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Question
Figure ( following ) shows a smooth track which consists of a straight inclined part of length l joining smoothly with the circular part. A particle of mass m is projected up the incline from its bottom. Find the minimum projection-speed \[\nu_0\] for which the particle reaches the top of the track.
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Solution
Net force on the particle at A and B \[F = \text{ mg }\sin \theta\] Work done to reach B from A, \[\text{W = FS = mg} \sin \theta \text{ l }\] Again, work done to reach B to C
\[= \text{mgh}\]
\[ = \text{mg R} \left( 1 - \cos \theta \right)\]
\[ = \text{mg}\left[ \text{l} \sin \theta + R \left( 1 - \cos \theta \right) \right]\]
\[\Rightarrow \frac{1}{2}\text{m}\nu_2^2 = mg \left[ \text{l} \sin \theta + R \left( 1 - \cos \theta \right) \right]\]
\[ \Rightarrow \nu_2 = \sqrt{2g \left[ R \left( 1 - \cos \theta \right) + \text{l} \sin \theta \right]}\]
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