The bulk modulus of a spherical object is $B$. If it is subjected to uniform pressure $P$, the fractional decr

Question

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

Accepted Answer

1. **Bulk Modulus Formula:** The Bulk Modulus ($B$) is defined as the ratio of hydraulic stress (pressure $P$) to the volumetric strain ($\frac{\Delta V}{V}$). Considering magnitude:
   $$ B = \frac{P}{\frac{\Delta V}{V}} \implies \frac{\Delta V}{V} = \frac{P}{B} $$
   (Reference: )
2. **Volume-Radius Relationship:** The volume of a sphere is given by $V = \frac{4}{3}\pi r^3$.
3. **Fractional Change:** For small changes, the fractional change in volume is three times the fractional change in radius:
   $$ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} $$
4. **Substitution:** Substitute the expression for volumetric strain into the Bulk Modulus equation:
   $$ 3 \frac{\Delta r}{r} = \frac{P}{B} $$
   $$ \frac{\Delta r}{r} = \frac{P}{3B} $$

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