the kinetic energy of a particle moving along a circle of radius R depends on the distance covered s as KE = as^2 where a is constant . find force acting on particle as function of s

Let us consider a particle starts from rest and attains kinetic energy  ( a s² ) after moving a distance s in circular path

 

By work-energy theorem , workdone W = ( a s² )

 

Force F_t = dW/ds = ( 2 a s )

 

Above force F_t is acting along tangential direction .

 

If kineteic energy at a distance s is (a s² ) , then we get velocity at a distance s as

 

(1/2) m v² = a s²   or    v² = ( 2 a s² ) /m

 

Radial force ( centripetal force ) , F_r =  m ( v² / R ) = ( 2 a s² ) / R

 

Radial force F_r and tangential force F_t are perpendicular to each other. Hence , resultant force F_R is given as

 

Direction of force makes angle θ with radial force so that , tanθ = F_t / F_r = ( 2 a s ) / [ ( 2 a s² ) / R ] = ( R / s )