Question

the bulk modu of a spherical object is b if it is subjected to uniform pressure p the decrease in the fractional radius is?

Answer 1

Answer:

volume of sphere,V=

3

4

πr

3

Differentiate both sides with respect to r

\begin{lgathered}\frac{dV}{dr}=4\pi r^2\\\\\Delta{V}=4\pi r^2\Delta{r}\\\\\text{dividing by},V=\frac{4}{3}\pi r^3\\\\\frac{\Delta V}{V} =3\frac{\Delta r}{r}\end{lgathered}

dr

dV

=4πr

2

ΔV=4πr

2

Δr

dividing by,V=

3

4

πr

3

V

ΔV

=3

r

Δr

Now, we know the formula,

Bulk modulus , B = \frac{-P}{\frac{\Delta{V}}{V}}

V

ΔV

−P

use ∆V/V = 3∆r/r from above derivation ,

B = -P/3(∆r/r)

-∆r/r = P/3B

Hence, fractional decreases in radius is P/3B

Answer 2

Answer:

$\frac{p}{3B}$

Explanation:

Given that,

Pressure = p

Bulk modulus = B

We know,

B = P/ΔV/V  ⇒ ΔV/V = $\frac{p}{B}$  _____(1)

V = $\frac{4}{3}$ π $R^{3}$  

ΔV/V = 3 ΔR/R  [While calculating fractional change, constants are eliminated.]

$\frac{p}{B}$ = 3 ΔR/R     [ From (1) ]

ΔR/R = $\frac{p}{3B}$