Question

Figure 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.

Answer 1

Given, $N=mg$

From the figure, the force of gravity and the centripetal force remains balanced on the topmost point.

$\frac{m{v}^2}{R}=mg$

thus,

$\Rightarrow v^2=gR$

For compression at point P, the block must be pushed with some amount of energy, this energy can be called as total energy.

$\therefore TotalenergyatpointA=energyatp$

$\frac{1}{2}k{x}^2=\frac{1}{2}m{v}^2+mgR$

$\Rightarrow \frac{1}{2}k{x}^2=\frac{mgR+2mgR}{2}(because{v}^2=gR)$

$\Rightarrow x^2=\frac{3mgR}{k}$

$\Rightarrow x=\sqrt{\frac{3mgR}{k}}$

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