A solid sphere of radius R made of a material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless pistion of area A floats on the surface of the liquid. When a mass M is placed on the piston to compress the liquid the fractional change in the radius of the sphere, `deltaR//R`, is .............

Answer

`Mg//3AB`

Explanation

When mass M is placed on the piston, the excess pressrue, p =Mg//A. As this pressure is equally applicable from all the direction on the sphere, hence there will be decrease in volume due to decrease in radius of sphere, Volume of the sphere, `V = (4)/(3)pi R^3`
Differentiating it, we get
`Delta V =(4)/(3) pi(3R^2) Delta R = 4pi R^2Delta R`
`:. (Delta V)/(V) = (4pi R^2 Delta R)/((4)/(3)pi R^3) = (3 Delta R)/(R )`
`We know that, B = (p)/(dV//V) = (Mg)/(A)//(3Delta R)/(R )`
` or (Delta R)/(R ) = (Mg)/(3BA)`