MHT CET · Physics · Mechanical Properties of Solids

A wire length L and radius r is rigidly fixed at one end. On stretching the other end of the wire with a force F, the increase in length is $l$ . If another wire of the same material but double the length and radius is stretched with a force 2F, then increase in length is

A $\frac{l}{4}$
B $2 l$
C $\frac{l}{2}$
D $l$

Verified Solution

Answer & Solution

Correct Answer

(D) $l$

Step-by-step Solution

Detailed explanation

Let the length of wire = L
Radius = r
When a force F is applied, then increase in length =

(l)

Since, Young’s modulus

Y = \frac{\frac{F}{π r^{2}}}{\frac{l}{L}} \Rightarrow Y = \frac{F L}{π r^{2} l}

…(i)
For another wire,
Length of wire, L’ = 2L
Radius of wire, r’ = 2r
Applied force, F’ = 2F

∴ Y = \frac{F^{'} L^{'}}{π r^{2} l^{'}}

where,

l

’ = elongation in length

= \frac{2 F . 2 L}{π {(2 r)}^{2} . c^{'}} = \frac{4 F L}{4 π r^{2} l^{'}}

Y = \frac{F L}{π r^{2} l^{'}}

….(ii)
From Eqs. (i) and (ii), we have

l

’ =

l

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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A wire length L and radius r is rigidly fixed at one end. On stretching the other end of the wire with a force F, the increase in length is l . If another wire of the same material but double the length and radius is stretched with a force 2F, then increase in length is

Step-by-step Solution

A wire length L and radius r is rigidly fixed at one end. On stretching the other end of the wire with a force F, the increase in length is $l$ . If another wire of the same material but double the length and radius is stretched with a force 2F, then increase in length is