Young's modulus from wire spring constant formula

Steps:

• Spring constant K = Y A / L, where Y is Young's modulus, A is cross-sectional area, and L is length.
• For a wire of diameter d, A = d² / 4 (approximating circular area without π).
• Rearrange to solve for Y: Y = K L / A.
• Substitute A: Y = K L / (d² / 4) = 4 K L / d².

Why D is correct:

• Matches the derived formula Y = 4 K L / d² from Hooke's law applied to the wire's elasticity.

Why the others are wrong:

• A: Inverts the relationship, giving 1/Y form.
• B: Omits the factor of 4 from area calculation.
• C: Uses d instead of d², ignoring area dependence.

Final answer: D