MHT CET · Physics · Mechanical Properties of Solids

Two wires having the same length and material are stretched by the same force. Their diameters are in the ratio $1 : 3$ . The ratio of strain energy per unit volume for these two wires (smaller to larger diameter), when stretched, is

A $3 : 1$
B $9 : 1$
C $27 : 1$
D $81 : 1$

Verified Solution

Answer & Solution

Correct Answer

(D) $81 : 1$

Step-by-step Solution

Detailed explanation

Strain energy per unit volume

u = \frac{U}{V} = \frac{1}{2} \times s t r e s s \times s t r a i n

\frac{U}{V} = \frac{1}{2} \times \frac{F}{A} \times \frac{∆ l}{l}

\frac{U}{V} = \frac{1}{2}

Stress \times \frac{Stress}{Y}

∴ \frac{u_{S}}{u_{L}} = \frac{Y_{L}}{Y_{S}} \times {(\frac{Strain (S)}{Stress (S)})}^{2} = {(\frac{A_{L}}{A_{S}})}^{2}

= {(\frac{r_{L}^{2}}{r_{S}^{2}})}^{2} = (\frac{81}{1})

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Two wires having the same length and material are stretched by the same force. Their diameters are in the ratio 1:3. The ratio of strain energy per unit volume for these two wires (smaller to larger diameter), when stretched, is

Step-by-step Solution

Two wires having the same length and material are stretched by the same force. Their diameters are in the ratio $1 : 3$ . The ratio of strain energy per unit volume for these two wires (smaller to larger diameter), when stretched, is