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 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 [see. Fig], where the radius of the track is horizontal

• A. $\sqrt{\frac{mgR}{3k}}$
• B. $\sqrt{\frac{3gR}{mk}}$
• C. $\sqrt{\frac{3mgR}{k}}$
• D. $\sqrt{\frac{3mg}{kR}}$

Answer 1

Answer: (C)

$\sqrt{\frac{3mgR}{k}}$

Answer 2

For the given condition, centrifugal force at P should be equal to mg i.e.$\frac{mv_{P}^{2}}{R} = mg\therefore v_{P} = \sqrt{Rg}$

From this we can easily calculate the required velocity at the lowest point of circular track.

$v_{p}^{2} = v_{L}^{2} - 2gR$ (by using formula : $v^{2} = u^{2} - 2gh$)

$v_{L} = \sqrt{v_{P}^{2} + 2gR} = \sqrt{Rg + 2gR} = \sqrt{3gR}$It means the block should possess kinetic energy $= \frac{1}{2}mv_{L}^{2} = \frac{1}{2}m \times 3gR$

And by the law of conservation of energy

$\frac{1}{2}kx^{2} = \frac{1}{2}3m \times ⥂ gR$ ⇒ $x = \sqrt{\frac{3mgR}{k}}$.