The bulk modulus of a spherical object is `B` if it is subjected to uniform pressure `p`, the fractional decrease in radius is:

Options

A. `P/B`
B. `(B)/(3P)`
C. `(3P)/(B)`
D. `(P)/(3B)`

Answer

To solve the problem of finding the fractional decrease in the radius of a spherical object when subjected to uniform pressure \( p \), given its bulk modulus \( B \), we can follow these steps:

### Step 1: Understand the Definition of Bulk Modulus
The bulk modulus \( B \) is defined as:
\[ B = -\frac{P}{\frac{dV}{V}} \]
where \( P \) is the applied pressure, \( dV \) is the change in volume, and \( V \) is the original volume.

### Step 2: Relate Volume Change to Radius Change
For a spherical object, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Differentiating this with respect to \( r \):
\[ dV = 4\pi r^2 dr \]

### Step 3: Express the Fractional Change in Volume
The fractional change in volume \( \frac{dV}{V} \) can be expressed as:
\[ \frac{dV}{V} = \frac{4\pi r^2 dr}{\frac{4}{3} \pi r^3} \]
This simplifies to:
\[ \frac{dV}{V} = \frac{3 dr}{r} \]

### Step 4: Substitute into the Bulk Modulus Equation
Now substituting \( \frac{dV}{V} \) back into the bulk modulus equation:
\[ B = -\frac{P}{\frac{3 dr}{r}} \]
Rearranging gives:
\[ \frac{3 dr}{r} = -\frac{P}{B} \]
Thus, we can express \( \frac{dr}{r} \) as:
\[ \frac{dr}{r} = -\frac{P}{3B} \]

### Step 5: Interpret the Result
The negative sign indicates that the radius is decreasing. Therefore, the fractional decrease in radius is:
\[ \frac{dr}{r} = \frac{P}{3B} \]

### Final Answer
Thus, the fractional decrease in radius when a spherical object is subjected to uniform pressure \( p \) is:
\[ \frac{P}{3B} \]