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,` ((dr)/(r)),` is.....

Options

A. `(Ka)/(mg)`
B. `(Ka)/(3mg)`
C. `(mg)/(3Ka)`
D. `(mg)/(Ka)`

Answer

C

Explanation

Bulk modulus `= ("volume stress")/("volume strain")`
`= (("Force")/("Area "))/( ( "change in volume")/("volume"))`
`therefore (Delta V )/( V) = (mg )/(ka)`
but, vlume of sphere `V = 4/3 pi r ^(3),`
`therefore (Delta V )/(V) =3 (Delta r )/(r)`
`therefore (Delta r )/( r ) = 1/3 xx (Delta V )/(V) = (mg )/(3Ka )`