Question

A rolling body of mass `m=4 kg`, radius `R` and radius of gyration `k=R//sqrt(3)` is placed as a plank which moves with an acceleration `a_(0) =1 ms^(-2)`. Find the frictional force acting on body if it rolls without sliding. (in `N`).

Answer 1

Correct Answer - 1
`FBD`: let the static friction `f_(s)` act towards right. It pushes the body with an acceleration a say and rotate the body with an angular acceleration `alpha` say. The other forces actin onthe body are `Nuarr` and `mguarr`

Force equation
`f_(s)=ma`........i
`N=mg`..........ii
Torque equation
`tau_(C)=f_(s)R=mk^(2)alpha` ...........iii
Rolling `a_(P)=a_(0)` where `a_(P)=a+Ralpha`
`a+ralpha=a_(0)`...........iv,
substituting a from eqn i `alpha` from eqn iii in eqn iv we have `(f_(s))/m+R(f_(s)R)/(mK^(2))=a_(0)`
`f_(s)=(ma_(0))/(1+(R^(2))/(k^(2)))=(4xx1)/((1+3))=1N`