સમાન લંબાઈ અને ત્રિજ્યાના બે તારને એકબીજા સાથે જોડેલા છે અને દળ લટાવેલ છે. બે તારના દ્રવ્યના યંગ મોડ્યુલસ અનુક્રમે \(Y_{1}\) અને \(Y_{2}\) છે. આ સંયોજન એક તાર તરીકે વર્તે તો તેનો યંગ મોડ્યુલસ કેટલો હશે?

A \(y=\frac{Y_{1} Y_{2}}{Y_{1}+Y_{2}}\)
B \(y=\frac{2 Y_{1} Y_{2}}{3\left(Y_{1}+Y_{2}\right)}\)
C \(Y=\frac{2 Y_{1} Y_{2}}{Y_{1}+Y_{2}}\)
D \({Y}=\frac{{Y}_{1} {Y}_{2}}{2\left({Y}_{1}+{Y}_{2}\right)}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(Y=\frac{2 Y_{1} Y_{2}}{Y_{1}+Y_{2}}\)

Step-by-step Solution

Detailed explanation

In series combination \(\Delta l =l_{1}+l_{2}\) \(Y =\frac{ F / A }{\Delta l / l} \Rightarrow \Delta l=\frac{ F l}{ AY }\) \(\Rightarrow \Delta l \propto \frac{l}{ Y }\) Equivalent length of rod after joining is \(=2 l\) As, lengths are same and force is also same in series…

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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\(f_i\)	\(4\)	\(4\)	\(\alpha\)	\(15\)	\(8\)	\(\beta\)	\(4\)	\(5\)