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

\includegraphics{figure_6} A particle of mass 5 kg is attached to a string \(AB\) and a rod \(BC\) at the point \(B\). The string \(AB\) is light and elastic with modulus \(\lambda\) N and natural length 2 m. The rod \(BC\) is light and of length 2 m. The end \(A\) of the string is attached to a fixed point and the end \(C\) of the rod is attached to another fixed point such that \(A\) is vertically above \(C\) with \(AC = 2\) m. When the particle rests in equilibrium, \(AB\) makes an angle of 50° with the downward vertical.

1. Determine, in terms of \(\lambda\), the tension in the string \(AB\). [3]
2. Calculate, in terms of \(\lambda\), the energy stored in the string \(AB\). [2]
3. Find, in terms of \(\lambda\), the thrust in the rod \(BC\). [4]

A bungee jumper of mass 80 kg steps off a high bridge with an elastic rope attached to her ankles. She is assumed to fall vertically from rest and the air resistance she experiences is modelled as a constant force of 32N. The rope has natural length 4 m and modulus of elasticity of 470 N. By considering energy, determine the total distance she falls before first coming to instantaneous rest. [6]

\includegraphics{figure_3} A light elastic string has natural length \(8a\) and modulus of elasticity \(5mg\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(AP = BP = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(AB\) it has speed \(\sqrt{80ag}\).

1. Find \(L\) in terms of \(a\). [5]
2. Find the initial acceleration of \(P\) in terms of \(g\). [3]

\(A\) 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. 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]

One end of a light spring of length 0.5 m is attached to a fixed point \(F\). A particle \(P\) of mass 2.5 kg is attached to the other end of the spring and hangs in equilibrium 0.55 m below \(F\). Another particle \(Q\), of mass 1.5 kg, is attached to \(P\), without moving it, and both particles are then released.

1. Show that the modulus of elasticity of the spring is 250 N. [2]
2. Prove that the motion is simple harmonic. [4]
3. Find
   1. the amplitude of the motion, [1]
   2. the greatest speed of the particles, [1]
   3. the period of the motion, [1]
   4. the time taken for the spring to be extended by 0.1 m for the first time. [3]

A light elastic string of natural length \(2a\) and modulus of elasticity \(\lambda\) is stretched between two points \(A\) and \(B\), which are \(3a\) apart on a smooth horizontal table. A particle of mass \(m\) is attached to the mid-point of the string, pulled aside to \(A\) and released.

1. Prove that, while one part of the string is taut and the other part is slack, the particle is describing simple harmonic motion. [2]
2. Find the speed of the particle when the slack part of the string becomes taut. [2]
3. Prove that the total time for the particle to reach the mid-point of the string for the first time is $$\sqrt{\frac{ma}{\lambda}} \left( \frac{\pi}{3} + \frac{1}{\sqrt{2}} \sin^{-1} \frac{1}{\sqrt{7}} \right).$$ [8]