JEE Main · 2018 · 08 Apr 2018 · Offline · Elasticity & SHM

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A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area a floats on the surface of the liquid, covering entire cross section of cylindrical container. When a mass

m

is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere,

(\frac{d r}{r})

, is:

(A) $\frac{m g}{K a}$
(B) $\frac{K a}{m g}$
(C) $\frac{K a}{3 m g}$
(D) $\frac{m g}{3 K a}$ ✓ Correct

Answer: (D)

Solution

The mass on the piston raises the liquid pressure by

P = \frac{m g}{a}

, and this pressure squeezes the sphere. The bulk modulus links pressure to the fractional volume change,

K = \frac{P}{Δ V / V}

, and for a sphere a small fractional change in radius changes the volume three times as much,

\frac{Δ V}{V} = 3 \frac{Δ r}{r}

. Combining the two,

\frac{Δ r}{r} = \frac{P}{3 K} = \frac{m g}{3 K a}

JEE Main · 2018 · 08 Apr 2018 · Offline · Single correct option

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JEE Main · 2018 · 08 Apr 2018 · Offline · Elasticity & SHM

Solution

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