UPSC 2024 Maths Optional Paper 1 Q5d — Step-by-Step Solution

10 marks · Section B

Stability of equilibrium (energy criterion) · Dynamics & Statics · asked 5× in 13 yrs · Read the full method →

Question

A solid hemisphere rests in equilibrium on a solid sphere of equal radius. Determine the stability of the equilibrium when (i) the curved surface rests on the sphere, and (ii) the flat surface rests on the sphere.

Technique

Apply the rolling-body stability criterion $1/h > 1/R_1+1/R_2$ ; identify $R_1$ , $R_2$ , and $h$ correctly for each configuration.

Solution

Two configurations of a hemisphere (radius a) on a fixed sphere of equal radius. Case (i): curved face down — CG at height h=5a/8 above the contact point, which exceeds R_1R_2/(R_1+R_2)=a/2, so the equilibrium is unstable. Case (ii): flat face down — CG at height h=3a/8<a, so the equilibrium is stable.

Stability criterion. A body of radius of curvature $R_1$ resting on a fixed body of radius $R_2$ is in stable equilibrium iff

\frac{1}{h}>\frac{1}{R_1}+\frac{1}{R_2},

where $h$ is the height of the upper body’s centre of gravity above the contact point. For a flat surface, $R\to\infty$ so $1/R=0$ .

CG of a uniform solid hemisphere of radius $a$ : at $3a/8$ from the flat face, on the axis.

Case (i) — Curved surface on the sphere

$R_1=a$ (curved face), $R_2=a$ (sphere).
The contact is at the bottom of the hemisphere’s curved face. The flat face is on top, so the CG is at height $a-3a/8=5a/8$ above the contact point. Thus $h=5a/8$ .

Stability requires $h<\dfrac{R_1 R_2}{R_1+R_2}=\dfrac{a}{2}$ .

But $5a/8>a/2$ : the condition fails. Equilibrium is unstable.

Case (ii) — Flat surface on the sphere

$R_1=\infty$ (flat), $R_2=a$ .
The flat face rests on the sphere’s top. The CG is at height $3a/8$ above the flat face (= above the contact point). Thus $h=3a/8$ .

Stability requires $h<a$ (since $1/R_1=0$ ).

$3a/8<a$ : the condition holds. Equilibrium is stable.

Answer

\boxed{\;\text{Case (i): unstable}\;(h=5a/8>a/2);\quad\text{Case (ii): stable}\;(h=3a/8<a).\;}