R ત્રિજયાના વર્તુળમાં ગતિ કરતા કણની ગતિઊર્

According to given problem $\frac{1}{2}m{v^2} = a{s^2}$ $ \Rightarrow v = s\sqrt {\frac{{2a}}{m}} $

So ${a_R} = \frac{{{v^2}}}{R} = \frac{{2a{s^2}}}{{mR}}$…$(i)$

Further more as ${a_t} = \frac{{dv}}{{dt}} = \frac{{dv}}{{ds}} \cdot \frac{{ds}}{{dt}} = v\frac{{dv}}{{ds}}$…$(ii)$   (By chain rule)

Which in light of equation $(i)$ i.e. $v = s\sqrt {\frac{{2a}}{m}} $ yields

${a_t} = \left[ {s\sqrt {\frac{{2a}}{m}} } \right]\,\left[ {\sqrt {\frac{{2a}}{m}} } \right] = \frac{{2as}}{m}$…$(iii)$

So that $a = \sqrt {a_R^2 + a_t^2} = \sqrt {{{\left[ {\frac{{2a{s^2}}}{{mR}}} \right]}^2} + {{\left[ {\frac{{2as}}{m}} \right]}^2}} $

Hence $a = \frac{{2as}}{m}\sqrt {1 + {{[s/R]}^2}} $

 $F = ma = 2as\sqrt {1 + {{[s/R]}^2}} $