Question

A spherical ball contracts in volume by $0.01\%$ when subjected to a normal uniform pressure of $100$ atmospheres. The bulk modulus of its material in dyne per centimeter square is:
A. $10 \times {10^{11}}$
B. $100 \times {10^2}$
C. $1 \times {10^{11}}$
D. $2 \times {10^{11}}$

Answer 1

In this question pressure is given in atmospheric unit so try to change it in Newton per meter square by multiplying it with $1.01 \times {10^5}$ and also to get the bulk modulus in dyne per centimeter square, multiply the resulting bulk modulus value by ten.

Formula used:
$B = \dfrac{P}{{\dfrac{{\Delta v}}{v}}}$ where $B$ is bulk modulus, $P$ is pressure exerted on the material, $\Delta v$ is change in volume of the object due to pressure and $v$ is the volume of the object.

Complete answer:
Given that,
Pressure exerted $= 100$ atmospheres $= 100 \times 1.01 \times {10^5}$ Newton per meter square
And the spherical ball contracts in volume $= 0.01\%$
$\therefore \dfrac{{\Delta v}}{v} = \dfrac{{0.01}}{{100}}$
As we know that bulk modulus, $B = \dfrac{P}{{\dfrac{{\Delta v}}{v}}}$, putting the values we get:
$B = \dfrac{{100 \times 1.01 \times {{10}^5}}}{{\dfrac{{0.01}}{{100}}}}$
$B = 1.01 \times {10^{11}}$Newton per meter square
Therefore bulk modulus of material in dyne per centimeter square $= 1.01 \times {10^{11}} \times 10 = 10.1 \times {10^{11}}$.

So, the correct answer is “Option A”.

Note:
Basically the bulk modulus of a substance is the measure of how resistant to compression a substance is and is given by the formula $B = \dfrac{P}{{\dfrac{{\Delta v}}{v}}}$ here we substituted the given mathematical values after conversion of units and got the value of bulk modulus as $10.1 \times {10^{11}}$ dynes per centimeter square.