Question

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 triple the radius and one third of length by 1 mm is:

• A. 8 J
• B. 27 J
• C. 54 J
• D. 18 J

Hint:

Work $\propto$ Radius squared / Length.

Answer 1

The Correct Option is C

Step 1: Energy stored in a stretched wire.
When a wire is stretched by a fixed amount $\Delta L$, the work stored is \[ W = \frac{1}{2}\,\frac{Y A}{L}\,(\Delta L)^2, \] where $Y$ is Young's modulus, $A$ the cross section area and $L$ the length.
Step 2: What stays the same.
Both wires are of the same material, so $Y$ is the same, and the stretch $\Delta L = 1$ mm is the same. So the work depends only on $\dfrac{A}{L}$.
Step 3: Write the ratio.
Since $A = \pi r^2$, $W \propto \dfrac{r^2}{L}$, so \[ \frac{W_2}{W_1} = \left(\frac{r_2}{r_1}\right)^2 \times \frac{L_1}{L_2}. \]
Step 4: Put in the changes.
The radius is tripled, $r_2 = 3r_1$, and the length is one third, $L_2 = \dfrac{L_1}{3}$: \[ \frac{W_2}{W_1} = 3^2 \times \frac{L_1}{L_1/3} = 9 \times 3 = 27. \]
Step 5: Find the new work.
$W_1 = 2$ J, so $W_2 = 27 \times 2 = 54$ J.
Step 6: Conclusion.
The work needed for the second wire is 54 J. \[ \boxed{54\ \text{J}} \]