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 ${\displaystyle \frac{21}{\lambda }}\log \{\lambda +\sqrt{(1+{\lambda }^2)}\}$, where ${\displaystyle \frac{1}{\lambda }}=\mu (2n+1)\cdot \mu$ 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 $\mathrm{A}\mathrm{B}$ 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 $r\{{\displaystyle \frac{3}{\log (2+\sqrt{3})}}+{\displaystyle \frac{4\pi }{3}}\}$.

[10M]