Question

A wheel of radius R, mass m and moment of inertia $I=m{R}^2$ is pulled along a horizontal surface by application of force F to a rope unwinding from the axle of radius r as shown in Figure. Friction is sufficient for the pure rolling of the wheel.
(a) What is the linear acceleration of the wheel?
(b) Calculate the frictional force that acts on the wheel.

Answer 1

$F-f=ma\to \left(1\right)$
$Fr+fR=1\propto =m{R}^2\propto$
$\Rightarrow \frac{Fr}{R}+f=m\propto R=ma\to \left(2\right)$
$Adding\left(1\right)\&\left(2\right),F(1+\frac{r}{R})=2ma\therefore q=\frac{F}{2m}(1+\frac{r}{R})$
Find f from any of the equations by substitution.