Questions — OCR Further Mechanics AS (51 questions)

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OCR Further Mechanics AS 2018 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{5960a9cf-2c51-4c07-9973-c29604762df7-2_540_269_395_897} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length 3.2 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 making an angle of \(15 ^ { \circ }\) with the vertical. It is then projected with velocity \(1.2 \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to \(O P\) and with a downwards component so that it begins to move in a vertical circle (see diagram). In the ensuing motion the string remains taut and the angle it makes with the downwards vertical through \(O\) is denoted by \(\theta ^ { \circ }\).
  1. Find the speed of \(P\) at the point on its path vertically below \(O\).
  2. Find the value of \(\theta\) at the point where \(P\) first comes to instantaneous rest.
OCR Further Mechanics AS 2018 June Q2
2 A particle \(P\) of mass 3.5 kg is moving down a line of greatest slope of a rough inclined plane. At the instant that its speed is \(2.1 \mathrm {~ms} ^ { - 1 } P\) is at a point \(A\) on the plane. At that instant an impulse of magnitude 33.6 Ns , directed up the line of greatest slope, acts on \(P\).
  1. Show that as a result of the impulse \(P\) starts moving up the plane with a speed of \(7.5 \mathrm {~ms} ^ { - 1 }\). While still moving up the plane, \(P\) has speed \(1.5 \mathrm {~ms} ^ { - 1 }\) at a point \(B\) where \(A B = 4.2 \mathrm {~m}\). The plane is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The frictional force exerted by the plane on \(P\) is modelled as constant.
  2. Calculate the work done against friction as \(P\) moves from \(A\) to \(B\).
  3. Hence find the magnitude of the frictional force acting on \(P\).
    \(P\) first comes to instantaneous rest at point \(C\) on the plane.
  4. Calculate \(A C\).
OCR Further Mechanics AS 2018 June Q3
3 A particle moves in a straight line with constant acceleration. Its initial and final velocities are \(u\) and \(v\) respectively and at time \(t\) its displacement from its starting position is \(s\). An equation connecting these quantities is \(s = k \left( u ^ { \alpha } + v ^ { \beta } \right) t ^ { \gamma }\), where \(k\) is a dimensionless constant.
  1. Use dimensional analysis to find the values of \(\alpha , \beta\) and \(\gamma\).
  2. By considering the case where the acceleration is zero, determine the value of \(k\).
OCR Further Mechanics AS 2018 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{5960a9cf-2c51-4c07-9973-c29604762df7-3_218_1335_251_367} Three particles \(A\), \(B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are \(1.2 \mathrm {~kg} , 1.8 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively (see diagram). The coefficient of restitution in collisions between any two of them is \(\frac { 3 } { 4 }\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving with a velocity of \(4.0 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\).
  1. Show that immediately after the collision between \(A\) and \(B\) the speed of \(B\) is \(2.8 \mathrm {~ms} ^ { - 1 }\).
  2. Find the velocity of \(A\) immediately after this collision.
    \(B\) subsequently collides with \(C\).
  3. Find, in terms of \(m\), the velocity of \(B\) after its collision with \(C\).
  4. Given that the direction of motion of \(B\) is reversed by the collision with \(C\), find the range of possible values of \(m\).
OCR Further Mechanics AS 2018 June Q5
5 The engine of a car of mass 1200 kg produces a maximum power of 40 kW .
In an initial model of the motion of the car the total resistance to motion is assumed to be constant.
  1. Given that the greatest steady speed of the car on a straight horizontal road is \(42 \mathrm {~ms} ^ { - 1 }\), find the magnitude of the resistance force. The car is attached to a trailer of mass 200 kg by a light rigid horizontal tow bar. The greatest steady speed of the car and trailer on the road is now \(30 \mathrm {~ms} ^ { - 1 }\). The resistance to motion of the trailer may also be assumed constant.
  2. Find the magnitude of the resistance force on the trailer. The car and trailer again travel along the road. At one instant their speed is \(15 \mathrm {~ms} ^ { - 1 }\) and their acceleration is \(0.57 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. (a) Find the power of the engine of the car at this instant.
    (b) Find the magnitude of the tension in the tow bar at this instant. In a refined model of the motion of the car and trailer the resistance to the motion of each is assumed to be zero until they reach a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the speed is \(10 \mathrm {~ms} ^ { - 1 }\) or above the same constant resistance forces as in the first model are assumed to apply to each. The car and trailer start at rest on the road and accelerate, using maximum power.
  4. Without carrying out any further calculations,
    (a) explain whether the time taken to attain a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) would be predicted to be lower, the same or higher using the refined model compared with the original model,
    (b) explain whether the greatest steady speed of the system would be predicted to be lower, the same or higher using the refined model compared with the original model.
OCR Further Mechanics AS 2018 June Q6
6 Two particles \(A\) and \(B\) are connected by a light inextensible string. Particle \(A\) has mass 1.2 kg and moves on a smooth horizontal table in a circular path of radius 0.6 m and centre \(O\). The string passes through a small smooth hole at \(O\). Particle \(B\) moves in a horizontal circle in such a way that it is always vertically below \(A\). The angle that the portion of the 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]{5960a9cf-2c51-4c07-9973-c29604762df7-4_519_803_484_632}
  1. Find the time taken for the particles to perform a complete revolution.
  2. Find the mass of \(B\). \section*{END OF QUESTION PAPER}
OCR Further Mechanics AS 2019 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{74bada9e-60cf-4ed4-abd0-4be155b7cf81-2_533_424_402_246} A smooth wire is shaped into a circle of radius 2.5 m which is fixed in a vertical plane with its centre at a point \(O\). A small bead \(B\) is threaded onto the wire. \(B\) is held with \(O B\) vertical and is then projected horizontally with an initial speed of \(8.4 \mathrm {~ms} ^ { - 1 }\) (see diagram).
  1. Find the speed of \(B\) at the instant when \(O B\) makes an angle of 0.8 radians with the downward vertical through \(O\).
  2. Determine whether \(B\) has sufficient energy to reach the point on the wire vertically above \(O\).
OCR Further Mechanics AS 2019 June Q2
2 A particle \(A\) of mass 3.6 kg is attached by a light inextensible string to a particle \(B\) of mass 2.4 kg .
\(A\) and \(B\) are initially at rest, with the string slack, on a smooth horizontal surface. \(A\) is projected directly away from \(B\) with a speed of \(7.2 \mathrm {~ms} ^ { - 1 }\).
  1. Calculate the speed of \(A\) after the string becomes taut.
  2. Find the impulse exerted on \(A\) at the instant that the string becomes taut.
  3. Find the loss in kinetic energy as a result of the string becoming taut.
OCR Further Mechanics AS 2019 June Q3
3 A car of mass 1500 kg has an engine with maximum power 60 kW . When the car is travelling at \(10 \mathrm {~ms} ^ { - 1 }\) along a straight horizontal road using maximum power, its acceleration is \(3.3 \mathrm {~ms} ^ { - 2 }\). In an initial model of the motion of the car it is assumed that the resistance to motion is constant.
  1. Using this initial model, find the greatest possible steady speed of the car along the road. In a refined model the resistance to motion is assumed to be proportional to the speed of the car.
  2. Using this refined model, find the greatest possible steady speed of the car along the road. The greatest possible steady speed of the car on the road is measured and found to be \(21.6 \mathrm {~ms} ^ { - 1 }\).
  3. Explain what this value means about the models used in parts (a) and (b).
OCR Further Mechanics AS 2019 June Q4
4 A student is studying the speed of sound, \(u\), in a gas under different conditions. He assumes that \(u\) depends on the pressure, \(p\), of the gas, the density, \(\rho\), of the gas and the wavelength, \(\lambda\), of the sound in the relationship \(u = k p ^ { \alpha } \rho ^ { \beta } \lambda ^ { \gamma }\), where \(k\) is a dimensionless constant. (The wavelength of a sound is the distance between successive peaks in the sound wave.)
  1. Use the fact that density is mass per unit volume to find \([ \rho ]\).
  2. Given that the units of \(p\) are \(\mathrm { Nm } ^ { - 2 }\), determine the values of \(\alpha , \beta\) and \(\gamma\).
  3. Comment on what the value of \(\gamma\) means about how fast sounds of different wavelengths travel through the gas. The student carries out two experiments, \(A\) and \(B\), to measure \(u\). Only the density of the gas varies between the experiments, all other conditions being unchanged. He finds that the value of \(u\) in experiment \(B\) is double the value in experiment \(A\).
  4. By what factor has the density of the gas in experiment \(A\) been multiplied to give the density of the gas in experiment \(B\) ?
    \includegraphics[max width=\textwidth, alt={}, center]{74bada9e-60cf-4ed4-abd0-4be155b7cf81-4_659_401_269_251} As shown in the diagram, \(A B\) is a long thin rod which is fixed vertically with \(A\) above \(B\). One end of a light inextensible string of length 1 m is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m _ { 1 } \mathrm {~kg}\). One end of another light inextensible string of length 1 m is also attached to \(P\). Its other end is attached to a small smooth ring \(R\), of mass \(m _ { 2 } \mathrm {~kg}\), which is free to move on \(A B\). Initially, \(P\) moves in a horizontal circle of radius 0.6 m with constant angular velocity \(\omega _ { \text {rads } } { } ^ { - 1 }\). The magnitude of the tension in string \(A P\) is denoted by \(T _ { 1 } \mathrm {~N}\) while that in string \(P R\) is denoted by \(T _ { 2 } \mathrm {~N}\).
OCR Further Mechanics AS 2019 June Q6
6 Particles \(A\) of mass \(2 m\) and \(B\) of mass \(m\) are on a smooth horizontal floor. \(A\) is moving with speed \(u\) directly towards a vertical wall, and \(B\) is at rest between \(A\) and the wall (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{74bada9e-60cf-4ed4-abd0-4be155b7cf81-5_224_828_354_244} A collides directly with \(B\). The coefficient of restitution in this collision is \(\frac { 1 } { 2 }\).
\(B\) then collides with the wall, rebounds, and collides with \(A\) for a second time.
  1. Show that the speed of \(B\) after its second collision with \(A\) is \(\frac { 1 } { 2 } u\). The first collision between \(A\) and \(B\) occurs at a distance \(d\) from the wall. The second collision between \(A\) and \(B\) occurs at a distance \(\frac { 1 } { 5 } d\) from the wall.
  2. Find the coefficient of restitution for the collision between \(B\) and the wall.
OCR Further Mechanics AS 2022 June Q1
1 Two stones, A and B , are sliding along the same straight line on a horizontal sheet of ice. Stone A, of mass 50 kg , is moving with a constant velocity of \(2.1 \mathrm {~ms} ^ { - 1 }\) towards stone B. Stone B, of mass 70 kg , is moving with a constant velocity of \(0.8 \mathrm {~ms} ^ { - 1 }\) towards stone A. A and B collide directly. Immediately after their collision stone A's velocity is \(0.35 \mathrm {~ms} ^ { - 1 }\) in the same direction as its velocity before the collision.
  1. Find the speed of stone B immediately after the collision.
  2. Find the coefficient of restitution for the collision.
  3. Find the total loss of kinetic energy caused by the collision.
  4. Explain whether the collision was perfectly elastic.
OCR Further Mechanics AS 2022 June Q2
2 A hockey puck of mass 0.2 kg is sliding down a rough slope which is inclined at \(10 ^ { \circ }\) to the horizontal. At the instant that its velocity is \(14 \mathrm {~ms} ^ { - 1 }\) directly down the slope it is hit by a hockey stick. Immediately after it is hit its velocity is \(24 \mathrm {~ms} ^ { - 1 }\) directly up the slope.
  1. Find the magnitude of the impulse exerted by the hockey stick on the puck. After it has been hit, the puck first comes to instantaneous rest when it has travelled 15 m up the slope. While the puck is moving up the slope, the resistance to its motion has constant magnitude \(R \mathrm {~N}\).
  2. Use an energy method to determine the value of \(R\).
OCR Further Mechanics AS 2022 June Q3
3 A smooth wire is shaped into a circle of radius 4.2 m which is fixed in a vertical plane with its centre at a point \(O\). A small bead \(B\) is threaded onto the wire. \(B\) is held so that \(O B\) makes an angle of \(\frac { 1 } { 3 } \pi\) radians with the downwards vertical through \(O\).
\(B\) is projected downwards along the wire with initial speed \(u \mathrm {~ms} ^ { - 1 }\) (see diagram). In its subsequent motion \(B\) describes complete circles about \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{98053e88-1aec-4b0d-ae5f-ece4ad340266-3_493_665_561_242} Given that the lowest speed of \(B\) in its motion is \(4 \mathrm {~ms} ^ { - 1 }\) determine the value of \(u\).
OCR Further Mechanics AS 2022 June Q4
4 A cyclist is riding a bicycle along a straight road which is inclined at an angle of \(4 ^ { \circ }\) to the horizontal. The cyclist is working at a constant rate of 250 W . The combined mass of the cyclist and bicycle is 80 kg and the resistance to their motion is a constant 70 N . Determine the maximum constant speed at which the cyclist can ride the bicycle
  • up the hill, and
  • down the hill.
OCR Further Mechanics AS 2022 June Q5
5 One end of a light inextensible string of length 3.5 m is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass \(0.45 \mathrm {~kg} . P\) moves with constant speed in a circular path on the plane with the string taut. The string will break if the tension in it exceeds 70 N . Determine the minimum possible time in which \(P\) can describe a complete circle about \(O\).
OCR Further Mechanics AS 2022 June Q6
6 A particle moves in a straight line with constant acceleration \(a\). Its initial velocity is \(u\) and at time \(t\) its velocity is \(v\). It is assumed that \(v\) depends only on \(u , a\) and \(t\).
  1. Assuming that this dependency is of the form \(\mathrm { u } ^ { \alpha } \mathrm { a } ^ { \beta } \mathrm { t } ^ { \gamma }\), use dimensional analysis to find \(\alpha\) and \(\gamma\) in terms of \(\beta\).
  2. By noting that the graph of \(v\) against \(t\) must be a straight line, determine the possible values of \(\beta\). You may assume that the units of the given quantities are the corresponding SI units.
  3. By considering \(v\) when \(t = 0\) seconds and when \(t = 1\) second, derive the equation of motion \(\mathrm { v } = \mathrm { u } + \mathrm { at }\), explaining your reasoning.
OCR Further Mechanics AS 2022 June Q7
7 Two particles, \(P\) and \(Q\), are on a smooth horizontal floor. \(P\), of mass 1 kg , is moving with speed \(1.79 \mathrm {~ms} ^ { - 1 }\) directly towards a vertical wall. \(Q\), of mass 2.74 kg , is between \(P\) and the wall, moving directly towards \(P\) with speed \(0.08 \mathrm {~ms} ^ { - 1 }\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{98053e88-1aec-4b0d-ae5f-ece4ad340266-4_232_830_1370_246}
\(P\) and \(Q\) collide directly and the coefficient of restitution for this collision is denoted by \(e\).
  1. Show that after this collision the speed of \(Q\) is given by \(0.42 + 0.5 e \mathrm {~ms} ^ { - 1 }\). After this collision, \(Q\) then goes on to collide directly with the wall. The coefficient of restitution for the collision between \(Q\) and the wall is also \(e\). There is then no subsequent collision between \(P\) and \(Q\).
  2. Determine the range of possible values of \(e\).
OCR Further Mechanics AS 2022 June Q8
8 As part of an industrial process a single pump causes the intake of a liquid chemical to the bottom end of a tube, draws it up the tube and then discharges it through a nozzle at the top end of the tube. The tube is straight and narrow, 35 m long and inclined at an angle of \(26 ^ { \circ }\) to the horizontal. The chemical arrives at the intake at the bottom end of the tube with a speed of \(6.2 \mathrm {~ms} ^ { - 1 }\). At the top end of the tube the chemical is discharged horizontally with a speed of \(14.3 \mathrm {~ms} ^ { - 1 }\) (see diagram). In total, the pump discharges 1500 kg of chemical through the nozzle each hour.
\includegraphics[max width=\textwidth, alt={}, center]{98053e88-1aec-4b0d-ae5f-ece4ad340266-5_405_1175_685_242} In order to model the changes to the mechanical energy of the chemical during the entire process of intake, drawing and discharge, the following modelling assumptions are made.
  • At any instant the total resistance to the motion of all the liquid in the tube is 40 N .
  • All other resistances to motion are ignored.
  • The liquid in the tube moves at a constant speed of \(6.2 \mathrm {~ms} ^ { - 1 }\).
    1. State one other modelling assumption which is required to model the changes to the mechanical energy of the liquid with the given information.
    2. Determine the power at which the pump is working, according to the model.
When the power at which the pump is working is measured it is in fact found to be 450 W .
    1. Find the difference between the total amount of energy output by the pump each hour and the total amount of mechanical energy gained by the chemical each hour.
    2. Give one reason why the model underestimates the power of the engine.
    •
    OCR Further Mechanics AS 2023 June Q1
    1 Two particles \(A\), of mass \(m \mathrm {~kg}\), and \(B\), of mass \(3 m \mathrm {~kg}\), are connected by a light inextensible string and placed together at rest on a smooth horizontal surface with the string slack. \(A\) is projected along the surface, directly away from \(B\), with a speed of \(2.4 \mathrm {~ms} ^ { - 1 }\).
    1. Find the speed of \(B\) immediately after the string becomes taut.
    2. Find, in terms of \(m\), the magnitude of the impulse exerted on \(B\) as a result of the string becoming taut.
    OCR Further Mechanics AS 2023 June Q2
    2
    \includegraphics[max width=\textwidth, alt={}, center]{b190b8c9-75b0-4ede-913f-cdecdb58180f-2_337_579_842_246} A small body \(P\) of mass 3 kg is at rest at the lowest point of the inside of a smooth hemispherical shell of radius 3.2 m and centre \(O\).
    \(P\) is projected horizontally with a speed of \(u \mathrm {~ms} ^ { - 1 }\). When \(P\) first comes to instantaneous rest \(O P\) makes an angle of \(60 ^ { \circ }\) with the downward vertical through \(O\).
    1. Find the value of \(u\).
    2. State one assumption made in modelling the motion of \(P\).
    OCR Further Mechanics AS 2023 June Q3
    3 A crate of mass 45 kg is sliding with a speed of \(0.8 \mathrm {~ms} ^ { - 1 }\) in a straight line across a smooth horizontal floor. One end of a light inextensible rope is attached to the crate. At a certain instant a builder takes the other end of the rope and starts to pull, applying a constant force of 80 N for 5 seconds. While the builder is pulling the crate, the rope makes a constant angle of \(40 ^ { \circ }\) above the horizontal. Both the rope and the velocity of the crate lie in the same vertical plane (see diagram).
    \includegraphics[max width=\textwidth, alt={}, center]{b190b8c9-75b0-4ede-913f-cdecdb58180f-3_250_817_605_246} It may be assumed that there is no resistance to the motion of the crate.
    1. Determine the work done by the builder in pulling the crate.
      1. Find the kinetic energy of the crate at the instant when the builder stops pulling the crate.
      2. Explain why the answers to part (a) and part (b)(i) are not equal.
    3. Find the average power developed by the builder in pulling the crate.
    4. Calculate the total impulse exerted on the crate by the builder.
    OCR Further Mechanics AS 2023 June Q4
    4 A rower is rowing a boat in a straight line across a lake. The combined mass of the rower, boat and oars is 240 kg . The maximum power that the rower can generate is 450 W . In a model of the motion of the boat it is assumed that the total resistance to the motion of the boat is 150 N at any instant when the boat is in motion.
    1. Find the maximum possible acceleration of the boat, according to the model, at an instant when its speed is \(0.5 \mathrm {~ms} ^ { - 1 }\). At one stage in its motion the boat is travelling at a constant speed and the rower is generating power at an average rate of 210 W , which is assumed to be constant. The boat passes a pole and then, after travelling 350 m , a second pole.
    2. Determine how long it takes, according to the model, for the boat to travel between the two poles.
    3. State a reason why the assumption that the rower's generated power is constant may be unrealistic.
    OCR Further Mechanics AS 2023 June Q5
    5 Two identical spheres, \(A\) and \(B\), each of mass 4 kg , are moving directly towards each other along the same straight line on a smooth horizontal surface until they collide. Before they collide, the speeds of \(A\) and \(B\) are \(5 \mathrm {~ms} ^ { - 1 }\) and \(3 \mathrm {~ms} ^ { - 1 }\) respectively. Immediately after they collide, the speed of \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\) and its direction of motion has been reversed.
      1. Determine the velocity of \(B\) immediately after \(A\) and \(B\) collide.
      2. Show that the coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\).
      3. Calculate the total loss of kinetic energy due to this collision. Sphere \(B\) goes on to strike a fixed wall directly. As a result of this collision \(B\) moves along the same straight line with a speed of \(4 \mathrm {~ms} ^ { - 1 }\).
    2. Find the coefficient of restitution between \(B\) and the wall, stating whether the collision between \(B\) and the wall is perfectly elastic.
    3. Determine the magnitude of the impulse that \(B\) exerts on \(A\) the next time that they collide.
    OCR Further Mechanics AS 2023 June Q6
    6 The physical quantity pressure, denoted by \(P\), can be calculated using the formula \(P = \frac { F } { A }\) where \(F\) is a force and \(A\) is an area.
    1. Find the dimensions of \(P\). An object of mass \(m\) is moving on a smooth horizontal surface subject to a system of forces which begin to act at time \(t = 0\). The initial velocity of the object is \(u\) and its velocity and acceleration at time \(t\) are denoted by \(v\) and \(a\) respectively. The object exerts a pressure \(P\) on the surface. The total work done by the forces is denoted by \(W\). A Mathematics class suggests three formulae to model the quantity \(W\).
      The first suggested formula is \(W = \frac { 1 } { 2 } m v ^ { 2 } - \frac { 1 } { 2 } m u ^ { 2 } + m P\).
    2. Use dimensional analysis to show that this formula cannot be correct. The second suggested formula is \(W = k u ^ { \alpha } v ^ { \beta } t ^ { \gamma }\) where \(k\) is a dimensionless constant.
    3. Use dimensional analysis to show that this formula cannot be correct for any values of \(\alpha , \beta\) and \(\gamma\). The third suggested formula is \(W = k u ^ { \alpha } a ^ { \beta } m ^ { \gamma } t ^ { \delta }\) where \(k\) is a dimensionless constant.
      1. Explain why it is not possible to use dimensional analysis to determine the values of \(\alpha\), \(\beta , \gamma\) and \(\delta\) for the third suggested formula.
      2. Given that \(\alpha = 3\), use dimensional analysis to determine the values of \(\beta , \gamma\) and \(\delta\) for the third suggested formula.
      3. By considering what the formula predicts for large values of \(t\), explain why the formula derived in part (d)(ii) is likely to be incorrect.