Question

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 the area floats on the surface of the liquid, covering the entire cross-section of the 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 `((dr)/r)`, is ______.

विकल्प

• `(Ka)/(mg)`

• `(Ka)/(3mg)`

• `(mg)/(3Ka)`

• `(mg)/(Ka)`

Answer 1

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 the area floats on the surface of the liquid, covering the entire cross-section of the 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 `((dr)/r)`, is `underlinebb((mg)/(3Ka))`.

Explanation:

Bulk modulus, K = `"volumetric stress"/"volumetric strain"`

K = `(mg)/(a((dV)/V))` `[∵ "Stress" = F/A = (mg)/a]`

⇒ `(dV)/V = (mg)/(Ka)` .........(i)

volume of sphere, V = `4/3pir^3`

Fractional change in volume `(dV)/V = (3dr)/r` ....(ii)

Using eq. (I) & (ii) `(3dr)/r = (mg)/(Ka)`

∴ `(dr)/r = (mg)/(3Ka)` (Fractional decrement in radius)