Questions — OCR MEI Further Mechanics Major (88 questions)

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OCR MEI Further Mechanics Major 2020 November Q12
12 marks Challenging +1.2
\includegraphics{figure_12} Fig. 12 shows a hemispherical bowl. The rim of this bowl is a circle with centre O and radius \(r\). The bowl is fixed with its rim horizontal and uppermost. A particle P, of mass \(m\), is connected by a light inextensible string of length \(l\) to the lowest point A on the bowl and describes a horizontal circle with constant angular speed \(\omega\) on the smooth inner surface of the bowl. The string is taut, and AP makes an angle \(\alpha\) with the vertical.
  1. Show that the normal contact force between P and the bowl is of magnitude \(mg + 2mr\omega^2\cos^2\alpha\). [9]
  2. Deduce that \(g < r\omega^2(k_1 + k_2\cos^2\alpha)\), stating the value of the constants \(k_1\) and \(k_2\). [3]
OCR MEI Further Mechanics Major Specimen Q1
4 marks Moderate -0.3
A particle P has position vector \(\mathbf{r}\) m at time \(t\) s given by \(\mathbf{r} = (t^3 - 3t^2)\mathbf{i} - (4t^2 + 1)\mathbf{j}\) for \(t \geq 0\). Find the magnitude of the acceleration of P when \(t = 2\). [4]
OCR MEI Further Mechanics Major Specimen Q2
3 marks Moderate -0.8
A particle of mass 5 kg is moving with velocity \(2\mathbf{i} + 5\mathbf{j}\) m s\(^{-1}\). It receives an impulse of magnitude 15 N s in the direction \(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}\). Find the velocity of the particle immediately afterwards. [3]
OCR MEI Further Mechanics Major Specimen Q3
5 marks Standard +0.3
The fixed points E and F are on the same horizontal level with EF = 1.6 m. A light string has natural length 0.7 m and modulus of elasticity 29.4 N. One end of the string is attached to E and the other end is attached to a particle of mass \(M\) kg. A second string, identical to the first, has one end attached to F and the other end attached to the particle. The system is in equilibrium in a vertical plane with each string stretched to a length of 1 m, as shown in Fig. 3. \includegraphics{figure_3}
  1. Find the tension in each string. [2]
  2. Find \(M\). [3]
OCR MEI Further Mechanics Major Specimen Q4
6 marks Standard +0.3
A fixed smooth sphere has centre O and radius \(a\). A particle P of mass \(m\) is placed at the highest point of the sphere and given an initial horizontal speed \(u\). For the first part of its motion, P remains in contact with the sphere and has speed \(v\) when OP makes an angle \(\theta\) with the upward vertical. This is shown in Fig. 4. \includegraphics{figure_4}
  1. By considering the energy of P, show that \(v^2 = u^2 + 2ga(1 - \cos\theta)\). [2]
  2. Show that the magnitude of the normal contact force between the sphere and particle P is $$mg(3\cos\theta - 2) - \frac{mv^2}{a}.$$ [2]
The particle loses contact with the sphere when \(\cos\theta = \frac{3}{4}\).
  1. Find an expression for \(u\) in terms of \(a\) and \(g\). [2]
OCR MEI Further Mechanics Major Specimen Q5
8 marks Standard +0.8
Fig. 5 shows a light inextensible string of length 3.3 m passing through a small smooth ring R. The ends of the string are attached to fixed points A and B, where A is vertically above B. The ring R has mass 0.27 kg and is moving with constant speed in a horizontal circle of radius 1.2 m. The distances AR and BR are 2 m and 1.3 m respectively. \includegraphics{figure_5}
  1. Show that the tension in the string is 6.37 N. [4]
  2. Find the speed of R. [4]
OCR MEI Further Mechanics Major Specimen Q6
10 marks Standard +0.8
Fig. 6 shows a pendulum which consists of a rod AB freely hinged at the end A with a weight at the end B. The pendulum is oscillating in a vertical plane. The total energy, \(E\), of the pendulum is given by $$E = \frac{1}{2}I\omega^2 - mgh\cos\theta,$$ where
  • \(\omega\) is its angular speed
  • \(m\) is its mass
  • \(h\) is the distance of its centre of mass from A
  • \(\theta\) is the angle the rod makes with the downward vertical
  • \(g\) is the acceleration due to gravity
  • \(I\) is a quantity known as the moment of inertia of the pendulum.
\includegraphics{figure_6}
  1. Use the expression for \(E\) to deduce the dimensions of \(I\). [4]
It is suggested that the period of oscillation, \(T\), of the pendulum is given by \(T = kI^\alpha(mg)^\beta h^\gamma\), where \(k\) is a dimensionless constant.
  1. Use dimensional analysis to find the values of \(\alpha\), \(\beta\) and \(\gamma\). [5]
A class experiment finds that, when all other quantities are fixed, \(T\) is proportional to \(\frac{1}{\sqrt{m}}\).
  1. Determine whether this result is consistent with your answer to part (ii). [1]
OCR MEI Further Mechanics Major Specimen Q7
9 marks Standard +0.3
A uniform ladder of length 8 m and weight 180 N stands on a rough horizontal surface and rests against a smooth vertical wall. The ladder makes an angle of 20° with the wall. A woman of weight 720 N stands on the ladder. Fig. 7 shows this situation modelled with the woman's weight acting at a distance \(x\) m from the lower end of the ladder. The system is in equilibrium. \includegraphics{figure_7}
  1. Show that the frictional force between the ladder and the horizontal surface is \(F\) N, where \(F = 90(1 + x)\tan 20°\). [4]
    1. State with a reason whether \(F\) increases, stays constant or decreases as \(x\) increases. [1]
    2. Hence determine the set of values of the coefficient of friction between the ladder and the surface for which the woman can stand anywhere on the ladder without it slipping. [4]
OCR MEI Further Mechanics Major Specimen Q8
16 marks Standard +0.3
A tractor has a mass of 6000 kg. When developing a power of 5 kW, the tractor is travelling at a steady speed of 2.5 m s\(^{-1}\) across a horizontal field.
  1. Calculate the magnitude of the resistance to the motion of the tractor. [2]
The tractor comes to horizontal ground where the resistance to motion is different. The power developed by the tractor during the next 10 s has an average value of 8 kW. During this time, the tractor accelerates uniformly from 2.5 m s\(^{-1}\) to 3 m s\(^{-1}\).
    1. Show that the work done against the resistance to motion during the 10 s is 71 750 J. [4]
    2. Assuming that the resistance to motion is constant, calculate its value. [3]
The tractor can usually travel up a straight track inclined at an angle \(\alpha\) to the horizontal, where \(\sin\alpha = \frac{1}{20}\), while accelerating uniformly from 3 m s\(^{-1}\) to 3.25 m s\(^{-1}\) over a distance of 100 m against a resistance to motion of constant magnitude of 2000 N. The tractor develops a fault which limits its maximum power to 16kW.
  1. Determine whether the tractor could now perform the same motion up the track. [You should assume that the mass of the tractor and the resistance to motion remain the same.] [7]
OCR MEI Further Mechanics Major Specimen Q9
14 marks Challenging +1.2
\includegraphics{figure_9} Fig. 9 shows the instant of impact of two identical uniform smooth spheres, A and B, each with mass \(m\). Immediately before they collide, the spheres are sliding towards each other on a smooth horizontal table in the directions shown in the diagram, each with speed \(v\). The coefficient of restitution between the spheres is \(\frac{1}{2}\).
  1. Show that, immediately after the collision, the speed of A is \(\frac{1}{8}v\). Find its direction of motion. [6]
  2. Find the percentage of the original kinetic energy that is lost in the collision. [7]
  3. State where in your answer to part (i) you have used the assumption that the contact between the spheres is smooth. [1]
OCR MEI Further Mechanics Major Specimen Q10
14 marks Standard +0.3
In this question take \(g = 10\). A smooth ball of mass 0.1 kg is projected from a point on smooth horizontal ground with speed 65 m s\(^{-1}\) at an angle \(\alpha\) to the horizontal, where \(\tan\alpha = \frac{3}{4}\). While it is in the air the ball is modelled as a particle moving freely under gravity. The ball bounces on the ground repeatedly. The coefficient of restitution for the first bounce is 0.4.
  1. Show that the ball leaves the ground after the first bounce with a horizontal speed of 52 m s\(^{-1}\) and a vertical speed of 15.6 m s\(^{-1}\). Explain your reasoning carefully. [4]
  2. Calculate the magnitude of the impulse exerted on the ball by the ground at the first bounce. [2]
Each subsequent bounce is modelled by assuming that the coefficient of restitution is 0.4 and that the bounce takes no time. The ball is in the air for \(T_1\) seconds between projection and bouncing the first time, \(T_2\) seconds between the first and second bounces, and \(T_n\) seconds between the \((n-1)\)th and \(n\)th bounces.
    1. Show that \(T_1 = \frac{39}{5}\). [2]
    2. Find an expression for \(T_n\) in terms of \(n\). [2]
  2. According to the model, how far does the ball travel horizontally while it is still bouncing? [3]
  3. According to the model, what is the motion of the ball after it has stopped bouncing? [1]
OCR MEI Further Mechanics Major Specimen Q11
16 marks Challenging +1.2
The region bounded by the \(x\)-axis and the curve \(y = \frac{1}{2}k(1-x^2)\) for \(-1 \leq x \leq 1\) is occupied by a uniform lamina, as shown in Fig. 11.1. \includegraphics{figure_11_1}
  1. In this question you must show detailed reasoning. Show that the centre of mass of the lamina is at \(\left(0, \frac{1}{5}k\right)\). [7]
A shop sign is modelled as a uniform lamina in the form of the lamina in part (i) attached to a rectangle ABCD, where AB = 2 and BC = 1. The sign is suspended by two vertical wires attached at A and D, as shown in Fig. 11.2. \includegraphics{figure_11_2}
  1. Show that the centre of mass of the sign is at a distance $$\frac{2k^2 + 10k + 15}{10k + 30}$$ from the midpoint of CD. [4]
The tension in the wire at A is twice the tension in the wire at D.
  1. Find the value of \(k\). [5]
OCR MEI Further Mechanics Major Specimen Q12
15 marks Challenging +1.2
Fig. 12 shows \(x\)- and \(y\)- coordinate axes with origin O and the trajectory of a particle projected from O with speed 28 m s\(^{-1}\) at an angle \(\alpha\) to the horizontal. After \(t\) seconds, the particle has horizontal and vertical displacements \(x\) m and \(y\) m. Air resistance should be neglected. \includegraphics{figure_12}
  1. Show that the equation of the trajectory is given by $$\tan^2\alpha - \frac{160}{x}\tan\alpha + \frac{160y}{x^2} + 1 = 0.$$ (*) [5]
    1. Show that if (*) is treated as an equation with \(\tan\alpha\) as a variable and with \(x\) and \(y\) as constants, then (*) has two distinct real roots for \(\tan\alpha\) when \(y < 40 - \frac{x^2}{160}\). [3]
    2. Show the inequality in part (ii)(A) as a locus on the graph of \(y = 40 - \frac{x^2}{160}\) in the Printed Answer Booklet and label it R. [1]
S is the locus of points \((x, y)\) where (*) has one real root for \(\tan\alpha\). T is the locus of points \((x, y)\) where (*) has no real roots for \(\tan\alpha\).
  1. Indicate S and T on the graph in the Printed Answer Booklet. [2]
  2. State the significance of R, S and T for the possible trajectories of the particle. [3]
A machine can fire a tennis ball from ground level with a maximum speed of 28 m s\(^{-1}\).
  1. State, with a reason, whether a tennis ball fired from the machine can achieve a range of 80 m. [1]