A uniform wire (Y = 2 × 10¹¹ N m⁻²) is subjected to a longitudinal tensile stress of 6 × 10⁷ N m⁻². If the overall change in volume of the wire is 0.02%, the fractional decrease in radius of the wire multiplied by 10⁵ is x. Find x.
Physics · Mixed Chapter
JEE Main Practice

Question

A uniform wire (Y = 2 × 10¹¹ N m⁻²) is subjected to a longitudinal tensile stress of 6 × 10⁷ N m⁻². If the overall change in volume of the wire is 0.02%, the fractional decrease in radius of the wire multiplied by 10⁵ is x. Find x.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Longitudinal strain εₗ = stress/Y = (6 × 10⁷)/(2 × 10¹¹) = 3 × 10⁻⁴. Step 2: Volume strain εᵥ = ΔV/V = 0.02/100 = 2 × 10⁻⁴. Step 3: For small strains in a cylindrical wire, εᵥ = εₗ + 2εᵣ, where εᵣ is the signed radial strain. Step 4: εᵣ = (εᵥ − εₗ)/2 = (2 × 10⁻⁴ − 3 × 10⁻⁴)/2 = −5 × 10⁻⁵. Step 5: Fractional decrease in radius = 5 × 10⁻⁵; therefore x = (5 × 10⁻⁵) × 10⁵ = 5. Final answer: 5.