The Question is from neet-physics , and it was created by the author VENKATESWARARAO

Question

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

VENKATESWARARAO · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
	**Primary Concept**: Bulk Modulus and its relationship with pressure and volume change
	**Related Topics**: Stress, Strain, Elasticity
	**Prerequisites**: Definition of Bulk Modulus, Understanding of fractional change in volume and its relation to change in radius for a sphere.
## SOLUTION PATHWAY
**Given Information**:
	Bulk modulus = B
	Pressure = p
	We need to find the fractional decrease in radius, which is -Δr/r
**Method & Steps**:
	1. Define Bulk modulus:
		Reasoning: Bulk modulus is defined as the ratio of pressure to the fractional change in volume.
		Formula/Rule:  B = -p / (ΔV/V)
	2.  Relate fractional volume change to fractional radius change.
		Reasoning: For a sphere, V = (4/3)πr³. Taking derivative, dV = 4πr²dr. Dividing by V, dV/V = 4πr²dr / ((4/3)πr³) = 3dr/r.  Therefore, ΔV/V ≈ 3Δr/r.
		Working: ΔV/V = 3Δr/r
	3. Substitute and solve for fractional change in radius
		Working: B = -p / (3Δr/r). Therefore, Δr/r = -p / (3B). Since we are looking for fractional *decrease* in radius, the negative sign is absorbed, and the fractional decrease is p/(3B)
		Result: Fractional decrease in radius = p/(3B)
 ## OPTION ANALYSIS
[A] ✓:  Correctly uses bulk modulus definition and sphere volume relation to arrive at the correct answer. The fractional decrease in radius is p/(3B).
[B] :x: This is the fractional change in volume, not radius.
[C] :x: This is the inverse relation of what we are looking for, also it is a ratio of B/3p instead of p/3B
[D] :x: This is not the correct relationship between bulk modulus, pressure, and fractional change in radius.
 ## LEARNING AIDS
	**Common Mistakes**:
		1. Confusing fractional change in volume with fractional change in radius: Always remember the relation ΔV/V = 3Δr/r for spheres.
		2.  Forgetting the negative sign: Bulk modulus is defined with a negative sign to make it positive for compression. Remember to account for this sign and for the problem statement asking for a *decrease*.
	**Quick Tips**:
		1.  Remember the definition of bulk modulus and the volume of a sphere.
		2.  Use differentials to quickly relate fractional changes.

VENKATESWARARAO · Answer

The correct Option is "$\frac{p}{{3B}}$"

VENKATESWARARAO · Answer

$\frac{p}{B}$

VENKATESWARARAO · Answer

$\frac{B}{{3p}}$

VENKATESWARARAO · Answer

$\frac{{3p}}{B}$

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