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. `(B)/(3p)`
B. `(3p)/(B)`
C. `(p)/(3B)`
D. `(p)/(B)`

Explanation

To find the fractional decrease in the radius of a spherical object when subjected to uniform pressure \( p \), we can follow these steps:

### Step 1: Understand the relationship between bulk modulus and volume change
The bulk modulus \( B \) is defined as:
\[ B = -\frac{p}{\frac{\Delta V}{V}} \]
where:
- \( p \) is the applied pressure,
- \( \Delta V \) is the change in volume,
- \( V \) is the original volume.

### Step 2: Rearranging the bulk modulus formula
From the definition, we can rearrange the formula to express the fractional change in volume:
\[ \frac{\Delta V}{V} = -\frac{p}{B} \]

### Step 3: Determine the volume of a sphere
The volume \( V \) of a sphere with radius \( r \) is given by:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 4: Relate volume change to radius change
The change in volume \( \Delta V \) can also be expressed in terms of the change in radius \( \Delta r \):
\[ \Delta V = V' - V = \frac{4}{3} \pi (r + \Delta r)^3 - \frac{4}{3} \pi r^3 \]
Using the binomial expansion for small changes, we can approximate:
\[ \Delta V \approx 4 \pi r^2 \Delta r \]
Thus, the fractional change in volume can be expressed as:
\[ \frac{\Delta V}{V} = \frac{4 \pi r^2 \Delta r}{\frac{4}{3} \pi r^3} = \frac{3 \Delta r}{r} \]

### Step 5: Equate the two expressions for fractional change
Now, we can set the two expressions for fractional change in volume equal to each other:
\[ \frac{3 \Delta r}{r} = -\frac{p}{B} \]

### Step 6: Solve for the fractional change in radius
Rearranging the equation gives us:
\[ \Delta r = -\frac{p}{3B} r \]
Thus, the fractional decrease in radius is:
\[ \frac{\Delta r}{r} = -\frac{p}{3B} \]

### Final Answer
The fractional decrease in radius when a spherical object is subjected to uniform pressure \( p \) is:
\[ \frac{\Delta r}{r} = -\frac{p}{3B} \]