6.02j Conservation with elastics: springs and strings

A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity \(5mg\). The other end of the spring is attached to a fixed point \(O\). The spring hangs vertically with \(P\) below \(O\). The particle \(P\) is pulled down vertically and released from rest when the length of the spring is \(\frac{7}{5}a\). Find the distance of \(P\) below \(O\) when \(P\) first comes to instantaneous rest. [4]

\includegraphics{figure_4} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. Sphere \(B\) is at rest on a smooth horizontal surface. Sphere \(A\) is moving on the surface with speed \(u\) at an angle of \(30°\) to the line of centres of \(A\) and \(B\) when it collides with \(B\) (see diagram). The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) after the collision is \(\frac{\sqrt{3}}{6}u(1 + e)\) and find the speed of \(A\) after the collision. [6]
2. Given that \(e = \frac{1}{2}\), find the loss of kinetic energy as a result of the collision. [3]

\includegraphics{figure_1} A light elastic string, of natural length \(3l\) and modulus of elasticity \(\lambda\), has its ends attached to two points \(A\) and \(B\), where \(AB = 3l\) and \(AB\) is horizontal. A particle \(P\) of mass \(m\) is attached to the mid-point of the string. Given that \(P\) rests in equilibrium at a distance \(2l\) below \(AB\), as shown in Figure 1,

1. show that \(\lambda = \frac{15mg}{16}\) [9]

The particle is pulled vertically downwards from its equilibrium position until the total length of the elastic string is \(7.8l\). The particle is released from rest.

1. Show that \(P\) comes to instantaneous rest on the line \(AB\). [6]

A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of \(5°\) to the horizontal. At a certain point \(P\) on the hill the car's speed is 20 m s\(^{-1}\). The point \(Q\) is 400 m further up the hill from \(P\), and at \(Q\) the car's speed is 15 m s\(^{-1}\).

1. Calculate the work done by the car's engine as the car moves from \(P\) to \(Q\), assuming that any resistances to the car's motion may be neglected. [4]

Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.

1. Given that the acceleration of the car at \(Q\) is zero, show that the power of the engine as the car passes through \(Q\) is 12.0 kW, correct to 3 significant figures. [3]
2. Given that the power of the car's engine at \(P\) is the same as at \(Q\), calculate the car's retardation at \(P\). [3]

Some tiles on a roof are being replaced. Each tile has a mass of 2 kg and the coefficient of friction between it and the existing roof is 0.75. The roof is at \(30°\) to the horizontal and the bottom of the roof is 6 m above horizontal ground, as shown in Fig. 4. \includegraphics{figure_4}

1. Calculate the limiting frictional force between a tile and the roof. A tile is placed on the roof. Does it slide? (Your answer should be supported by a calculation.) [5]
2. The tiles are raised 6 m from the ground, the only work done being against gravity. They are then slid 4 m up the roof and placed at the point A shown in Fig. 4.
   1. Show that each tile gains 156.8 J of gravitational potential energy. [3]
   2. Calculate the work done against friction per tile. [2]
   3. What average power is required to raise 10 tiles per minute from the ground to A? [2]
3. A tile is kicked from A directly down the roof. When the tile is at B, \(x\) m from the edge of the roof, its speed is \(4 \text{ m s}^{-1}\). It subsequently hits the ground travelling at \(9 \text{ m s}^{-1}\). In the motion of the tile from B to the ground, the work done against sliding and other resistances is 90 J. Use an energy method to find \(x\). [5]

A particle \(P\) of mass \(m\) kg is attached to one end of a light elastic string of natural length \(l\) m and modulus of elasticity \(\lambda\) N. The other end of the string is attached to a fixed point \(O\). \(P\) is released from rest at \(O\) and falls vertically downwards under gravity. The greatest distance below \(O\) reached by \(P\) is \(2l\) m.

1. Show that \(\lambda = 4mg\). [3 marks]
2. Find, in terms of \(l\) and \(g\), the speed with which \(P\) passes through the point \(A\), where \(OA = \frac{5l}{4}\) m. [6 marks]

A particle \(P\) is attached to one end of a light inextensible string of length \(l\) m. The other end of the string is attached to a fixed point \(O\). When \(P\) is hanging at rest vertically below \(O\), it is given a horizontal speed \(u\) ms\(^{-1}\) and starts to move in a vertical circle. Given that the string becomes slack when it makes an angle of 120° with the downward vertical through \(O\),

1. show that \(u^2 = \frac{7gl}{2}\). [10 marks]
2. Find, in terms of \(l\), the greatest height above \(O\) reached by \(P\) in the subsequent motion. [7 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_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_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]

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]

A light elastic string, of natural length \(a\) and modulus of elasticity \(5ma\omega^2\), lies unstretched along a straight line on a smooth horizontal plane. A particle of mass \(m\) is attached to one end of the spring. At time \(t = 0\), the other end of the spring starts to move with constant speed \(U\) along the line of the spring and away from the particle. As the particle moves along the plane it is subject to a resistance of magnitude \(2m\omega v\), where \(v\) is its speed. At time \(t\), the extension of the spring is \(x\) and the displacement of the particle from its initial position is \(y\). Show that

1. \(x + y = Ut\), [2]
2. \(\frac{d^2x}{dt^2} + 2\omega \frac{dx}{dt} + 5\omega^2 x = 2\omega U\). [7]

1. Find \(x\) in terms of \(\omega\), \(U\) and \(t\). [8]

In a school experiment, a particle, of mass \(m\) kilograms, is released from rest at a point \(h\) metres above the ground. At the instant it reaches the ground, the particle has velocity \(v \text{ m s}^{-1}\)

1. Show that $$v = \sqrt{2gh}$$ [2 marks]
2. A student correctly used \(h = 18\) and measured \(v\) as 20 The student's teacher claims that the machine measuring the velocity must have been faulty. Determine if the teacher's claim is correct. Fully justify your answer. [3 marks]

In this question use \(g = 9.81 \text{ m s}^{-2}\) A conical pendulum is made from an elastic string and a sphere of mass 0.2 kg The string has natural length 1.6 metres and modulus of elasticity 200 N The sphere describes a horizontal circle of radius 0.5 metres at a speed of \(v \text{ m s}^{-1}\) The angle between the elastic string and the vertical is \(\alpha\)

1. Show that $$62.5 - 200 \sin \alpha = 1.962 \tan \alpha$$ [5 marks]
2. Use your calculator to find \(\alpha\) [1 mark]
3. Find the value of \(v\) [4 marks]

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]

\includegraphics{figure_7} A particle P of mass \(m\) is attached to one end of a light elastic string of natural length \(6a\) and modulus of elasticity \(3mg\). The other end of the string is fixed to a point O on a smooth plane, which is inclined at an angle of \(30°\) to the horizontal. The string lies along a line of greatest slope of the plane and P rests in equilibrium on the inclined plane at a point A, as shown in Fig. 7. P is now pulled a further distance \(2a\) down the line of greatest slope through A and released from rest. At time \(t\) later, the displacement of P from A is \(x\), where the positive direction of \(x\) is down the plane.

1. Show that, until the string slackens, \(x\) satisfies the differential equation $$\frac{d^2x}{dt^2} + \frac{gx}{2a} = 0.$$ [6]
2. Determine, in terms of \(a\) and \(g\), the time at which the string slackens. [5]
3. Find, in terms of \(a\) and \(g\), the speed of P when the string slackens. [2]

3 A light elastic string has natural length 0.8 m and modulus of elasticity 16 N . One end of the string is attached to a fixed point \(O\), and a particle \(P\) of mass 0.4 kg is attached to the other end of the string. The particle \(P\) hangs in equilibrium vertically below \(O\).

1. Show that the extension of the string is 0.2 m . \(P\) is projected vertically downwards from the equilibrium position. \(P\) first comes to instantaneous rest at the point where \(O P = 1.4 \mathrm {~m}\).
2. Calculate the speed at which \(P\) is projected.
3. Find the speed of \(P\) at the first instant when the string subsequently becomes slack.