2004

1) A heavy uniform chain AB hangs freely under gravity with A fixed and \(B\) attached by a light string BC to a fixed point C at the same level as A. The chain \(\mathrm{AB}\) and the string \(\mathrm{BC}\) make angles \(60^{\circ}\) and \(30^{\circ}\) respectively with the horizontal. Find the ratio of the length of the string to that of the chain.

[10M]

2) Three forces each of magnitude \(P\) and acting in the positive directions of the axes have their lines of action \(-y=z=2-z=x-2-x=y=2\)

[12M]

TBC

2003

1) A uniform rod \(AB\) of length \(2a\) is hinged at \(A\), a string atached to the middle point \(G\) of the rod passes over a smooth pulley at \(\mathrm{C}\) at a height a, vertically above \(\mathrm{A}\), and supports a weight P having freely, find the positions of equitibrium and determine their nature as to stability or unstability.

[10M]

2002

1) A smooth rod passes through a smooth ring at the focus of an cllipse 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 the positions of equilibrium and show that is length must be at least \(\dfrac{3 a}{4}+\dfrac{a}{4} \sqrt{1+8 e^{2}},\) where \(2 a\) is the major axis and \(e\) the eccentricity.

[10M]

2001

1) A solid sphere rests inside a fixed rough hemispherical bowl of twice its radius. Show that however large a weight is attached to the highest point of the sphere, the equilibrium is stable.

[10M]

2) A smooth parabolic wire is fixed with its axis vertical and vertex downéards and in it is placed a uniform rod of length \(2 l\) with its ends resting on wire. Show that, for equilibrium, the rod is either horizontal or makes with horizontal an angle \(\theta\) given by \(\cos ^{2} \theta=\dfrac{2 a}{l}, 4 a\) being the latus rectum of the parabola.

[14M]

2000

1) Use the principle of virtual work to find the inclination of a rod to the horizontal, when the rod lies in equilibrium with its ends on two smooth planes inclined at angles \(30^{\circ}\) and \(60^{\circ}\) to the horizontal, the planes intersecting in a horizontal line.