Questions Further Mechanics (96 questions)

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OCR Further Mechanics 2021 June Q2
11 marks Challenging +1.2
2 One end of a light inextensible string of length 0.8 m is attached to a fixed point, \(O\). The other end is attached to a particle \(P\) of mass \(1.2 \mathrm {~kg} . P\) hangs in equilibrium at a distance of 1.5 m above a horizontal plane. The point on the plane directly below \(O\) is \(F\). \(P\) is projected horizontally with speed \(3.5 \mathrm {~ms} ^ { - 1 }\). The string breaks when \(O P\) makes an angle of \(\frac { 1 } { 3 } \pi\) radians with the downwards vertical through \(O\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{0428f2f2-12c4-4e89-93ab-8cfe2c5aca4a-02_757_889_1482_251}
  1. Find the magnitude of the tension in the string at the instant before the string breaks.
  2. Find the distance between \(F\) and the point where \(P\) first hits the plane.
OCR Further Mechanics 2021 June Q3
17 marks Standard +0.3
3 This question is about modelling the relation between the pressure, \(P\), volume, \(V\), and temperature, \(\theta\), of a fixed amount of gas in a container whose volume can be varied. The amount of gas is measured in moles; 1 mole is a dimensionless constant representing a fixed number of molecules of gas. Gas temperatures are measured on the Kelvin scale; the unit for temperature is denoted by K . You may assume that temperature is a dimensionless quantity. A gas in a container will always exert an outwards force on the walls of the container. The pressure of the gas is defined to be the magnitude of this force per unit area of the walls, with \(P\) always positive. An initial model of the relation is given by \(P ^ { \alpha } V ^ { \beta } = n R \theta\), where \(n\) is the number of moles of gas present and \(R\) is a quantity called the Universal Gas Constant. The value of \(R\), correct to 3 significant figures, is \(8.31 \mathrm { JK } ^ { - 1 }\).
  1. Show that \([ P ] = \mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 2 }\) and \([ R ] = \mathrm { ML } ^ { 2 } \mathrm {~T} ^ { - 2 }\).
  2. Hence show that \(\alpha = 1\) and \(\beta = 1\). 5 moles of gas are present in the container which initially has volume \(0.03 \mathrm {~m} ^ { 3 }\) and which is maintained at a temperature of 300 K .
  3. Find the pressure of the gas, as predicted by the model. An improved model of the relation is given by \(\left( P + \frac { a n ^ { 2 } } { V ^ { 2 } } \right) ( V - n b ) = n R \theta\), where \(a\) and \(b\) are constants.
  4. Determine the dimensions of \(b\) and \(a\). The values of \(a\) and \(b\) (in appropriate units) are measured as being 0.14 and \(3.2 \times 10 ^ { - 5 }\) respectively.
  5. Find the pressure of the gas as predicted by the improved model. Suppose that the volume of the container is now reduced to \(1.5 \times 10 ^ { - 4 } \mathrm {~m} ^ { 3 }\) while keeping the temperature at 300 K .
  6. By considering the value of the pressure of the gas as predicted by the improved model, comment on the validity of this model in this situation.
OCR Further Mechanics 2021 June Q2
8 marks Challenging +1.2
2 Three particles, \(A , B\) and \(C\), of masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively, are at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Collisions between \(A\) and \(B\) are perfectly elastic. The coefficient of restitution for collisions between \(B\) and \(C\) is \(e\). \(A\) is projected towards \(B\) with a speed of \(5 u \mathrm {~ms} ^ { - 1 }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{709f3a7a-d857-4813-98ab-de6b41a3a8dc-02_190_885_1151_260} Show that only two collisions occur.
OCR Further Mechanics 2021 June Q3
9 marks Challenging +1.2
3 A particle \(P\) of mass 8 kg moves in a straight line on a smooth horizontal plane. At time \(t \mathrm {~s}\) the displacement of \(P\) from a fixed point \(O\) on the line is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). Initially, \(P\) is at rest at \(O\). \(P\) is acted on by a horizontal force, directed along the line away from \(O\), with magnitude proportional to \(\sqrt { 9 + v ^ { 2 } }\). When \(v = 1.25\), the magnitude of this force is 13 N .
  1. Show that \(\frac { 1 } { \sqrt { 9 + v ^ { 2 } } } \frac { \mathrm {~d} v } { \mathrm {~d} t } = \frac { 1 } { 2 }\).
  2. Find an expression for \(v\) in terms of \(t\) for \(t \geqslant 0\).
  3. Find an expression for \(x\) in terms of \(t\) for \(t \geqslant 0\).
OCR Further Mechanics 2021 June Q4
13 marks Standard +0.8
4 Particles \(A , B\) and \(C\) of masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively are joined by light rigid rods to form a triangular frame. The frame is placed at rest on a horizontal plane with \(A\) at the point \(( 0,0 )\), \(B\) at the point ( \(0.6,0\) ) and \(C\) at the point ( \(0.4,0.2\) ), where distances in the coordinate system are measured in metres (see Fig. 1). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{709f3a7a-d857-4813-98ab-de6b41a3a8dc-03_311_661_338_258} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \(G\), which is the centre of mass of the frame, is at the point \(( \bar { x } , \bar { y } )\).
  1. - Show that \(\bar { x } = 0.38\).
    A rough plane, \(\Pi\), is inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 3 } { 5 }\). The frame is placed on \(\Pi\) with \(A B\) vertical and \(B\) in contact with \(\Pi . C\) is in the same vertical plane as \(A B\) and a line of greatest slope of \(\Pi . C\) is on the down-slope side of \(A B\). The frame is kept in equilibrium by a horizontal light elastic string whose natural length is \(l \mathrm {~m}\) and whose modulus of elasticity is \(g \mathrm {~N}\). The string is attached to \(A\) at one end and to a fixed point on \(\Pi\) at the other end (see Fig. 2). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{709f3a7a-d857-4813-98ab-de6b41a3a8dc-03_605_828_1525_248} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The coefficient of friction between \(B\) and \(\Pi\) is \(\mu\).
  2. Show that \(l = 0.3\).
  3. Show that \(\mu \geqslant \frac { 14 } { 27 }\).
OCR Further Mechanics 2023 June Q1
8 marks Standard +0.3
One end of a light inextensible string of length \(0.8\) m is attached to a particle \(P\) of mass \(m\) 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 \(5.3\) m s\(^{-1}\) so that it moves in a vertical circular path with centre \(O\) (see diagram). \includegraphics{figure_1} 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 \(4.5\) m s\(^{-1}\). [3]
  2. Find the magnitude and direction of the radial acceleration of \(P\) at this instant. [3]
  3. Find the magnitude of the tangential acceleration of \(P\) at this instant. [2]
OCR Further Mechanics 2023 June Q2
11 marks Standard +0.3
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. [1]
  2. By considering the dimensions of both Stress and Strain determine the dimensions of \(E\). [2]
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, \(\rho\).
  1. Use dimensional analysis to suggest a formula for \(c\) in terms of \(E\), \(V\) and \(\rho\). [5]
  2. The speed of sound in a certain material is \(500\) m s\(^{-1}\).
    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. [1]
    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. [1]
  3. Suggest one possible limitation caused by using dimensional analysis to set up the model in part (c). [1]
OCR Further Mechanics 2023 June Q3
7 marks Challenging +1.2
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 \(4\) m s\(^{-1}\) at an angle of \(60°\) to the line of centres, • the velocity of \(B\) is \(6\) m s\(^{-1}\) along the line of centres (see diagram). \includegraphics{figure_3} 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. [7]
OCR Further Mechanics 2023 June Q4
9 marks Standard +0.3
\(ABCD\) is a uniform lamina in the shape of a kite with \(BA = BC = 0.37\) m, \(DA = DC = 0.91\) m and \(AC = 0.7\) m (see diagram). The centre of mass of \(ABCD\) is \(G\). \includegraphics{figure_4}
  1. Explain why \(G\) lies on \(BD\). [1]
  2. Show that the distance of \(G\) from \(B\) is \(0.36\) m. [4]
The lamina \(ABCD\) is freely suspended from the point \(A\).
  1. Determine the acute angle that \(CD\) makes with the horizontal, stating which of \(C\) or \(D\) is higher. [4]
OCR Further Mechanics 2023 June Q5
13 marks Challenging +1.3
A particle \(P\) of mass \(2\) kg moves along the \(x\)-axis. At time \(t = 0\), \(P\) passes through the origin \(O\) with speed \(3\) m s\(^{-1}\). At time \(t\) seconds the displacement of \(P\) from \(O\) is \(x\) m and the velocity of \(P\) is \(v\) m s\(^{-1}\), 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 \((v^2 + 1)\) N acting in the negative \(x\)-direction.
  1. Find an expression for \(v\) in terms of \(x\). [5]
  2. Determine the distance travelled by \(P\) while its speed drops from \(3\) m s\(^{-1}\) to \(2\) m s\(^{-1}\). [2]
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 \(t' = 0\), \(Q\) is at the origin and its speed is \(3\) m s\(^{-1}\).
  1. 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 \(x < 9\). [3]
  2. Sketch the velocity-time graph for \(P\). You do not need to indicate any values on your sketch. [1]
  3. Determine the maximum displacement of \(P\) from \(O\) during \(P\)'s motion. [2]
OCR Further Mechanics 2023 June Q6
12 marks Challenging +1.2
A particle \(P\) of mass \(3\) kg is moving on a smooth horizontal surface under the influence of a variable horizontal force \(\mathbf{F}\) N. At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P\), \(\mathbf{v}\) m s\(^{-1}\), is given by $$\mathbf{v} = (32\sinh(2t))\mathbf{i} + (32\cosh(2t) - 257)\mathbf{j}.$$
    1. By considering kinetic energy, determine the work done by \(\mathbf{F}\) over the interval \(0 \leqslant t \leqslant \ln 2\). [5]
    2. Explain the significance of the sign of the answer to part (a)(i). [1]
  2. Determine the rate at which \(\mathbf{F}\) is working at the instant when \(P\) is moving parallel to the \(\mathbf{i}\)-direction. [6]
OCR Further Mechanics 2023 June Q7
7 marks Challenging +1.2
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{figure_7}
  1. Determine the angular speed of \(A\) and the angular speed of \(B\). [5]
At the start of the motion, \(A\), \(O\) and \(B\) all lie in the same vertical plane.
  1. Find the first subsequent time when \(A\), \(O\) and \(B\) all lie in the same vertical plane. [2]
OCR Further Mechanics 2023 June Q8
8 marks Challenging +1.2
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 \(2.4\) m s\(^{-1}\). 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. [8]
OCR Further Mechanics 2018 September Q1
5 marks Moderate -0.3
A car of mass 850 kg is being driven uphill along a straight road inclined at \(7°\) to the horizontal. The resistance to motion is modelled as a constant force of magnitude 140 N. At a certain instant the car's speed is \(12 \text{ms}^{-1}\) and its acceleration is \(0.4 \text{ms}^{-2}\).
  1. Calculate the power of the car's engine at this instant. [3]
  2. Find the constant speed at which the car could travel up the hill with the engine generating this power. [2]
OCR Further Mechanics 2018 September Q2
6 marks Standard +0.3
A particle of mass 0.8 kg is moving in a straight line on a smooth horizontal surface with constant speed \(12 \text{ms}^{-1}\) when it is struck by a horizontal impulse. Immediately after the impulse acts, the particle is moving with speed \(9 \text{ms}^{-1}\) at an angle of 50° to its original direction of motion (see diagram). \includegraphics{figure_2} Find
  1. the magnitude of the impulse, [3]
  2. the angle that the impulse makes with the original direction of motion of the particle. [3]
OCR Further Mechanics 2018 September Q3
6 marks Standard +0.3
Assume that the earth moves round the sun in a circle of radius \(1.50 \times 10^8\) km at constant speed, with one complete orbit taking 365 days. Given that the mass of the earth is \(5.97 \times 10^{24}\) kg,
  1. calculate the magnitude of the force exerted by the sun on the earth, giving your answer in newtons, [5]
  2. state the direction in which this force acts. [1]
OCR Further Mechanics 2018 September Q4
13 marks Standard +0.8
\(A\) and \(B\) are two points a distance of 5 m apart on a horizontal ceiling. A particle \(P\) of mass \(m\) kg is attached to \(A\) and \(B\) by light elastic strings. The particle hangs in equilibrium at a distance of 4 m from \(A\) and 3 m from \(B\) so that angle \(APB = 90°\) (see diagram). \includegraphics{figure_4} The string joining \(P\) to \(A\) has natural length 2 m and modulus of elasticity \(\lambda_A\) N. The string joining \(P\) to \(B\) also has natural length 2 m but has modulus of elasticity \(\lambda_B\) N.
    1. Show that \(\lambda_B = \frac{3}{4}\lambda_A\). [4]
    2. Find an expression for \(\lambda_A\) in terms of \(m\) and \(g\). [3]
  2. Find, in terms of \(m\) and \(g\), the total elastic potential energy stored in the strings. [2]
The string joining \(P\) to \(A\) is detached from \(A\) and a second particle, \(Q\), of mass \(0.3m\) kg is attached to the free end of the string. \(Q\) is then gently lowered into a position where the system hangs vertically in equilibrium.
  1. Find the distance of \(Q\) below \(B\) in this equilibrium position. [4]
OCR Further Mechanics 2018 September Q5
10 marks Standard +0.3
One end of a non-uniform rod is freely hinged to a fixed point so that the rod can rotate about the point. When the rod rotates with angular velocity \(\omega\) it can be shown that the kinetic energy \(E\) of the rod is given by \(E = \frac{1}{2}I\omega^2\), where \(I\) is a quantity called the moment of inertia of the rod.
  1. Deduce the dimensions of \(I\). [3]
  2. Given that the rod has mass \(m\) and length \(r\), suggest an expression for \(I\), explaining any additional symbols that you use. [3]
A student notices that the formula \(E = \frac{1}{2}I\omega^2\) looks similar to the formula \(E = \frac{1}{2}mv^2\) for the kinetic energy of a particle, with angular velocity for the rod corresponding to velocity for the particle, and moment of inertia corresponding to mass. Assuming a similar correspondence between angular acceleration (i.e. \(\frac{d\omega}{dt}\)) and acceleration, the student thinks that an equation for angular motion of the rod corresponding to Newton's second law for the particle should be \(F = I\alpha\), where \(F\) is the force applied to the rod and \(\alpha\) is the resulting angular acceleration.
  1. Use dimensional analysis to show that the student's suggestion is incorrect. [2]
  2. State the dimensions of a quantity \(x\) for which the equation \(Fx = I\alpha\) would be dimensionally consistent. [1]
  3. Explain why the fact that the equation in part (iv) is dimensionally consistent does not necessarily mean that it is correct. [1]
OCR Further Mechanics 2018 September Q6
10 marks Standard +0.8
A particle \(P\) of mass \(m\) moves along the positive \(x\)-axis. When its displacement from the origin \(O\) is \(x\) its velocity is \(v\), where \(v \geqslant 0\). It is subject to two forces: a constant force \(T\) in the positive \(x\) direction, and a resistive force which is proportional to \(v^2\).
  1. Show that \(v^2 = \frac{1}{k}\left(T - Ae^{-\frac{2kx}{m}}\right)\) where \(A\) and \(k\) are constants. [5]
\(P\) starts from rest at \(O\).
  1. Find an expression for the work done against the resistance to motion as \(P\) moves from \(O\) to the point where \(x = 1\). [4]
  2. Find an expression for the limiting value of the velocity of \(P\) as \(x\) increases. [1]
OCR Further Mechanics 2018 September Q7
9 marks Standard +0.8
A uniform solid hemisphere has radius 0.4 m. A uniform solid cone, made of the same material, has base radius 0.4 m and height 1.2 m. A solid, \(S\), is formed by joining the hemisphere and the cone so that their circular faces coincide. \(O\) is the centre of the joint circular face and \(V\) is the vertex of the cone. \(G\) is the centre of mass of \(S\).
  1. Explain briefly why \(G\) lies on the line through \(O\) and \(V\). [1]
  2. Show that the distance of \(G\) from \(O\) is 0.12 m. (The volumes of a hemisphere and cone are \(\frac{2}{3}\pi r^3\) and \(\frac{1}{3}\pi r^2 h\) respectively.) [5]
\includegraphics{figure_7} \(S\) is suspended from two light vertical strings, one attached to \(V\) and the other attached to a point on the circumference of the joint circular face, and hangs in equilibrium with \(OV\) horizontal (see diagram).
  1. The weight of \(S\) is \(W\). Find the magnitudes of the tensions in the strings in terms of \(W\). [3]
OCR Further Mechanics 2018 September Q8
16 marks Challenging +1.8
A point \(O\) is situated a distance \(h\) above a smooth horizontal plane, and a particle \(A\) of mass \(m\) is attached to \(O\) by a light inextensible string of length \(h\). A particle \(B\) of mass \(2m\) is at rest on the plane, directly below \(O\), and is attached to a point \(C\) on the plane, where \(BC = l\), by a light inextensible string of length \(l\). \(A\) is released from rest with the string \(OA\) taut and making an acute angle \(\theta\) with the downward vertical (see diagram). \includegraphics{figure_8} \(A\) moves in a vertical plane perpendicular to \(CB\) and collides directly with \(B\). As a result of this collision, \(A\) is brought to rest and \(B\) moves on the plane in a horizontal circle with centre \(C\). After \(B\) has made one complete revolution the particles collide again.
  1. Show that, on the next occasion that \(A\) comes to rest, the string \(OA\) makes an angle \(\phi\) with the downward vertical through \(O\), where \(\cos \phi = \frac{3 + \cos \theta}{4}\). [9]
\(A\) and \(B\) collide again when \(AO\) is next vertical.
  1. Find the percentage of the original energy of the system that remains immediately after this collision. [5]
  2. Explain why the total momentum of the particles immediately before the first collision is the same as the total momentum of the particles immediately after the second collision. [1]
  3. Explain why the total momentum of the particles immediately before the first collision is different from the total momentum of the particles immediately after the third collision. [1]