The Question is from neet-physics properties-solids-liquids rigidity-modules, and it was created by the author SUBHANI

Question

A wire of length L and cross-sectional area A is stretched by a force F. If the Young's modulus of the material is Y, and the Poisson's ratio is σ, what is the rigidity modulus (G) of the material?

SUBHANI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [A]
## CONCEPT FRAMEWORK
	**Primary Concept**: Relationship between Young's modulus, Rigidity modulus, and Poisson's ratio.
	**Related Topics**: Material properties, Elasticity, Stress, Strain.
	**Prerequisites**: Understanding of Young's modulus, Rigidity modulus, and Poisson's ratio.
## SOLUTION PATHWAY
**Given Information**:
	Young's modulus = Y
	Poisson's ratio = \u03c3
	We need to find the Rigidity modulus G.
**Method & Steps**:
	1.  The relationship between Young's modulus (Y), rigidity modulus (G), and Poisson's ratio (\u03c3) is given by:
		Reasoning: This is a standard formula in material science relating the elastic constants.
		Formula/Rule: $Y = 2G(1 + \sigma)$
	2.  Rearrange the formula to solve for G:
		Working:  Divide both sides of the equation by 2(1+\u03c3)
		Key Point:  We need to isolate G on one side of the equation.
	3.  The final result is:
		Result:  $G = \frac{Y}{2(1+\sigma)}$
		Verification:  The derived formula matches option A.
 ## OPTION ANALYSIS
[A] ✓: The correct formula relating Y, G and \u03c3 is  $G = \frac{Y}{2(1+\sigma)}$
[B] :x: This option is missing the factor of 2 in the denominator. Incorrect formula.
[C] :x: This option has (1 - \u03c3) instead of (1 + \u03c3) in the denominator. Incorrect formula.
[D] :x: This option is missing the factor of 2 in the denominator and has (1 - \u03c3) instead of (1 + \u03c3) in the denominator. Incorrect formula.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Confusing the relationship between Y, G and \u03c3. Remember the correct formula.
		2.[Error 2]:  Forgetting the factor of 2 in the denominator. Always double-check the formula.
	**Quick Tips**:
		1.[Helpful shortcut]: Remember the formula relating Y, G, and Poisson's ratio directly: $Y = 2G(1 + \sigma)$. This will allow you to quickly derive the formula for G.
		2.[Memory aid]: Think of G as the shear modulus which is always smaller than Young's modulus. Hence, you divide by a number larger than one i.e. 2(1+\u03c3).

SUBHANI · Answer

The correct Option is "$G = \frac{Y}{2(1+σ)}$"

SUBHANI · Answer

$G = \frac{Y}{(1+σ)}$

SUBHANI · Answer

$G = \frac{Y}{2(1-σ)}$

SUBHANI · Answer

$G = \frac{Y}{(1-σ)}$

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