2007

1) A heavy chain of length \(2 /\) has one end tied at \(\mathrm{A}\) and the other is attached to a small heavy ring which can slide on a rough horizontal rod which passes through A. If the weight of the ring be \(n\) times the weight of the chain, show that its greatest possible distance from \(\mathrm{A}\) is \(\dfrac{21}{\lambda} \log \left\{\lambda+\sqrt{\left(1+\lambda^{2}\right)}\right\}\), where \(\dfrac{1}{\lambda}=\mu(2 n+1) \cdot \mu\) being the coefficient of friction.

[13M]

2006

1) A uniform beam rests tangentially upon a smooth curve in a vertical plane and one end of the beam rests against a smooth vertical wall, If the beam is in equilibrium in any position, find the equation to the curve.

[13M]

2005

1) A regular hexagon \(ABCDEF\) consists of six equal rods which are each of weight \(W\) and are freely jointed together. The hexagon rests in a vertical plane and \(\mathrm{AB}\) is in contact with a horizontal table. If \(\mathrm{C}\) and \(\mathrm{F}\) be connected by a light string, prove that the tension in the string is \(W \sqrt{3}\).

[10M]

2) Show that the length of an endless chain which will hang oyer a circular pulley of radius \(r\) so as to be in contact with two- thirds of the circumference of the pulley is \(r\left\{\dfrac{3}{\log (2+\sqrt{3})}+\dfrac{4 \pi}{3}\right\}\).

[10M]