Question

Figure (8-E15) 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,
normal force on the track at point P,
N = mg
As shown in the figure,



$$\begin{gathered} \frac{m{\nu }^2}{\mathrm{R}}=mg \\ \Rightarrow {\nu }^2=g\mathrm{R}...\left(i\right) \end{gathered}$$

Total energy at point A = Total energy at point P
$$\begin{gathered} \mathrm{i}.\mathrm{e}.\frac{1}{2}k{x}^2=\frac{1}{2}m{\nu }^2+mg\mathrm{R} \\ \Rightarrow x^2=\frac{mg\mathrm{R}+2mg\mathrm{R}}{k} \\ [\mathrm{because},{\nu }^2=g\mathrm{R}] \\ \Rightarrow x^2=3mg\mathrm{R}/k \\ \Rightarrow x=\sqrt{\frac{\left(3mg\mathrm{R}\right)}{k}} \end{gathered}$$