First slide

Elastic Modulie

Question

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

Moderate

A

$\frac{p}{B}$
B

$\frac{B}{3 p}$
C

$\frac{3 p}{B}$
D

$\frac{p}{3 B}$

Solution

The object is spherical and the bulk modulus is represented by B. It is the ratio of normal stress to the volumetric strain.
Hence, $B = \frac{F / A}{ΔV / V} (∵ p = \frac{F}{A})$
$\Rightarrow \frac{ΔV}{V} = \frac{p}{B} = |\frac{ΔV}{V}| = \frac{p}{B}$
Here, p is applied pressure on the object and $\frac{ΔV}{V}$ is volume strain.
Fractional decrease in volume, $\frac{ΔV}{V} = 3 \frac{ΔR}{R} (∵ V = \frac{4}{3} {πR}^{3})$
Volume of the sphere decreases due to the decrease in its radius, hence $\frac{ΔV}{V} = \frac{3 ΔR}{R} = \frac{P}{B} \Rightarrow \frac{ΔR}{R} = \frac{P}{3 B}$

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