Question

A wheel of mass m and radius r is in accelerated rolling motion without slip under a steady axis torque T. If the coefficient of kinetic friction is $\mu$ the friction force from the ground on the wheel is

• A. $\mu mg$
• B. $\frac{T}{r}$
• C. Zero
• D. None of these

Answer 1

The correct option is D None of these

Assuming point contact
Because coefficient of rolling resistance is not given.

as wheel is in accelerated motion so pt A is in acceleration.

$a_A=r\alpha$ (for pure rolling) ...(i)

$T\to$ Torque given about A. (Axie)

Friction will oppose slipping of point B, so acting rightward.

Torque $=I\alpha$

Torque is about A or line perpendicular to wheel & passing through A.

$T-fr=I\alpha =\frac{m{r}^2}{2}\alpha$ ...(ii)

[As mg passes through A]

Newton's 2nd law in horizontal direction
$f=m{a}_A$ ...(iii)

From equations (i) and (iii)

$f=mr\alpha$ ...(iv)

From equations (ii) and (iv)
$T-m{r}^2\alpha =\frac{m{r}^2}{2}\alpha$

$T=\frac{3}{2}m{r}^2.\alpha$

$\alpha =\frac{2T}{3m{r}^2}$

we know that
$f=mr\alpha =mr\times \frac{2T}{3m{r}^2}=\frac{2T}{3r}$