Question

A sphere contracts in volume by 0.01% when taken to the bottom of sea 1 km deep. Find bulk modulus of the material of sphere

• A. $9.8\times {10}^{6}M/M^2$
• B. $1.2\times {10}^{10}N/M^2$
• C. $9.8\times {10}^{10}N/M^2$
• D. $9.8\times {10}^{11}N/M^2$

Answer 1

The correct option is C $9.8\times {10}^{10}N/M^2$

Given that,

Sphere contracts in volume $0.01$%

Change in Depth = $1$ km

Acceleration due to gravity = $9.8$ m/s²

Density of water = $1000$ kg/m³

Now, the volume of sphere is $V$

So, $dV=0.01$% of $V$

Now,

$dV=\frac{0.01V}{100}$

$\frac{dV}{V}=\frac{0.01}{100}$

$\frac{dV}{V}={10}^{-4}$

Now, the change in pressure

$dP=\rho g\left(dh\right)$

$dP={10}^3\times 9.8\times {10}^3$

$dP=9.8\times {10}^{6}N/m^2$

Now, the bulk modulus is

$K=\frac{dP}{\frac{dV}{V}}$

$K=\frac{9.8\times {10}^6}{{10}^{-4}}$

$K=9.8\times {10}^{10}N/m^2$

Hence, the bulk modulus of the material of sphere is $9.8\times {10}^{10}N/m^2$