Elastic string equilibrium and statics

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]

\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]