एक ऊध्व्वाधर चिकने अर्द्धवृत्तीय पथ के बिन्दु \(X\) से एक कण को इस प्रकार छोड़ा जाता हैं कि \(OX\) ऊध्र्वाधर से कोण \(\theta\) बनाता हैं जैसा कि चित्र में दर्शाया गया हैं। कण के ऊपर पथ की अभिलम्ब प्रतिक्रिया बिन्दु \(Y\) पर समाप्त हो जाती हैं जहाँ \(OY\) क्षैतिज से कोण \(\phi\) बनाता है। तब :

A \(\sin \,\phi = \,\cos \,\phi \)
B \(\sin \,\phi = \frac{1}{2}\,\cos \,\theta \)
C \(\sin \,\phi = \frac{2}{3}\,\cos \,\theta \)
D \(\sin \,\phi = \frac{3}{4}\,\cos \,\theta \)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\sin \,\phi = \frac{2}{3}\,\cos \,\theta \)

Step-by-step Solution

Detailed explanation

\(\frac{\mathrm{mv}^{2}}{\mathrm{r}}=\mathrm{mg} \sin \phi\) \(...(i)\) \(\mathrm{mg} \mathrm{r} \cos \theta=\frac{1}{2} \mathrm{mv}^{2}+\mathrm{rg} \sin \phi\) \(\frac{\mathrm{v}^{2}}{\mathrm{rg}}=2 \cos \theta-2 \sin \phi \ldots \ldots(\mathrm{ii})\)…

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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