Question

Two wires A and B having same length and material are stretched by the same force. Their diameters are in the ratio 1 : 3. The ratio of energy density of wire A to that of wire B when stretched, is

• A. 27 : 1
• B. 9 : 1
• C. 81 : 1
• D. 3 : 1

Hint:

Energy density in a stretched material is proportional to the square of the diameter of the wire. This helps to calculate energy ratios when dimensions change.

Answer 1

The Correct Option is C

Step 1: Understanding the energy density.
The energy density \( u \) in a stretched wire is given by: \[ u = \frac{1}{2} \sigma \epsilon^2 \] where \( \sigma \) is the stress and \( \epsilon \) is the strain. Stress is proportional to the square of the diameter, and strain is inversely proportional to the square of the diameter. Therefore, the energy density will vary as the square of the diameter. Step 2: Applying the ratio of diameters.
Since the diameters are in the ratio 1 : 3, the energy density ratio will be the square of the diameter ratio: \[ \frac{u_A}{u_B} = \left( \frac{d_A}{d_B} \right)^2 = \left( \frac{1}{3} \right)^2 = 81 : 1 \] Step 3: Conclusion.
Thus, the correct answer is (C) 81 : 1.