The bulk modulus of a spherical object is $B$. If it is subjected to uniform pressure $p$, the fract

Question

The bulk modulus of a spherical object is $B$. If it is subjected to uniform pressure $p$, the fractional decrease in its surface area is

Professor PC Thomas Classes · Answer

(a) : The definition of Bulk modulus is $B = \frac{p}{\Delta V / V}$, so the fractional decrease in volume is $\frac{\Delta V}{V} = \frac{p}{B}$. Since volume $V = \frac{4}{3} \pi r^3$, taking the relative error gives $\frac{\Delta V}{V} = 3 \frac{\Delta r}{r}$. Thus, the fractional decrease in radius is $\frac{\Delta r}{r} = \frac{p}{3B}$. Surface area is $A = 4 \pi r^2$, so the fractional decrease in surface area is $\frac{\Delta A}{A} = 2 \frac{\Delta r}{r} = 2 \left(\frac{p}{3B}\right) = \frac{2p}{3B}$.