A rolling body of mass m = 4 kg, radius R and radius of gyration k=R/3 is placed on a plank which moves with an acceleration a0=1ms−2. Find the frictional force acting on the body if it rolls without sliding (in N).

FBD: Let the static friction fs act towards right. It pushes the body with an acceleration a, say and rotate the body with an angular acceleration α, say. The other forces acting on the body are N↑ and mg ↓. Force equation :        fs=ma                       ...(i)N=mg                       ...(ii)Torque equation:        τC=fsR=mk2α           ...(iii)Rolling : aP=a0 where aP=a+Rα      a+Rα=a0          ...(iv)Substituting a from equation (i), α from equation (iii) in equation (iv), we have fsm+RfsRmk2=a0        fs=ma01+R2k2=4×1(1+3)=1N