Question

If the work done in stretching a wire by 1 mm is 2 J, the work necessary for stretching another wire of the same material but with double radius of cross-section and half the length by 1 mm is

• A. 16 J
• B. 8 J
• C. \( \frac{1}{16} \) J
• D. \( \frac{1}{8} \) J

Hint:

When calculating work done in stretching a wire, remember that the work is proportional to the length and inversely proportional to the cross-sectional area.

Answer 1

The Correct Option is B

The work done in stretching a wire is given by:

\[ W = \frac{1}{2} \frac{F^2}{Y A} \] where \( W \) is the work, \( F \) is the force, \( Y \) is the Young's modulus, and \( A \) is the cross-sectional area of the wire. The force required to stretch the wire is related to the cross-sectional area and the material's Young's modulus. Since the radius of the wire is doubled and the length is halved, the cross-sectional area \( A \) is quadrupled (since \( A = \pi r^2 \), and the radius is doubled). The work done is proportional to both the cross-sectional area \( A \) and the square of the length. Hence, changing the dimensions of the wire affects the work required to stretch it.

Using these relationships and applying them to the second wire, we find that the work required is 8 J.

Hence, the correct answer is (b).