Question

Find out bulk stress on the spherical object of radius $\dfrac{{10}}{\pi }\;{\text{cm}}$ if area and mass of piston is $50\;{\text{c}}{{\text{m}}^2}$ and $50\;{\text{kg}}$ respectively for a cylinder filled with gas.
A. $4 \times {10^5}\;{\text{N/}}{{\text{m}}^{\text{2}}}$
B. $6 \times {10^5}\;{\text{N/}}{{\text{m}}^{\text{2}}}$
C. $2 \times {10^5}\;{\text{N/}}{{\text{m}}^{\text{2}}}$
D. $5 \times {10^5}\;{\text{N/}}{{\text{m}}^{\text{2}}}$

Answer 1

In this question, the concept of the bulk stress is used that is for the spherical body the bulk modulus is equal to pressure exerted by the gas on the spherical object. First draw the diagram as all the data given to understand the problem and then discuss bulk stress. Then find out the bulk stress on the spherical object by using all the data given.

Complete step by step solution:
As per the given question, it is given that a spherical object is placed in a cylinder with gas and it is covered by a piston. The radius of the spherical object is $\dfrac{{10}}{\pi }\;{\text{cm}}$ and the area of cross-section of the piston is $50\;{\text{c}}{{\text{m}}^{\text{2}}}$ and the mass of the piston is $50\;{\text{kg}}$.
Now we have to find out bulk stress on the object. As we know bulk stress is the pressure given on the spherical object by the cylinder filled with a gas. So, the bulk modulus is equal to pressure exerted by the gas on the spherical object. The pressure of gas will be the atmospheric pressure and the pressure given by the piston.
Now we write the expression for the pressure exerted by the piston on the object due to its weight as,
${P_p} = \dfrac{{mg}}{A}$
As we know, $m$is the mass of the piston, $g$ is the acceleration due to gravity, and $A$ is the area of the piston.
Now, we calculate the bulk stress as,
${p_{gas}} = {p_{atm}} + \dfrac{{mg}}{A}$
Here, ${p_{gas}}$ is bulk stress and ${p_{atm}}$ is the atmospheric pressure
As we know that the value of the atmospheric pressure is $1 \times {10^5}\;{\text{N/}}{{\text{m}}^{\text{2}}}$.
Now, we substitute the value of atmospheric pressure, mass of the piston, acceleration due to gravity, and the area of the cross-section of the piston in the bulk modulus equation as,
${p_{gas}} = 1 \times {10^5} + \dfrac{{50 \times 10}}{{50 \times {{10}^{ - 4}}}}$
$\Rightarrow {p_{gas}} = 2 \times {10^5}\;{\text{N/}}{{\text{m}}^{\text{2}}}$
$\therefore$ The bulk stress on the spherical object is $2 \times {10^5}N/{M^2}$.

Hence option (C) is the correct.

Note: As we know that the bulk stress is termed as volume stress when the volume of the body changes due to deforming force. Bulk stress works in all the directions of a body thus we can say bulk stress is equal to the applied pressure on the spherical body.