एक हुप जिसकी किज्या \(r\) एवं द्रव्यमान \(m\) है एक खुरदरी क्षैतिज सतह पर रखी गयी है, तथा यह कोणीय वेग \(\omega_{0}\) से घुणी गाति कर रही है। हुप के केन्द्र का आरंभिक वेग शून्य है। हुप के केन्द्र का वेग क्या होगा जब यह फिसलना बंद्कर देता है ?

A \(r\)\({\omega _0}\)
B \(\frac{{{\rm{r}}{\omega _0}}}{4}\)
C \({\rm{\;}}\frac{{{\rm{r}}{\omega _0}}}{3}\)
D \({\rm{\;}}\frac{{{\rm{r}}{\omega _0}}}{2}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \({\rm{\;}}\frac{{{\rm{r}}{\omega _0}}}{2}\)

Step-by-step Solution

Detailed explanation

From conservation of angular momentum about any fix point on the surface, \(\begin{array}{l} m{r^2}{\omega _0} = 2m{r^2}\omega \\ \Rightarrow \,\omega = {\omega _0}\backslash 2 \Rightarrow v = \frac{{{\omega _0}r}}{2} \end{array}\)

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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