The Question is from neet-physics properties-solids-liquids , and it was created by the author ABDUL JILANI BASHA

Question

A solid sphere of radius R made of a material with bulk modulus K and rigidity modulus $\eta$ is subjected to uniform external pressure P. The fractional decrease in its radius $\frac{\Delta R}{R}$ is:

ABDUL JILANI BASHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [B]
## CONCEPT FRAMEWORK
	**Primary Concept**: Bulk Modulus and Volume Strain
	**Related Topics**: Stress, Strain, Elasticity
	**Prerequisites**: Definition of Bulk Modulus, Relationship between pressure and volume change
## SOLUTION PATHWAY
**Given Information**:
	* Initial radius: R
	* Bulk modulus: K
	* External pressure: P
	* Fractional change in radius: $\frac{\Delta R}{R}$
**Method & Steps**:
	1.  Relate pressure to change in volume using bulk modulus:
		Reasoning: Bulk modulus is defined as the ratio of pressure to the fractional change in volume. 
		Formula/Rule: $K = - \frac{P}{\frac{\Delta V}{V}}$
	2.  Express the change in volume in terms of the change in radius. For a sphere, $V = \frac{4}{3}\pi R^3$.  Differentiating, we get $dV = 4\pi R^2 dR$. So $\frac{dV}{V} = \frac{4\pi R^2 dR}{\frac{4}{3}\pi R^3} = 3\frac{dR}{R}$
		Working: $\frac{\Delta V}{V} = 3\frac{\Delta R}{R}$
		Key Point: We are looking for the fractional decrease, so $\Delta R$ will be negative.
	3. Substitute the expression for $\frac{\Delta V}{V}$ in the bulk modulus equation and solve for $\frac{\Delta R}{R}$:
		Working: $K = - \frac{P}{3\frac{\Delta R}{R}}$  =>  $\frac{\Delta R}{R} = -\frac{P}{3K}$. Since we are looking for fractional decrease, it will be positive.
		Result: $\frac{\Delta R}{R} = \frac{P}{3K}$
		Verification: The units are consistent, pressure/bulk modulus, which is dimensionless.
 ## OPTION ANALYSIS
[A] :x: Incorrect. This option uses rigidity modulus which is not related to volume change under pressure.
[B] ✓: Correct. This is the correct expression for fractional decrease in radius.
[C] :x: Incorrect. The rigidity modulus should not appear in the equation. It is relevant for shear deformation, not volume change.
[D] :x: Incorrect. The expression is inverted and contains the wrong modulus.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Forgetting the negative sign in the bulk modulus formula. Always remember that an increase in pressure leads to a decrease in volume and vice versa.
		2.[Error 2]: Confusing bulk modulus with rigidity modulus. Bulk modulus is for volume change, while rigidity modulus is for shear deformation.
	**Quick Tips**:
		1.[Helpful shortcut]:  Remember that fractional change in volume is 3 times the fractional change in radius for a sphere.
		2.[Memory aid]: The bulk modulus K is related to pressure and volume change, similar to how Young's modulus is related to tensile/compressive stress and length change.

ABDUL JILANI BASHA · Answer

$\frac{PK}{3\eta}$

ABDUL JILANI BASHA · Answer

The correct Option is "$\frac{P}{3K}$"

ABDUL JILANI BASHA · Answer

$\frac{P\eta}{3K}$

ABDUL JILANI BASHA · Answer

$\frac{3KP}{\eta}$

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