Questions — OCR Further Mechanics (94 questions)

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OCR Further Mechanics 2019 June Q1
1 The region bounded by the \(x\)-axis, the curve \(\mathrm { y } = \sqrt { 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x - 20 }\) and the lines \(x = 1\) and \(x = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution \(R\). The centre of mass of \(R\) is the point \(G\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-2_569_463_507_280}
  1. Explain why the \(y\)-coordinate of \(G\) is 0 .
  2. Find the \(x\)-coordinate of \(G\).
    \(P\) is a point on the edge of the curved surface of \(R\) where \(x = 4 . R\) is freely suspended from \(P\) and hangs in equilibrium.
  3. Find the angle between the axis of symmetry of \(R\) and the vertical.
OCR Further Mechanics 2019 June Q2
3 marks
2 A solenoid is a device formed by winding a wire tightly around a hollow cylinder so that the wire forms (approximately) circular loops along the cylinder (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-2_161_691_1681_246} When the wire carries an electrical current a magnetic field is created inside the solenoid which can cause a particle which is moving inside the solenoid to accelerate. A student is carrying out experiments on particles moving inside solenoids. His professor suggests that, for a particle of mass \(m\) moving with speed \(v\) inside a solenoid of length \(h\), the acceleration \(a\) of the particle can be modelled by a relationship of the form \(a = \mathrm { km } ^ { \alpha } \mathrm { v } ^ { \beta } \mathrm { h } ^ { \gamma }\), where \(k\) is a constant. The professor tells the student that \([ k ] = \mathrm { MLT } ^ { - 1 }\).
  1. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
  2. The mass of an electron is \(9.11 \times 10 ^ { - 31 } \mathrm {~kg}\) and the mass of a proton is \(1.67 \times 10 ^ { - 27 } \mathrm {~kg}\). For an electron and a proton moving inside the same solenoid with the same speed, use the model to find the ratio of the acceleration of the electron to the acceleration of the proton. [3]
  3. The professor tells the student that \(a\) also depends on the number of turns or loops of wire, \(N\), that the solenoid has. Explain why dimensional analysis cannot be used to determine the dependence of \(a\) on \(N\). [1
OCR Further Mechanics 2019 June Q3
3 A particle \(Q\) of mass \(m \mathrm {~kg}\) is acted on by a single force so that it moves with constant acceleration \(\mathbf { a } = \binom { 1 } { 2 } \mathrm {~ms} ^ { - 2 }\). Initially \(Q\) is at the point \(O\) and is moving with velocity \(\mathbf { u } = \binom { 2 } { - 5 } \mathrm {~ms} ^ { - 1 }\). After \(Q\) has been moving for 5 seconds it reaches the point \(A\).
  1. Use the equation \(\mathbf { v . v } = \mathbf { u . u } + 2 \mathbf { a x }\) to show that at \(A\) the kinetic energy of \(Q\) is 37 m J .
    1. Show that the power initially generated by the force is - 8 mW .
    2. The power in part (b)(i) is negative. Explain what this means about the initial motion of \(Q\).
    1. Find the time at which the power generated by the force is instantaneously zero.
    2. Find the minimum kinetic energy of \(Q\) in terms of \(m\).
OCR Further Mechanics 2019 June Q4
4 A right circular cone \(C\) of height 4 m and base radius 3 m has its base fixed to a horizontal plane. One end of a light elastic string of natural length 2 m and modulus of elasticity 32 N is fixed to the vertex of \(C\). The other end of the string is attached to a particle \(P\) of mass 2.5 kg .
\(P\) moves in a horizontal circle with constant speed and in contact with the smooth curved surface of \(C\). The extension of the string is 1.5 m .
  1. Find the tension in the string.
  2. Find the speed of \(P\).
OCR Further Mechanics 2019 June Q5
5 A particle \(P\) of mass 4.5 kg is free to move along the \(x\)-axis. In a model of the motion it is assumed that \(P\) is acted on by two forces:
  • a constant force of magnitude \(f \mathrm {~N}\) in the positive \(x\) direction;
  • a resistance to motion, \(R \mathrm {~N}\), whose magnitude is proportional to the speed of \(P\).
At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). When \(t = 0 , P\) is at the origin \(O\) and is moving in the positive direction with speed \(u \mathrm {~ms} ^ { - 1 }\), and when \(v = 5 , R = 2\).
  1. Show that, according to the model, \(\frac { d v } { d t } = \frac { 10 f - 4 v } { 45 }\).
    1. By solving the differential equation in part (a), show that \(\mathrm { v } = \frac { 1 } { 2 } \left( 5 \mathrm { f } - ( 5 \mathrm { f } - 2 \mathrm { u } ) \mathrm { e } ^ { - \frac { 4 } { 45 } \mathrm { t } } \right)\).
    2. Describe briefly how, according to the model, the speed of \(P\) varies over time in each of the following cases.
      • \(\mathrm { u } < 2.5 \mathrm { f }\)
  3. \(\mathrm { u } = 2.5 \mathrm { f }\)
  4. \(u > 2.5 f\)
  5. In the case where \(\mathrm { u } = 2 \mathrm { f }\), find in terms of \(f\) the exact displacement of \(P\) from \(O\) when \(t = 9\).
OCR Further Mechanics 2019 June Q6
6 Two particles \(A\) and \(B\), of masses \(m \mathrm {~kg}\) and 1 kg respectively, are connected by a light inextensible string of length \(d \mathrm {~m}\) and placed at rest on a smooth horizontal plane a distance of \(\frac { 1 } { 2 } d \mathrm {~m}\) apart. \(B\) is then projected horizontally with speed \(v \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to \(A B\).
  1. Show that, at the instant that the string becomes taut, the magnitude of the instantaneous impulse in the string, \(I \mathrm { Ns }\), is given by \(\mathrm { I } = \frac { \sqrt { 3 } \mathrm { mv } } { 2 ( 1 + \mathrm { m } ) }\).
  2. Find, in terms of \(m\) and \(v\), the kinetic energy of \(B\) at the instant after the string becomes taut. Give your answer as a single algebraic fraction.
  3. In the case where \(m\) is very large, describe, with justification, the approximate motion of \(B\) after the string becomes taut.
OCR Further Mechanics 2019 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-4_330_1061_989_267} The flat surface of a smooth solid hemisphere of radius \(r\) is fixed to a horizontal plane on a planet where the acceleration due to gravity is denoted by \(\gamma\). \(O\) is the centre of the flat surface of the hemisphere. A particle \(P\) is held at a point on the surface of the hemisphere such that the angle between \(O P\) and the upward vertical through \(O\) is \(\alpha\), where \(\cos \alpha = \frac { 3 } { 4 }\).
\(P\) is then released from rest. \(F\) is the point on the plane where \(P\) first hits the plane (see diagram).
  1. Find an exact expression for the distance \(O F\). The acceleration due to gravity on and near the surface of the planet Earth is roughly \(6 \gamma\).
  2. Explain whether \(O F\) would increase, decrease or remain unchanged if the action were repeated on the planet Earth. \section*{END OF QUESTION PAPER}
OCR Further Mechanics 2022 June Q1
1 A car has mass 1200 kg . The total resistance to the car's motion is constant and equal to 250 N .
  1. The car is driven along a straight horizontal road with its engine working at 10 kW . Find the acceleration of the car at the instant that its speed is \(5 \mathrm {~ms} ^ { - 1 }\). The maximum power that the car's engine can generate is 20 kW .
  2. Find the greatest constant speed at which the car can be driven along a straight horizontal road. The car is driven up a straight road which is inclined at an angle \(\theta\) above the horizontal where \(\sin \theta = 0.05\).
  3. Find the greatest constant speed at which the car can be driven up this road.
OCR Further Mechanics 2022 June Q2
2 The coordinates of two points, \(A\) and \(B\), are \(( - 1,6 )\) and \(( 5,12 )\) respectively, where the units of the coordinate axes are metres. A particle \(P\) moves from \(A\) to \(B\) under the action of several forces. The force \(\mathbf { F } = 7 \mathbf { i } - 2 \mathbf { j } \mathbf { N }\) is one of the forces acting on \(P\).
  1. Calculate the work done by \(\mathbf { F }\) on \(P\) as \(P\) moves from \(A\) to \(B\). At the instant when \(P\) reaches \(B\) its velocity is \(- \mathbf { i } - 5 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).
  2. Find the power generated by \(\mathbf { F }\) at the instant that \(P\) reaches \(B\). One end of a light elastic string was attached to the origin of the coordinate system and the other to \(P\) when \(P\) was at \(A\), before it moved to \(B\). The natural length of the string is 8 m and its modulus of elasticity is 24 N .
  3. At the instant that \(P\) reaches \(B\), find the following.
    • The tension in the string
    • The elastic potential energy stored in the string
OCR Further Mechanics 2022 June Q3
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
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
    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
    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
    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
    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
    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
    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
    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
    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 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 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
    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
    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
    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
    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 \(\theta = 90 ^ { \circ }\).