The bulk modulus of a spherical object is B. If it is subjected to uniform pressure p,the fractional decrease in radius is?

The fractional decrease in radius of a spherical object subjected to uniform pressure can be calculated using the formula for the bulk modulus. The bulk modulus, denoted as B, is a measure of the resistance of a material to uniform compression. It relates the change in pressure to the resulting fractional change in volume.

The formula for the fractional decrease in radius (Δr/r) is given by:

Δr/r = -ΔV/V = -[3B(Δp)/V]

where Δr is the change in radius, r is the initial radius, ΔV is the change in volume, V is the initial volume, Δp is the change in pressure, and B is the bulk modulus.

In this case, the uniform pressure applied is p, so the change in pressure is Δp = p - 0 = p (as the initial pressure is assumed to be zero).

Assuming the object is an isotropic, homogeneous sphere, the initial volume V is given by V = (4/3)πr^3, and the change in volume ΔV is related to the change in radius Δr as ΔV = (4/3)π[(r + Δr)^3 - r^3].

Substituting these values into the formula, we have:

Δr/r = -[3B(Δp)/V]
= -[3B(p)/((4/3)πr^3)]
= -[9Bp/(4πr^3)]

Therefore, the fractional decrease in radius is -[9Bp/(4πr^3)].