6.02h Elastic PE: 1/2 k x^2

A small ball is attached to one end of a spring. The ball is modelled as a particle of mass 0.1 kg and the spring is modelled as a light elastic spring \(AB\), of natural length 0.5 m and modulus of elasticity 2.45 N. The particle is attached to the end \(B\) of the spring. Initially, at time \(t = 0\), \(A\) is held at rest and the particle hangs at rest in equilibrium below \(A\) at the point \(E\). The end \(A\) then begins to move along the line of the spring in such a way that, at time \(t\) seconds, \(t \leq 1\), the downward displacement of \(A\) from its initial position is \(2 \sin 2t\) metres. At time \(t\) seconds, the extension of the spring is \(x\) metres and the displacement of the particle below \(E\) is \(y\) metres.

1. Show, by referring to a simple diagram, that \(y + 0.2 = x + 2 \sin 2t\). [3]
2. Hence show that \(\frac{d^2y}{dt^2} + 49y = 98 \sin 2t\). [5]

Given that \(y = \frac{98}{45} \sin 2t\) is a particular integral of this differential equation,

1. find \(y\) in terms of \(t\). [5]
2. Find the time at which the particle first comes to instantaneous rest. [4]

A particle \(P\) of mass \(m\) kg is attached to the end \(A\) of a light elastic string \(AB\), of natural length \(a\) metres and modulus of elasticity \(9ma\) newtons. Initially the particle and the string lie at rest on a smooth horizontal plane with \(AB = a\) metres. At time \(t = 0\) the end \(B\) of the string is set in motion and moves at a constant speed \(U\) m s\(^{-1}\) in the direction \(AB\). The air resistance acting on \(P\) has magnitude \(6mv\) newtons, where \(v\) m s\(^{-1}\) is the speed of \(P\). At time \(t\) seconds, the extension of the string is \(x\) metres and the displacement of \(P\) from its initial position is \(y\) metres. Show that, while the string is taut,

1. \(x + y = Ut\) [2]
2. \(\frac{d^2x}{dt^2} + 6\frac{dx}{dt} + 9x = 6U\) [5]

You are given that the general solution of the differential equation in (b) is $$x = (A + Bt)e^{-3t} + \frac{2U}{3}$$ where \(A\) and \(B\) are arbitrary constants.

1. Find the value of \(A\) and the value of \(B\). [5]
2. Find the speed of \(P\) at time \(t\) seconds. [2]

\includegraphics{figure_2} A railway truck of mass \(M\) approaches the end of a straight horizontal track and strikes a buffer. The buffer is parallel to the track, as shown in Figure 2. The buffer is modelled as a light horizontal spring \(PQ\), which is fixed at the end \(P\). The spring has a natural length \(a\) and modulus of elasticity \(Mn^2a\), where \(n\) is a positive constant. At time \(t = 0\), the spring has length \(a\) and the truck strikes the end \(Q\) with speed \(U\). A resistive force whose magnitude is \(Mkv\), where \(v\) is the speed of the truck at time \(t\), and \(k\) is a positive constant, also opposes the motion of the truck. At time \(t\), the truck is in contact with the buffer and the compression of the buffer is \(x\).

1. Show that, while the truck is compressing the buffer $$\frac{\text{d}^2x}{\text{d}t^2} + k\frac{\text{d}x}{\text{d}t} + n^2x = 0$$ (4)

It is given that \(k = \frac{5n}{2}\)

1. Find \(x\) in terms of \(U\), \(n\) and \(t\). (7)

1. Find, in terms of \(U\) and \(n\), the greatest value of \(x\). (5)

A particle of mass \(m\) kg is attached to one end of a light elastic string of natural length \(a\) metres and modulus of elasticity \(5ma\) newtons. The other end of the string is attached to a fixed point \(O\) on a smooth horizontal plane. The particle is held at rest on the plane with the string stretched to a length \(2a\) metres and then released at time \(t = 0\). During the subsequent motion, when the particle is moving with speed \(v\) m s\(^{-1}\), the particle experiences a resistance of magnitude \(4mv\) newtons. At time \(t\) seconds after the particle is released, the length of the string is \((a + x)\) metres, where \(0 \leqslant x \leqslant a\).

1. Show that, from \(t = 0\) until the string becomes slack, $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 4\frac{\mathrm{d}x}{\mathrm{d}t} + 5x = 0$$ [3]
2. Hence express \(x\) in terms of \(a\) and \(t\). [6]
3. Find the speed of the particle at the instant when the string first becomes slack, giving your answer in the form \(ka\), where \(k\) is a constant to be found correct to 2 significant figures. [4]

\includegraphics{figure_3} Two uniform rods \(AB\) and \(BC\), each of length \(a\) and mass \(m\), are rigidly joined together so that \(AB\) is perpendicular to \(BC\). The rod \(AB\) is freely hinged to a fixed point at \(A\). The rods can rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda mg\) is attached to \(B\). The other end of the string is attached to a fixed point \(D\) vertically above \(A\), where \(AD = a\). The string \(BD\) makes an angle \(\theta\) radians with the downward vertical (see diagram).

1. Taking \(D\) as the reference level for gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$V = \frac{1}{2}mga(\sin 2\theta - 3\cos 2\theta) + \frac{1}{2}\lambda mga(2\cos \theta - 1)^2 - 2mga.$$ [5]
2. Given that \(\theta = \frac{1}{3}\pi\) is a position of equilibrium, find the exact value of \(\lambda\). [4]
3. Find \(\frac{d^2V}{d\theta^2}\) and hence determine whether the position of equilibrium at \(\theta = \frac{1}{3}\pi\) is stable or unstable. [4]

\includegraphics{figure_4} **Figure 1** A uniform lamina of mass \(M\) is in the shape of a right-angled triangle \(OAB\). The angle \(OAB\) is \(90°\), \(OA = a\) and \(AB = 2a\), as shown in Figure 1.

1. Prove, using integration, that the moment of inertia of the lamina \(OAB\) about the edge \(OA\) is \(\frac{8}{3}Ma^2\). (You may assume without proof that the moment of inertia of a uniform rod of mass \(m\) and length \(2l\) about an axis through one end and perpendicular to the rod is \(\frac{4}{3}ml^2\).) [6]

The lamina \(OAB\) is free to rotate about a fixed smooth horizontal axis along the edge \(OA\) and hangs at rest with \(B\) vertically below \(A\). The lamina is then given a horizontal impulse of magnitude \(J\). The impulse is applied to the lamina at the point \(B\), in a direction which is perpendicular to the plane of the lamina. Given that the lamina first comes to instantaneous rest after rotating through an angle of \(120°\),

1. find an expression for \(J\), in terms of \(M\), \(a\) and \(g\). [7]

Use \(g\) as 9.81 m s\(^{-2}\) in this question. A light elastic string has one end attached to a fixed point A on a smooth plane inclined at 25° to the horizontal. The other end of the string is attached to a wooden block of mass 2.5 kg, which rests on the plane. The elastic string has natural length 3 metres and modulus of elasticity 125 newtons. The block is pulled down the line of greatest slope of the plane to a point 4.5 metres from A and then released.

1. Find the elastic potential energy of the string at the point when the block is released. [1 mark]
2. Calculate the speed of the block when the string becomes slack. [4 marks]
3. Determine whether the block reaches the point A in the subsequent motion, commenting on any assumptions that you make. [3 marks]

An elastic string has modulus of elasticity 20 newtons and natural length 2 metres. The string is stretched so that its extension is 0.5 metres. Find the elastic potential energy stored in the string. Circle your answer. 1.25 J \quad\quad 5.5 J \quad\quad 5 J \quad\quad 10 J [1 mark]

In this question use \(g = 10 \text{ m s}^{-2}\) A light spring is attached to the base of a long tube and has a mass \(m\) attached to the other end, as shown in the diagram. The tube is filled with oil. When the compression of the spring is \(c\) metres, the thrust in the spring is \(9mc\) newtons. \includegraphics{figure_14} The mass is held at rest in a position where the compression of the spring is \(\frac{20}{9}\) metres. The mass is then released from rest. During the subsequent motion the oil causes a resistive force of \(6mv\) newtons to act on the mass, where \(v \text{ m s}^{-1}\) is the speed of the mass. At time \(t\) seconds after the mass is released, the displacement of the mass above its starting position is \(x\) metres.

1. Find \(x\) in terms of \(t\). [10 marks]
2. State, giving a reason, the type of damping which occurs. [1 mark]

In this question use \(g = 9.8\) m s\(^{-2}\) A particle \(P\) of mass \(m\) is attached to two light elastic strings, \(AP\) and \(BP\). The other ends of the strings, \(A\) and \(B\), are attached to fixed points which are 4 metres apart on a rough horizontal surface at the bottom of a container. The coefficient of friction between \(P\) and the surface is 0.68 • When the extension of string \(AP\) is \(e_A\) metres, the tension in \(AP\) is \(24me_A\) • When the extension of string \(BP\) is \(e_B\) metres, the tension in \(BP\) is \(10me_B\) • The natural length of string \(AP\) is 1 metre • The natural length of string \(BP\) is 1.3 metres \includegraphics{figure_15}

1. Show that when \(AP = 1.5\) metres, the tension in \(AP\) is equal to the tension in \(BP\). [1 mark]
2. \(P\) is held at the point between \(A\) and \(B\) where \(AP = 1.9\) metres, and then released from rest. At time \(t\) seconds after \(P\) is released, \(AP = (1.5 + x)\) metres. \includegraphics{figure_15b} Show that when \(P\) is moving towards \(A\), $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 34x = 6.664$$ [3 marks]
3. The container is then filled with oil, and \(P\) is again released from rest at the point between \(A\) and \(B\) where \(AP = 1.9\) metres. At time \(t\) seconds after \(P\) is released, the oil causes a resistive force of magnitude \(10mv\) newtons to act on the particle, where \(v\) m s\(^{-1}\) is the speed of the particle. Find \(x\) in terms of \(t\) when \(P\) is moving towards \(A\). [9 marks]

In this question use \(g\) as \(10\,\text{m}\,\text{s}^{-2}\) A smooth plane is inclined at \(30°\) to the horizontal. The fixed points \(A\) and \(B\) are 3.6 metres apart on the line of greatest slope of the plane, with \(A\) higher than \(B\) A particle \(P\) of mass 0.32 kg is attached to one end of each of two light elastic strings. The other ends of these strings are attached to the points \(A\) and \(B\) respectively. The particle \(P\) moves on a straight line that passes through \(A\) and \(B\) \includegraphics{figure_2} The natural length of the string \(AP\) is 1.4 metres. When the extension of the string \(AP\) is \(e_A\) metres, the tension in the string \(AP\) is \(7e_A\) newtons. The natural length of the string \(BP\) is 1 metre. When the extension of the string \(BP\) is \(e_B\) metres, the tension in the string \(BP\) is \(9e_B\) newtons. The particle \(P\) is held at the point between \(A\) and \(B\) which is 0.2 metres from its equilibrium position and lower than its equilibrium position. The particle \(P\) is then released from rest. At time \(t\) seconds after \(P\) is released, its displacement towards \(B\) from its equilibrium position is \(x\) metres.

1. Show that during the subsequent motion the object satisfies the equation $$\ddot{x} + 50x = 0$$ Fully justify your answer. [5 marks]
2. The experiment is repeated in a large tank of oil. During the motion the oil causes a resistive force of \(kv\) newtons to act on the particle, where \(v\,\text{m}\,\text{s}^{-1}\) is the speed of the particle. The oil causes critical damping to occur.
   1. Show that \(k = \frac{16\sqrt{2}}{5}\) [3 marks]
   2. Find \(x\) in terms of \(t\), giving your answer in exact form. [6 marks]
   3. Calculate the maximum speed of the particle. [5 marks]

In this question use \(g = 9.8\) m s\(^{-2}\) Two light elastic strings each have one end attached to a small ball \(B\) of mass 0.5 kg The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart with \(A\) vertically above \(C\) The whole system is in a thin tube of oil, as shown in the diagram below. \includegraphics{figure_18} The string connecting \(A\) and \(B\) has natural length 2 metres, and the tension in this string is \(7e\) newtons when the extension is \(e\) metres. The string connecting \(B\) and \(C\) has natural length 3 metres, and the tension in this string is \(3e\) newtons when the extension is \(e\) metres.

1. Find the extension of each string when the system is in equilibrium. [3 marks]
2. It is known that in a large bath of oil, the oil causes a resistive force of magnitude \(4.5v\) newtons to act on the ball, where \(v\) m s\(^{-1}\) is the speed of the ball. Use this model to answer part (b)(i) and part (b)(ii).
   1. The ball is pulled a distance of 0.6 metres downwards from its equilibrium position towards \(C\), and released from rest. Show that during the subsequent motion the particle satisfies the differential equation $$\frac{d^2x}{dt^2} + 9\frac{dx}{dt} + 20x = 0$$ where \(x\) metres is the displacement of the particle below the equilibrium position at time \(t\) seconds after the particle is released. [3 marks]
   2. Find \(x\) in terms of \(t\) [5 marks]
3. State one limitation of the model used in part (b) [1 mark]

A spring of natural length 50 cm and modulus of elasticity \(\lambda\) newtons has an elastic potential energy of 4 J when compressed by 5 cm. Find the value of \(\lambda\) Circle your answer. [1 mark] 8 16 800 1600

In this question use \(g = 9.8\) m s\(^{-2}\) A light elastic string has natural length 3 metres and modulus of elasticity 18 newtons. One end of the elastic string is attached to a particle of mass 0.25 kg The other end of the elastic string is attached to a fixed point \(O\) The particle is released from rest at a point \(A\), which is 4.5 metres vertically below \(O\)

1. Calculate the elastic potential energy of the string when the particle is at \(A\) [2 marks]
2. The point \(B\) is 3 metres vertically below \(O\) Calculate the gravitational potential energy gained by the particle as it moves from \(A\) to \(B\) [2 marks]
3. Find the speed of the particle at \(B\) [3 marks]
4. The point \(C\) is 3.6 metres vertically below \(O\) Explain, showing any calculations that you make, why the speed of the particle is increasing the first time that the particle is at \(C\) [3 marks]

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]

\includegraphics{figure_13} A step-ladder has two sides AB and AC, each of length \(4a\). Side AB has weight \(W\) and its centre of mass is at the half-way point; side AC is light. The step-ladder is smoothly hinged at A and the two parts of the step-ladder, AB and AC, are connected by a light taut rope DE, where D is on AB, E is on AC and AD = AE = \(a\). A man of weight \(4W\) stands at a point F on AB, where BF = \(x\). The system is in equilibrium with B and C on a smooth horizontal floor and the sides AB and AC are each at an angle \(\theta\) to the vertical, as shown in Fig. 13.

1. By taking moments about A for side AB of the step-ladder and then for side AC of the step-ladder show that the tension in the rope is $$W\left(1 + \frac{2x}{a}\right)\tan\theta.$$ [7]

The rope is elastic with natural length \(\frac{1}{2}a\) and modulus of elasticity \(W\).

1. Show that the condition for equilibrium is that $$x = \frac{1}{2}a(8\cos\theta - \cot\theta - 1).$$ [5]

In this question you must show detailed reasoning.

1. Hence determine, in terms of \(a\), the maximum value of \(x\) for which equilibrium is possible. [5]

END OF QUESTION PAPER

In this question take \(g = 10\). A particle P of mass 0.15 kg is attached to one end of a light elastic string of modulus of elasticity 13.5 N and natural length 0.45 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 0.5 m. [3]

At time \(t = 0\), P is projected vertically downwards from A with a speed of 1.25 m s\(^{-1}\). Throughout the subsequent motion, P experiences a variable resistance \(R\) newtons which is of magnitude 0.6 times its speed (in m s\(^{-1}\)).

1. Given that the downward displacement of P from A at time \(t\) seconds is \(x\) metres, show that, while the string remains taut, \(x\) satisfies the differential equation $$\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 200x = 0.$$ [3]
2. Verify that \(x = \frac{5}{56}e^{-2t}\sin(14t)\). [6]
3. Determine whether the string becomes slack during the motion. [5]

A particle P of mass \(m\) is fixed to one end of a light spring of natural length \(a\) and modulus of elasticity \(man^2\), where \(n > 0\). The other end of the spring is attached to the ceiling of a lift. The lift is at rest and P is hanging vertically in equilibrium.

1. Find, in terms of \(g\) and \(n\), the extension in the spring. [3]

At time \(t = 0\) the lift begins to accelerate upwards from rest. At time \(t\), the upward displacement of the lift from its initial position is \(y\) and the extension of the spring is \(x\).

1. Express, in terms of \(g\), \(n\), \(x\) and \(y\), the upward displacement of P from its initial position at time \(t\). [2]
2. Given that \(\ddot{y} = kt\), where \(k\) is a positive constant, express the upward acceleration of P in terms of \(\ddot{x}\), \(k\) and \(t\). [1]
3. Show that \(x\) satisfies the differential equation $$\ddot{x} + n^2 x = kt + g.$$ [3]
4. Verify that \(x = \frac{1}{n^2}(knt + gn - k \sin(nt))\). [4]
5. By considering \(\ddot{x}\) comment on the motion of P relative to the ceiling of the lift for all times after the lift begins to move. [2]

One end of a light spring is attached to a fixed point. A mass of 2 kg is attached to the other end of the spring. The spring hangs vertically in equilibrium. The extension of the spring is 0.05 m.

1. Find the stiffness of the spring. [2]
2. Find the energy stored in the spring. [2]
3. Find the dimensions of stiffness of a spring. [1]

A particle P of mass \(m\) is performing complete oscillations with amplitude \(a\) on the end of a light spring with stiffness \(k\). The spring hangs vertically and the maximum speed \(v\) of P is given by the formula $$v = Cm^{\alpha}a^{\beta}k^{\gamma},$$ where C is a dimensionless constant.

1. Use dimensional analysis to determine \(\alpha\), \(\beta\), and \(\gamma\). [4]

A particle P of mass \(3m\) kg is attached to one end of a light elastic string of modulus of elasticity \(4mg\) N and natural length 0.4 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. Find the distance OA. [2]

At time \(t = 0\) seconds, P is given a speed of \(2.5 \text{ m s}^{-1}\) vertically downwards from A.

1. Show that P initially performs simple harmonic motion with amplitude \(a\) m, where \(a\) is to be determined correct to 3 significant figures. [5]
2. Determine the smallest distance between P and O in the subsequent motion. [3]

A particle P of mass \(0.5\) kg is attached to a fixed point O by a light elastic string of natural length \(3\) m and modulus of elasticity \(75\) N. P is released from rest at O and is allowed to fall freely. Determine the length of the string when P is at its lowest point in the subsequent motion. [5]

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]

A light elastic string of natural length \(1.5\) m and modulus of elasticity \(490\) N has one end attached to a fixed point \(A\) and the other end attached to a particle \(P\) of mass \(30\) kg. Initially, \(P\) is held at rest vertically below \(A\) such that the distance \(AP\) is \(0.6\) m. It is then allowed to fall vertically.

1. Calculate the distance \(AP\) when \(P\) is instantaneously at rest for the first time, giving your answer correct to 2 decimal places. [8]
2. Estimate the distance \(AP\) when \(P\) is instantaneously at rest for the second time and clearly state one assumption that you have made in making your estimate. [2]

One end of a light elastic string, of natural length 2.5 m and modulus of elasticity \(30g\) N, is fixed to a point O. A particle \(P\), of mass 2 kg, is attached to the other end of the string. Initially, \(P\) is held at rest at the point O. It is then released and allowed to fall under gravity.

1. Show that, while the string is taut, $$v^2 = g(5 + 2x - 6x^2),$$ where \(v\text{ ms}^{-1}\) denotes the velocity of the particle when the extension in the string is \(x\) m. [6]
2. Calculate the maximum extension of the string. [3]
3. 1. Find the extension of the string when \(P\) attains its maximum speed.
   2. Hence determine the maximum speed of \(P\). [5]

One end of a light elastic string, of natural length \(0.2\) m and modulus of elasticity \(5g\) N, is attached to a fixed point \(O\). The other end is attached to a particle of mass \(4\) kg. The particle hangs in equilibrium vertically below \(O\).

1. Show that the extension of the string is \(0.16\) m. [2]
2. The particle is pulled down vertically and held at rest so that the extension of the string is \(0.28\) m. The particle is then released. Determine the speed of the particle as it passes through the equilibrium position. [8]