is 8.1 m .
    2. Determine the value of \(u\).
    OCR Further Mechanics 2024 June Q3
    7 marks Standard +0.3
    3 The mass of a truck is 6000 kg and the maximum power that its engine can generate is 90 kW . In a model of the motion of the truck it is assumed that while it is moving the total resistance to its motion is constant. At first the truck is driven along a straight horizontal road. The greatest constant speed that it can be driven at when it is using maximum power is \(25 \mathrm {~ms} ^ { - 1 }\).
    1. Find the value of the resistance to motion. The truck is being driven along the horizontal road with the engine working at 60 kW .
    2. Find the acceleration of the truck at the instant when its speed is \(10 \mathrm {~ms} ^ { - 1 }\). The truck is now driven down a straight road which is inclined at an angle \(\theta\) below the horizontal. The greatest constant speed that the truck can be driven at maximum power is \(40 \mathrm {~ms} ^ { - 1 }\).
    3. Determine the value of \(\theta\).
    OCR Further Mechanics 2024 June Q4
    15 marks Standard +0.8
    4 A particle, \(P\), of mass 6 kg is attached to one end of a light inextensible rod of length 2.4 m . The other end of the rod is smoothly hinged at a fixed point \(O\) and the rod is free to rotate in any direction. Initially, \(P\) is at rest, vertically below \(O\), when it is projected horizontally with a speed of \(12 \mathrm {~ms} ^ { - 1 }\). It subsequently describes complete vertical circles with \(O\) as the centre.
    \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-3_611_517_536_246} The angle that the rod makes with the downward vertical through \(O\) at each instant is denoted by \(\theta\) and \(A\) is the point which \(P\) passes through where \(\theta = 40 ^ { \circ }\) (see diagram).
    1. Find the tangential acceleration of \(P\) at \(A\), stating its direction.
    2. Determine the radial acceleration of \(P\) at \(A\), stating its direction.
    3. Find the magnitude of the force in the rod when \(P\) is at \(A\), stating whether the rod is in tension or compression. The motion is now stopped when \(P\) is at \(A\), and \(P\) is then projected in such a way that it now describes horizontal circles at a constant speed with \(\theta = 40 ^ { \circ }\) (see diagram).
      \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-3_403_524_1877_242}
    4. Find the speed of \(P\).
    5. Explain why, wherever \(P\) 's motion is initiated from and whatever its initial velocity, it is not possible for \(P\) to describe horizontal circles at constant speed with \(\theta = 90 ^ { \circ }\).
    OCR Further Mechanics 2024 June Q5
    6 marks Standard +0.3
    5 In this question you may assume that if \(x\) and \(y\) are any physical quantities then \(\left[ \frac { \mathrm { dy } } { \mathrm { dx } } \right] = \left[ \frac { \mathrm { y } } { \mathrm { x } } \right]\). A machine drives a piston of mass \(m\) into a vertical cylinder. The equation below is suggested to model the power developed by the machine, \(P\), while it is not doing any other external work. $$\mathrm { P } = \mathrm { k } _ { 1 } \mathrm { mv } \frac { \mathrm { dv } } { \mathrm { dt } } + \mathrm { k } _ { 2 } \mathrm { mgv } + \mathrm { k } _ { 3 } \mathrm { E }$$ in which
    • \(v\) is the velocity of the piston at a given time,
    • \(g\) is the acceleration due to gravity,
    • \(E\) is the rate at which heat energy is lost to the surroundings,
    • \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are dimensionless constants.
    Determine whether the equation is dimensionally consistent. Show all the steps in your argument.
    OCR Further Mechanics 2024 June Q6
    12 marks Challenging +1.2
    6 Two identical spheres, \(A\) and \(B\), each of mass \(m \mathrm {~kg}\), are moving directly towards each other along the same straight line on a smooth horizontal surface until they collide. Just before they collide, the speeds of \(A\) and \(B\) are \(20 \mathrm {~ms} ^ { - 1 }\) and \(10 \mathrm {~ms} ^ { - 1 }\) respectively. The coefficient of restitution between \(A\) and \(B\) is \(e\).
    1. By finding, in terms of \(e\), an expression for the velocity of \(B\) after the collision, show that the direction of motion of \(B\) is reversed by the collision. After the collision between \(A\) and \(B\), which is not perfectly elastic, \(B\) goes on to collide directly with a fixed, vertical wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 5 } e\). After the collision between \(B\) and the wall, there are no further collisions between \(A\) and \(B\).
    2. Determine the range of possible values of \(e\).
      \(7 \quad\) A body \(B\) of mass 1.5 kg is moving along the \(x\)-axis. At the instant that it is at the origin, \(O\), its velocity is \(u \mathrm {~ms} ^ { - 1 }\) in the positive \(x\)-direction. At any instant, the resistance to the motion of \(B\) is modelled as being directly proportional to \(v ^ { 2 }\) where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of \(B\) at that instant. The resistance to motion is the only horizontal force acting on \(B\). At an instant when \(B\) 's velocity is \(2 \mathrm {~ms} ^ { - 1 }\), the resistance to its motion is 24 N .
    OCR Further Mechanics 2024 June Q8
    8 marks Challenging +1.8
    8 A shape, \(S\), is formed by attaching a particle of mass \(2 m \mathrm {~kg}\) to the vertex of a uniform solid cone of mass \(8 m \mathrm {~kg}\). The height of the cone is \(h \mathrm {~m}\) and the radius of the base of the cone is 1.1 m .
    1. Explain why the centre of mass of \(S\) must lie on the central axis of the cone. Two strings are attached to \(S\), one at the vertex of the cone and one at \(A\) which is a point on the edge of the base of \(S\). The other ends of the strings are attached to a horizontal ceiling in such a way that the strings are both vertical. The string attached to \(S\) at \(A\) is inextensible and has length 1.6 m . The string attached to \(S\) at the vertex is elastic with modulus of elasticity 8 mgN . Shape \(S\) is in equilibrium with its axis horizontal (see diagram).
      \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-6_654_1541_879_244}
    2. Determine the natural length of the elastic string.
    OCR Further Mechanics 2020 November Q1
    5 marks Standard +0.3
    1 A force of \(\binom { 2 } { 10 } \mathrm {~N}\) is the only horizontal force acting on a particle \(P\) of mass 1.25 kg as it moves in a horizontal plane. Initially \(P\) is at the origin, \(O\), and 5 seconds later it is at the point \(A ( 50,140 )\). The units of the coordinate system are metres.
    1. Calculate the work done by the force during these 5 seconds.
    2. Calculate the average power generated by the force during these 5 seconds. The speed of \(P\) at \(O\) is \(10 \mathrm {~ms} ^ { - 1 }\).
    3. Calculate the speed of \(P\) at \(A\).
    OCR Further Mechanics 2020 November Q2
    6 marks Standard +0.8
    2 A bungee jumper of mass 80 kg steps off a high bridge with an elastic rope attached to her ankles. She is assumed to fall vertically from rest and the air resistance she experiences is modelled as a constant force of 32 N . The rope has natural length 4 m and modulus of elasticity 470 N . By considering energy, determine the total distance she falls before first coming to instantaneous rest.
    OCR Further Mechanics 2020 November Q3
    7 marks Challenging +1.2
    3 One end of a light inextensible string of length 0.75 m is attached to a particle \(A\) of mass 2.8 kg . The other end of the string is attached to a fixed point \(O\). \(A\) is projected horizontally with speed \(6 \mathrm {~ms} ^ { - 1 }\) from a point 0.75 m vertically above \(O\) (see Fig. 3). When \(O A\) makes an angle \(\theta\) with the upward vertical the speed of \(A\) is \(v \mathrm {~ms} ^ { - 1 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{831ba5da-df19-43bb-b163-02bbddb4e2b8-2_388_220_1790_244} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure}
    1. Show that \(v ^ { 2 } = 50.7 - 14.7 \cos \theta\).
    2. Given that the string breaks when the tension in it reaches 200 N , find the angle that \(O A\) turns through between the instant that \(A\) is projected and the instant that the string breaks.
    OCR Further Mechanics 2020 November Q4
    15 marks Standard +0.3
    4 The resistive force, \(F\), on a sphere falling through a viscous fluid is thought to depend on the radius of the sphere, \(r\), the velocity of the sphere, \(v\), and the viscosity of the fluid, \(\eta\). You are given that \(\eta\) is measured in \(\mathrm { N } \mathrm { m } ^ { - 2 } \mathrm {~s}\).
    1. By considering its units, find the dimensions of viscosity. A model of the resistive force suggests the following relationship: \(\mathbf { F } = 6 \pi \eta ^ { \alpha } \mathbf { r } ^ { \beta } \mathbf { v } ^ { \gamma }\).
    2. Explain whether or not it is possible to use dimensional analysis to verify that the constant \(6 \pi\) is correct.
    3. Use dimensional analysis to find the values of \(\alpha , \beta\) and \(\gamma\). A sphere of radius \(r\) and mass \(m\) falls vertically from rest through the fluid. After a time \(t\) its velocity is \(v\).
    4. By setting up and solving a differential equation, show that \(\mathrm { e } ^ { - \mathrm { kt } } = \frac { \mathrm { g } - \mathrm { kv } } { \mathrm { g } }\) where \(\mathrm { k } = \frac { 6 \pi \eta \mathrm { r } } { \mathrm { m } }\). As the time increases, the velocity of the sphere tends towards a limit called the terminal velocity.
    5. Find, in terms of \(g\) and \(k\), the terminal velocity of the sphere. In a sequence of experiments the sphere is allowed to fall through fluids of different viscosity, ranging from small to very large, with all other conditions being constant. The terminal velocity of the sphere through each fluid is measured.
    6. Describe how, according to the model, the terminal velocity of the sphere changes as the viscosity of the fluid through which it falls increases.
    OCR Further Mechanics 2020 November Q5
    9 marks Challenging +1.2
    5 The cover of a children's book is modelled as being a uniform lamina \(L . L\) occupies the region bounded by the \(x\)-axis, the curve \(\mathrm { y } = 6 + \sin \mathrm { x }\) and the lines \(x = 0\) and \(x = 5\) (see Fig. 5.1). The centre of mass of \(L\) is at the point ( \(\mathrm { x } , \mathrm { y }\) ). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{831ba5da-df19-43bb-b163-02bbddb4e2b8-4_659_540_397_244} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
    \end{figure}
    1. Show that \(\bar { X } = 2.36\), correct to 3 significant figures.
    2. Find \(\bar { y }\), giving your answer correct to 3 significant figures. The cover of the book weighs 6 N . \(A\) is the point on the cover with coordinates \(( 3 , \bar { y } )\) and \(B\) is the point on the cover with coordinates \(( 5 , \bar { y } )\). A small badge of weight 2 N is attached to the cover at \(A\). The side of \(L\) along the \(y\)-axis is attached to the rest of the book and the book is placed on a rough horizontal plane. The attachment of the cover to the book is modelled as a hinge. The cover is held in equilibrium at an angle of \(\frac { 1 } { 3 } \pi\) radians to the horizontal by a force of magnitude \(P N\) acting at \(B\) perpendicular to the cover (see Fig. 5.2). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{831ba5da-df19-43bb-b163-02bbddb4e2b8-4_444_899_1889_246} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
      \end{figure}
    3. State two additional modelling assumptions, one about the attachment of the cover and one about the badge, which are necessary to allow the value of \(P\) to be determined.
    4. Using the modelling assumptions, determine the value of \(P\) giving your answer correct to 3 significant figures.
    OCR Further Mechanics 2020 November Q6
    12 marks Challenging +1.8
    6 Two smooth circular discs \(A\) and \(B\) are moving on a horizontal plane. The masses of \(A\) and \(B\) are 3 kg and 4 kg respectively. At the instant before they collide
    • the velocity of \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line joining their centres,
    • the velocity of \(B\) is \(5 \mathrm {~ms} ^ { - 1 }\) towards \(A\) along the line joining their centres (see Fig. 6).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{831ba5da-df19-43bb-b163-02bbddb4e2b8-5_490_1047_470_244} \captionsetup{labelformat=empty} \caption{Fig. 6}
    \end{figure} Given that the velocity of \(A\) after the collision is perpendicular to the velocity of \(A\) before the collision, find
    1. the coefficient of restitution between \(A\) and \(B\),
    2. the total loss of kinetic energy as a result of the collision.