PYQs

1) A body consists of a cone and underlying hemisphere. The base of the cone and the top of the hemisphere have same radius \(a\). The whole body rests on a rough horizontal table with hemisphere in contact with the table. Show that the greatest height of the cone, so that the equilibrium may be stable, is \(\sqrt{3}a\).

[2019, 15M]

2) A uniform solid hemisphere rests on a rough plane inclined to the horizon at an angle \(\phi\) with its curved surface touching the plane. Find the greatest admissible value of the inclination \(\phi\) for equilibrium. If \(\phi\) be less than this value, is the equilibrium stable?

[2017, 17M]

3) A heavy hemispherical shell of radius \(a\) has a particle attached to a point on the rim, and rests with the curved surface in contact with a rough sphere of radius \(b\) at the highest point. Prove that if \(\dfrac{b}{a}>\sqrt{5}-1\), the equilibrium is stable, whatever be the weight of the particle.

[2012, 20M]

4) A solid right cylinder cone whose height is \(h\) and radius of whose base is \(r\), is placed on an inclined plane. It is prevented from sliding. If the inclination \(\theta\) of the plane (to the horizontal) be gradually decreased, find when the cone will topple over. For a cone semi-vertical angle is \(30^{\circ}\), determine the critical value of \(\theta\) which when exceeded, the cone will topple over.

[2008, 15M]

5) A uniform beam of length \(l\) rests with its ends on two smooth planes which intersect in a horizontal line. If the inclinations of the planes to the horizontal are \(\alpha\) and \(\beta(\beta>\alpha)\), show that the inclination \(\theta\) of the beam to the horizontal, in one of the equilibrium positions, is given by \(\tan \theta=\dfrac{1}{2}(\cot \alpha-\cot \beta)\) and show that the beam is unstable in this position.

[2007, 15M]

6) A sphere of weight \(W\) and radius \(a\) lies within a fixed spherical shell of radius \(b\). A particle of weight \(w\) is fixed to the upper end of the vertical diameter. Prove that the equilibrium is stable if \(\dfrac{W}{w}> \dfrac{b-2a}{a}\).

[2003, 12M]

7) A solid cylinder floats in a liquid with its axis vertical. Let \(\sigma\) be the ratio of the specific gravity of the cylinder to that of the liquid. Prove that the equilibrium is stable if the ratio of the radius of the base to the height is greater than \(\sqrt{2 \sigma(1-\sigma)}\).

[2002, 15M]

8) A right circular cylinder floating with its axis horizontal and in the surface, is displaced in the vertical plane through the axis. Discuss its stability of equilibrium.

[2001, 15M]