6.02i Conservation of energy: mechanical energy principle

Two light elastic strings, each of length \(l\) m and modulus of elasticity \(\lambda\) N, are attached to a particle \(P\) of mass \(m\) kg. The other ends of the strings are attached to fixed points \(A\) and \(B\) on the same horizontal level, where \(AB = 2l\) m. \(P\) is held vertically below the mid-point of \(AB\), with each string taut and inclined at \(30°\) to the horizontal, and released from rest. Given that \(P\) comes to instantaneous rest when each string makes an angle of \(60°\) with the horizontal, show that \(\lambda = \frac{3mg}{6 - 2\sqrt{3}}\). \includegraphics{figure_1} [10 marks]

A particle \(P\) is projected horizontally with speed \(u\) ms\(^{-1}\) from the highest point of a smooth sphere of radius \(r\) m and centre \(O\). It moves on the surface in a vertical plane, and at a particular instant the radius \(OP\) makes an angle \(\theta\) with the upward vertical, as shown. At this instant \(P\) has speed \(v\) ms\(^{-1}\) and the magnitude of the reaction between \(P\) and the sphere is \(X\) N. \includegraphics{figure_2}

1. Assuming that \(u^2 < gr\), show that
   1. \(v^2 = u^2 + 2gr(1 - \cos \theta)\), [2 marks]
   2. \(X = mg\left(3\cos \theta - 2 - \frac{u^2}{gr}\right)\). [4 marks]
2. Show that \(P\) leaves the surface of the sphere when \(\cos \theta = \frac{u^2 + 2gr}{3gr}\). [3 marks]
3. Discuss what happens if \(u^2 \geq gr\). [2 marks]

A small bead is threaded onto a smooth circular hoop, of radius \(r\) m, fixed in a vertical plane. It is projected with speed \(u\) ms\(^{-1}\) from the lowest point of the hoop. Find \(u\) in terms of \(g\) and \(r\) if

1. the bead just reaches the highest point of the hoop, [3 marks]
2. the reaction on the bead is zero when it is at the highest point of the hoop. [4 marks]

Two particles \(A\) and \(B\), of masses \(M\) kg and \(m\) kg respectively, are connected by a light inextensible string passing over a smooth fixed pulley. \(B\) is placed on a smooth horizontal table and \(A\) hangs freely, as shown. \(B\) is attached to a spring of natural length \(l\) m and modulus of elasticity \(\lambda\) N, whose other end is fixed to a vertical wall. \includegraphics{figure_3} The system starts to move from rest when the string is taut and the spring neither extended nor compressed. \(A\) does not reach the ground, nor does \(B\) reach the pulley, during the motion.

1. Show that the maximum extension of the spring is \(\frac{2Mgl}{\lambda}\) m. [3 marks]
2. If \(M = 3\), \(m = 1.5\) and \(\lambda = 35l\), find the speed of \(A\) when the extension in the spring is \(0.5\) m. [6 marks]

The figure shows a swing consisting of two identical vertical light springs attached symmetrically to a light horizontal cross-bar and supported from a strong fixed horizontal beam. When a mass of 24 kg is placed at the mid-point of the cross-bar, both springs extend by 30 cm to the position \(A\), as shown. \includegraphics{figure_6} Each spring has natural length \(l\) m and modulus of elasticity \(\lambda\) N.

1. Show that \(\lambda = 392l\). [2 marks]

The 24 kg mass is left on the bar and the bar is then displaced downwards by a further 20 cm.

1. Prove that the system comprising the bar and the mass now performs simple harmonic motion with the centre of oscillation at the level \(A\). [5 marks]
2. Calculate the number of oscillations made per second in this motion. [3 marks]
3. Find the maximum acceleration which the mass experiences during the motion. [2 marks]

\includegraphics{figure_5} Each of two identical strings has natural length \(1.5\) m and modulus of elasticity \(18\) N. One end of one of the strings is attached to \(A\) and one end of the other string is attached to \(B\), where \(A\) and \(B\) are fixed points which are \(3\) m apart and at the same horizontal level. \(M\) is the mid-point of \(AB\). A particle \(P\) of mass \(m\) kg is attached to the other end of each of the strings. \(P\) is held at rest at the point \(0.8\) m vertically above \(M\), and then released. The lowest point reached by \(P\) in the subsequent motion is \(2\) m below \(M\) (see diagram).

1. Find the maximum tension in each of the strings during \(P\)'s motion. [3]
2. By considering energy,
   1. show that the value of \(m\) is \(0.42\), correct to 2 significant figures, [5]
   2. find the speed of \(P\) at \(M\). [3]

A bungee jumper of weight \(W\) N is joined to a fixed point \(O\) by a light elastic rope of natural length \(20\) m and modulus of elasticity \(32\,000\) N. The jumper starts from rest at \(O\) and falls vertically. The jumper is modelled as a particle and air resistance is ignored.

1. Given that the jumper just reaches a point \(25\) m below \(O\), find the value of \(W\). [5]
2. Find the maximum speed reached by the jumper. [4]
3. Find the maximum value of the deceleration of the jumper during the downward motion. [3]

\includegraphics{figure_6} A particle \(P\) of weight \(6\) N is attached to the highest point \(A\) of a fixed smooth sphere by a light elastic string. The sphere has centre \(O\) and radius \(0.8\) m. The string has natural length \(\frac{1}{10}\pi\) m and modulus of elasticity \(9\) N. \(P\) is released from rest at a point \(X\) on the sphere where \(OX\) makes an angle of \(\frac{1}{3}\pi\) radians with the upwards vertical. \(P\) remains in contact with the sphere as it moves upwards to \(A\). At time \(t\) seconds after the release, \(OP\) makes an angle of \(\theta\) radians with the upwards vertical (see diagram). When \(\theta = \frac{1}{4}\pi\), \(P\) passes through the point \(Y\).

1. Show that as \(P\) moves from \(X\) to \(Y\) its gravitational potential energy increases by \(2.4(\sqrt{3} - \sqrt{2})\) J and the elastic potential energy in the string decreases by \(0.4\pi\) J. [5]
2. Verify that the transverse acceleration of \(P\) is zero when \(\theta = \frac{1}{4}\pi\), and hence find the maximum speed of \(P\). [6]

\includegraphics{figure_3} A small object \(P\) is attached to one end of each of two vertical light elastic strings. One string is of natural length \(0.4\) m and has modulus of elasticity \(10\) N; the other string is of natural length \(0.5\) m and has modulus of elasticity \(12\) N. The upper ends of both strings are attached to a fixed horizontal beam and \(P\) hangs in equilibrium \(0.6\) m below the beam (see diagram).

1. Show that the weight of \(P\) is \(7.4\) N and find the total elastic potential energy stored in the two strings when \(P\) is hanging in equilibrium. [6]

\(P\) is then held at a point \(0.7\) m below the beam with the strings vertical. \(P\) is released from rest.

1. Show that, throughout the subsequent motion, \(P\) performs simple harmonic motion, and find the period. [7]

\includegraphics{figure_7} One end of a light inextensible string of length \(0.5\) m is attached to a fixed point \(O\). A particle \(P\) of mass \(0.2\) kg is attached to the other end of the string. \(P\) is projected horizontally from the point \(0.5\) m below \(O\) with speed \(u\text{ ms}^{-1}\). When the string makes an angle of \(\theta\) with the downward vertical the particle has speed \(v\text{ ms}^{-1}\) (see diagram).

1. Show that, while the string is taut, the tension, \(T\) N, in the string is given by $$T = 5.88\cos \theta + 0.4u^2 - 3.92.$$ [5]
2. Find the least value of \(u\) for which the particle will move in a complete circle. [3]
3. If in fact \(u = 3.5\text{ ms}^{-1}\), find the speed of the particle at the point where the string first becomes slack. [4]

END OF QUESTION PAPER

\includegraphics{figure_4} \(A\) and \(C\) are two fixed points, \(1.5\) m apart, on a smooth horizontal plane. A light elastic string of natural length \(0.4\) m and modulus of elasticity \(20\) N has one end fixed to point \(A\) and the other end fixed to a particle \(B\). Another light elastic string of natural length \(0.6\) m and modulus of elasticity \(15\) N has one end fixed to point \(C\) and the other end fixed to the particle \(B\). The particle is released from rest when \(ABC\) forms a straight line and \(AB = 0.4\) m (see diagram). Find the greatest kinetic energy of particle \(B\) in the subsequent motion. [7]

\includegraphics{figure_5} One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs at rest. \(P\) is then projected horizontally from this position with speed \(2\sqrt{ag}\). When the string makes an angle \(\theta\) with the upward vertical \(P\) has speed \(v\) (see diagram). The tension in the string is \(T\).

1. Find an expression for \(T\) in terms of \(m\), \(g\) and \(\theta\), and hence find the height of \(P\) above its initial level when the string becomes slack. [6]

\(P\) is now projected horizontally from the same initial position with speed \(U\).

1. Find the set of values of \(U\) for which the string does not remain taut in the subsequent motion. [5]

A particle \(P\) of mass \(m\) kg is attached to one end of a light elastic string of modulus of elasticity \(24mg\) N and natural length \(0.6\) m. The other end of the string is attached to a fixed point \(O\); the particle \(P\) rests in equilibrium at a point \(A\) with the string vertical.

1. Show that the distance \(OA\) is \(0.625\) m. [2]

Another particle \(Q\), of mass \(3m\) kg, is released from rest from a point \(0.4\) m above \(P\) and falls onto \(P\). The two particles coalesce.

1. Show that the combined particle initially moves with speed \(2.1\) m s\(^{-1}\). [3]
2. Show that the combined particle initially performs simple harmonic motion, and find the centre of this motion and its amplitude. [5]
3. Find the time that elapses between \(Q\) being released from rest and the combined particle first reaching the highest point of its subsequent motion. [7]

\includegraphics{figure_3} A uniform rod \(AB\), of mass \(m\) and length \(2a\), can rotate freely in a vertical plane about a fixed smooth horizontal axis through \(A\). The fixed point \(C\) is vertically above \(A\) and \(AC = 4a\). A light elastic string, of natural length \(2a\) and modulus of elasticity \(\frac{1}{4}mg\), joins \(B\) to \(C\). The rod \(AB\) makes an angle \(\theta\) with the upward vertical at \(A\), as shown in Fig. 3.

1. Show that the potential energy of the system is $$-mga[\cos \theta + \sqrt{(5 - 4 \cos \theta)}] + \text{constant}.$$ [9]
2. Hence determine the values of \(\theta\) for which the system is in equilibrium. [6]

\includegraphics{figure_1} Figure 1 shows a uniform rod \(AB\), of mass \(m\) and length \(4a\), resting on a smooth fixed sphere of radius \(a\). A light elastic string, of natural length \(a\) and modulus of elasticity \(\frac{1}{4}mg\), has one end attached to the lowest point \(C\) of the sphere and the other end attached to \(A\). The points \(A\), \(B\) and \(C\) lie in a vertical plane with \(\angle BAC = 2\theta\), where \(\theta < \frac{\pi}{4}\). Given that \(AC\) is always horizontal,

1. show that the potential energy of the system is $$\frac{mga}{8}(16\sin 2\theta + 3\cot^2 \theta - 6\cot \theta) + \text{constant}.$$ [7]
2. show that there is a value of \(\theta\) for which the system is in equilibrium such that \(0.535 < \theta < 0.545\). [6]
3. Determine whether this position of equilibrium is stable or unstable. [3]

A particle \(P\) of mass \(m\) is attached to the mid-point of a light elastic string, of natural length \(2L\) and modulus of elasticity \(2mk^2L\), where \(k\) is a positive constant. The ends of the string are attached to points \(A\) and \(B\) on a smooth horizontal surface, where \(AB = 3L\). The particle is released from rest at the point \(C\), where \(AC = 2L\) and \(ACB\) is a straight line. During the subsequent motion \(P\) experiences air resistance of magnitude \(2mkv\), where \(v\) is the speed of \(P\). At time \(t\), \(AP = 1.5L + x\).

1. Show that \(\frac{d^2x}{dt^2} + 2k\frac{dx}{dt} + 4k^2x = 0\). [6]
2. Find an expression, in terms of \(t\), \(k\) and \(L\), for the distance \(AP\) at time \(t\). [8]

\includegraphics{figure_1} A smooth wire \(PMQ\) is in the shape of a semicircle with centre \(O\) and radius \(a\). The wire is fixed in a vertical plane with \(PQ\) horizontal and the mid-point \(M\) of the wire vertically below \(O\). A smooth bead \(B\) of mass \(m\) is threaded on the wire and is attached to one end of a light elastic string. The string has modulus of elasticity \(4mg\) and natural length \(\frac{3}{4}a\). The other end of the string is attached to a fixed point \(P\) which is a distance \(a\) vertically above \(O\), as shown in Fig. 1.

1. Show that, when \(\angle BFO = \theta\), the potential energy of the system is $$\frac{1}{16}mga(8 \cos \theta - 5)^2 - 2mga \cos^2\theta + \text{constant}.$$ [6]
2. Hence find the values of \(\theta\) for which the system is in equilibrium. [6]
3. Determine the nature of the equilibrium at each of these positions. [5]

A particle of mass \(m\) is attached to one end \(P\) of a light elastic spring \(PQ\), of natural length \(a\) and modulus of elasticity \(man^2\). At time \(t = 0\), the particle and the spring are at rest on a smooth horizontal table, with the spring straight but unstretched and uncompressed. The end \(Q\) of the spring is then moved in a straight line, in the direction \(PQ\), with constant acceleration \(f\). At time \(t\), the displacement of the particle in the direction \(PQ\) from its initial position is \(x\) and the length of the spring is \((a + y)\).

1. Show that \(x + y = \frac{1}{2}ft^2\). [2]
2. Hence show that $$\frac{d^2x}{dt^2} + n^2x = \frac{1}{2}n^2ft^2.$$ [6]

You are given that the general solution of this differential equation is $$x = A\cos nt + B\sin nt + \frac{1}{2}ft^2 - \frac{f}{n^2},$$ where \(A\) and \(B\) are constants.

1. Find the values of \(A\) and \(B\). [6]
2. Find the maximum tension in the spring. [4]

A particle \(P\) of mass \(m\) is suspended from a fixed point by a light elastic spring. The spring has natural length \(a\) and modulus of elasticity \(2m\omega^2a\), where \(\omega\) is a positive constant. At time \(t = 0\) the particle is projected vertically downwards with speed \(U\) from its equilibrium position. The motion of the particle is resisted by a force of magnitude \(2m\omega v\), where \(v\) is the speed of the particle. At time \(t\), the displacement of \(P\) downwards from its equilibrium position is \(x\).

1. Show that \(\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\omega \frac{\mathrm{d}x}{\mathrm{d}t} + 2\omega^2x = 0\). [5] Given that the solution of this differential equation is \(x = e^{-\omega t}(A \cos \omega t + B \sin \omega t)\), where \(A\) and \(B\) are constants,
2. find \(A\) and \(B\). [4]
3. Find an expression for the time at which \(P\) first comes to rest. [3]

A non-uniform rod \(BC\) has mass \(m\) and length \(3l\). The centre of mass of the rod is at distance \(l\) from \(B\). The rod can turn freely about a fixed smooth horizontal axis through \(B\). One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\frac{mg}{6}\), is attached to \(C\). The other end of the string is attached to a point \(P\) which is at a height \(3l\) vertically above \(B\).

1. Show that, while the string is stretched, the potential energy of the system is $$mgl(\cos^2 \theta - \cos \theta) + \text{constant},$$ where \(\theta\) is the angle between the string and the downward vertical and \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\). [6]
2. Find the values of \(\theta\) for which the system is in equilibrium with the string stretched. [6]

\includegraphics{figure_1} A uniform rod \(PQ\) has mass \(m\) and length \(2l\). A small smooth light ring is fixed to the end \(P\) of the rod. This ring is threaded on to a fixed horizontal smooth straight wire. A second small smooth light ring \(R\) is threaded on to the wire and is attached by a light elastic string, of natural length \(l\) and modulus of elasticity \(kmg\), to the end \(Q\) of the rod, where \(k\) is a constant.

1. Show that, when the rod \(PQ\) makes an angle \(\theta\) with the vertical, where \(0 < \theta \leq \frac{\pi}{3}\), and \(Q\) is vertically below \(R\), as shown in Figure 1, the potential energy of the system is $$mgl[2k\cos^2\theta - (2k + 1)\cos\theta] + \text{constant}.$$ [7]

Given that there is a position of equilibrium with \(\theta > 0\),

1. show that \(k > \frac{1}{2}\). [5]

\includegraphics{figure_4} A light elastic spring has natural length \(l\) and modulus of elasticity \(4mg\). One end of the spring is attached to a point \(A\) on a plane that is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{3}{4}\). The other end of the spring is attached to a particle \(P\) of mass \(m\). The plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac{1}{4}\). The particle \(P\) is held at a point \(B\) on the plane where \(B\) is below \(A\) and \(AB = l\), with the spring lying along a line of greatest slope of the plane, as shown in Figure 4. At time \(t = 0\), the particle is projected up the plane towards \(A\) with speed \(\frac{1}{2}\sqrt{gl}\). At time \(t\), the compression of the spring is \(x\).

1. Show that $$\frac{d^2x}{dt^2} + 4\omega^2x = -g, \text{ where } \omega = \sqrt{\frac{g}{l}}.$$ [6]

1. Find \(x\) in terms of \(l\), \(\omega\) and \(t\). [7]

1. Find the distance that \(P\) travels up the plane before first coming to rest. [4]