2000

1) If \(T\) is the tension at any point \(P\) of a catenary and \(T_{0}\) that at the lowest point \(C\), then show that \(T^{2}-T_{0}^{2}=W^{2}\), where \(W\) is the weight of the arc \(CP\) of the catenary.

[12M]

2) A telephonic wire weighing 0.04 lb per foot has a horizontal span of 150 feel and sag of 1.5 feet. Find the length of the wire and also find maximum tension.

[12M]

1999

1) A perfectly rough plane is inclined at an angle \(\alpha\) to the horizon. Show that the least eccentricity of the ellipse which can rest on the plane is \(\left[ \dfrac{2sin\alpha}{1+sin\alpha} \right]^{1/2}\)

[10M]

2) A string of length \(a\) forms the shorter diagonal of a rhombus of four uniform rods, each of length \(b\) and weigth \(W\), are hinged together. If one of the rods be supported in a horizontal position, prove that the tension of the string is \(\dfrac{2W(2b^2-a^2)}{b(4b^2-a^2)^{1/2}}\)

[10M]

3) A uniform chain, of length \(l\) and weigth \(W\), hangs between two fixed points at the same level, and weigth \(W'\) is attached at the middle point. If \(K\) be the sag in the middle, prove that the pull on either point of support is \(\dfrac{K}{2l}W+\dfrac{l}{4K}W'+\dfrac{l}{8K}W.\)

[10M]

1998

1) A heavy elastic string whose natural length is \(2\pi a\) is, placed round a smooth cone whose axis vertical and whose semi-vertical angle is \(\alpha\). If \(W\) be the weight and \(\lambda\) the modulus of elasticity of the string, prove that it be in equilibrium when in the form of the circle whose radius is \(a \left( 1+\dfrac{W}{2\pi\lambda}cot\alpha \right)\).

[10M]

2) Show how to cut out of a uniform cylinder a cone, whose base coincides with that of a cylinder, so that the centre of gravity of the remaining solid may coincide with vertex of the cone.

[10M]

3) One end of an inextensible string is fixed to a point O and to the other end is tied a particle of mass \(m\). The particle is projected from its position of equilibrium vertically below O with a horizontal velocity so as to carry it right round the circle. Prove that the sum of the tensions at the ends of a diameter is constant.

[10M]

1997

1) A heavy uniform chain rests on a rough cycloid whose axis vertical and verter upwards, one end of the chain being at the vertex and the other at a cusp. If the equilibrium is limiting. show that \(\left(1+\mu^{2}\right) e^{\mu x / 2}=3\).

[10M]

2) A solid frustum of a paraboloid of revolution of height hand latus rectum \(4 \mathrm{a}\), rests with its vertex on the vertex of another parboloid (inverted) of revolution whose latus rectum is \(4 b\). Show that the equilibrium is state if \(h<\dfrac{3 a b}{a+b}\).

[10M]

1996

1) A body of weight \(W\) is placed on a rough inclined plane whose inclination to the horizon is \(\alpha\) greater than the angle of friction \(\lambda\). The body is supported by a force acting in a vertical plane through the line of greatest slope and makes an angle \(\theta\) with the inclined plane. Find the limits between which the force must lie.

[15M]

2) A body consisting of a cone and a hemisphere on the same base rests on a rough horizontal table, the hemisphere being in contact with the table. Show that the greatest height of the cone, so that the equilibrium may be stable, is \(\sqrt{3}\) times the radius of the sphere.

[15M]