એક ખેંચ્યા વગર ની સ્પ્રિંગ લંબાઈ \(l\) અને દળ \(m\) ધરાવે છે અને તેનો એક છેડો દ્રઢ આધાર સાથે જોડેલો છે.ધારો કે સ્પ્રિંગ ને એકસમાન તાર થી બનાવેલી છે તો તેના એક છેડાને સમાન વેગ \(v\) થી ખેંચવામાં આવે છે તો તેણે મેળવેલી ગતિઉર્જા કેટલી થશે?

A \(\frac {1}{2}\,mv^2\)
B \(mv^2\)
C \(\frac {1}{3}\,mv^2\)
D \(\frac {1}{6}\,mv^2\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac {1}{6}\,mv^2\)

Step-by-step Solution

Detailed explanation

We can find the effective mass of the spring by finding its kinetic energy. This requires adding all the length elements' kinetic energy, and requires the following integral: \(T=\int_{m} \frac{1}{2} u^{2} d m\) since the spring is uniform, \(d m=\left(\frac{d y}{L}\right) m,\)…

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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