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Question
Following figure following shows a smooth track, a part of which is a circle of radius R. A block of mass m is pushed against a spring of spring constant k fixed at the left end and is then released. Find the initial compression of the spring so that the block presses the track with a force mg when it reaches the point P, where the radius of the track is horizontal.
Solution
Given,
normal force on the track at point P,
N = mg
As shown in the figure,
\[\frac{\text{ m}\nu^2}{\text{R}} = \text{mg}\]
\[ \Rightarrow \nu^2 = \text{ gR . . . (i) }\]
Total energy at point A = Total energy at point P
\[\text{ i . e} . \frac{1}{2}\text{kx}^2 = \frac{1}{2}\text{m} \nu^2 + \text{mgR}\]
\[ \Rightarrow \text{x}^2 = \frac{\text{mgR + 2mgR}}{\text{k}}\]
\[ [\text{ because, } \nu^2 = \text{ gR }]\]
\[ \Rightarrow x^2 = 3 \text{ mgR}/\text{ k } \]
\[ \Rightarrow x = \sqrt{\frac{\left( 3\text{ mgR } \right)}{\text{k}}}\]
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