Question

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

• A. $\frac{B}{3p} $
• B. $\frac{3p}{B} $
• C. $\frac{p}{3B} $
• D. $\frac{p}{B} $

Answer 1

The Correct Option is C

To determine the fractional decrease in the radius of a spherical object subjected to uniform pressure, we must understand the relationship between bulk modulus and change in volume under pressure.

The bulk modulus, B, is defined as the ratio of pressure applied to the fractional change in volume:

B = -\frac{p}{\frac{\Delta V}{V}}

where:

• p is the applied pressure,
• \Delta V is the change in volume, and
• V is the original volume.

For a sphere, the volume V is given by V = \frac{4}{3}\pi r^3, where r is the radius.

The change in volume \Delta V can be expressed in terms of the change in radius \Delta r:

\Delta V = 4\pi r^2 \Delta r

This means that the fractional change in volume is:

\frac{\Delta V}{V} = \frac{3 \Delta r}{r}

By substituting the expression for fractional volume change into the formula for the bulk modulus, we obtain:

B = -\frac{p}{\frac{3 \Delta r}{r}}

Simplifying the above equation gives:

3B \Delta r = -pr

Solving for the fractional change in radius \frac{\Delta r}{r}, we have:

\frac{\Delta r}{r} = -\frac{p}{3B}

The negative sign indicates a decrease in radius. Therefore, the correct expression for the fractional decrease in radius is:

\frac{p}{3B}

Thus, the correct answer is \frac{p}{3B}.