According to given problem \(\frac{1}{2}mv^{2} = as^{2}\) \(\Rightarrow v = s_1 \sqrt{\frac{2a}{m}}\)

So \(\mathbf{a}_{R} = \frac{v^{2}}{R} = \frac{2as}{mR}\) …(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. \(v = s_1 \sqrt{\frac{2a}{m}}\) yields

\(a_{t} = \left[ s \sqrt{\frac{2a}{m}} \right] \left[ \sqrt{\frac{2a}{m}} \right] = \frac{2as}{m}\) …(iii)

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

Hence \(\mathbf{a} = \frac{2as}{m} \sqrt{1 + \left(s/R\right)^2}\)

∴ ∴ \(F = ma = 2as \sqrt{1 + \left(s/R\right)^2}\)