MHT CET · Physics · Waves and Sound

A sonometer wire is in unison with a tuning fork, when it is stretched by weight w and corresponding resonating length is $L_{1}$ . If the weight reduced to $(\frac{w}{4})$ , the corresponding resonating length becomes $L_{2}$ . The ratio $(\frac{L_{1}}{L_{2}})$

A $4 : 1$
B $1 : 4$
C $1 : 2$
D $2 : 1$

Verified Solution

Answer & Solution

Correct Answer

(D) $2 : 1$

Step-by-step Solution

Detailed explanation

When sonometer is stretched by weight w (tension), then frequency of vibration in the string

v = \frac{1}{2 l} \sqrt{\frac{T}{m}}

v = \frac{1}{2 L_{1}} \sqrt{\frac{w}{m}}

… (i)
where,

L_{1}

is the resonating length and m is mass per unit length of the string.
Similarly, when weight reduces to

\frac{w}{4}

and
Resonating lengths is

L_{2},

then

v = \frac{1}{2 L_{2}} \sqrt{\frac{\frac{w}{4}}{m}} \Rightarrow v = \frac{1}{4 L_{2}} \sqrt{\frac{w}{m}}

…(ii)
From Eqs. (i) and (ii), we have

\frac{1}{2 L_{1}} \sqrt{\frac{w}{m}} \times \frac{1}{4 L_{2}} \sqrt{\frac{w}{m}} \Rightarrow L_{1} = 2 L_{2}

∴ \frac{L_{1}}{L_{2}} = 2

∴ L_{1} : L_{2} = 2 : 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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A sonometer wire is in unison with a tuning fork, when it is stretched by weight w and corresponding resonating length is L1 . If the weight reduced to w4 , the corresponding resonating length becomes L2 . The ratio L1L2

Step-by-step Solution

A sonometer wire is in unison with a tuning fork, when it is stretched by weight w and corresponding resonating length is $L_{1}$ . If the weight reduced to $(\frac{w}{4})$ , the corresponding resonating length becomes $L_{2}$ . The ratio $(\frac{L_{1}}{L_{2}})$