Question

A uniform ring is connected to a light axle with light spokes so as to form a wheel. The wheel is placed on a horizontal surface with its plane vertical and a constant horizontal force $F$ is applied to the axle. Surface AB is rough and surface right of B is smooth. The wheel does not slip when it moves from A to B. The wheel moves from A to B in time $T$.

• A. Rotational KE remains constant right of B.
• B. Energy is dissipated from A to B.
• C. Linear acceleration right of B is more than linear acceleration left of B.
• D. The wheel undergoes constant non-zero acceleration to the right of B.

Answer 1

Answer: (A), (C), (D)

The correct options are
A Rotational KE remains constant right of B.
C Linear acceleration right of B is more than linear acceleration left of B.
D The wheel undergoes constant non-zero acceleration to the right of B.

A) Since there is no friction after B, there is no torque due to friction. Hence, angular velocity (Rotational KE) remains constant.

B) Since the ring is already rolling without slipping, the frictional force is static and non dissipative.

C) and D)

Left of B,

$F-f=m{a}_1$ (force balance)
$f\times r=m{r}^2\alpha$ (torque balance)
$a_1=r\alpha$ (no slipping)
$\Rightarrow f=mr\alpha =m{a}_1$
$\Rightarrow F=f+m{a}_1=2m{a}_1$
$\Rightarrow a_1=\frac{F}{2m}$

Right of B,


$F=m{a}_2\Rightarrow a_2=\frac{F}{m}\left(constant\right)$
$\Rightarrow a_2>a_1$