1995

1) Two uniform rods AB and AC, smoothly jointed at A, are in equilibrium in a vertical plane. The ends $\mathrm{B}$ and $\mathrm{C}$ rest on a smooth horizontal plane and the middle points of $\mathrm{A}\mathrm{B}$ and $\mathrm{A}\mathrm{C}$ are connected by a string. Show that the tension of the string is ${\displaystyle \frac{W}{(\tan B+\tan C)}},$ where $\mathrm{W}$ is the total weight of the rods and $B$ and $C$ are the inclinations to the horizontal of the rods $AB$ and $AC$

[20M]

2) Prove that for the common catenary the radius of curvature al any point of the curve is equal to the length of the normal intercepted between the curve and the directrix.

[20M]

1994

1) Show that the length of an endless chain which will hang over a circular pulley of radius a so as to be in contact with two-thirds of the circumference of the pulley is $a\{{\displaystyle \frac{3}{\log (2+\sqrt{3})}}+{\displaystyle \frac{4\pi }{3}}\}$

[10M]

2) A smooth rod passes through a smooth fing at the focus of an ellipse whose major axis is horizontal and rests with its lower end on the quadrant of the curve which is farthest removed from the focus. Find its position of equilibrium and show that its length must at least be $({\displaystyle \frac{3a}{4}}+{\displaystyle \frac{a}{4}}\sqrt{1+8e^2})$ where $2a$ is the major axis and $e,$ the eccentricity

[10M]

1993

1) The end links of a uniform chain slide along a fixed rough horizontal rod. Prove that the ratio of the maximum span to the length of the chain is

where $\mu$ is the coefficient of friction.

[10M]

2) A solid hemisphere is supported by a string fixed to a point on its rim and to point on a smooth vertical wall with which the curved surface of the sphere is in contact If $\theta$ and $\varphi$ are the inclinations of the string and the plane base of the hemisphere to the vertical, prove that $\tan \varphi ={\displaystyle \frac{3}{8}}+\tan \theta$.

[10M]

1992

1) Two equal rods, each of weight $w$ and length $l,$ are hinged together and placed astride a smooth horizontal cylindrical peg of radius $\mathrm{r}$. Then the lower ends are tied together by a string and the rods are left at the same inclination $\varphi$ to the horizontal. Find the tension in the string and if the string is slack, show that $\varphi$ satisfies the equation

[10M]

2) Define central axis for a system of forces acting on a rigid body. A force $\mathrm{F}$ acts along the axis of $x$ and another force $nF$ along a generator of the cylinder $x^2+y^2=a^2$, Show that the central axis lies on the cylinder. $$ n^{2}(n x-z)^{2}+\left(1+n^{2}\right)^{2} y^{2}=n^{4} a^{2} $$

[10M]