Question

A wheel (to be considered as a ring) of mass m and radius R rolls without sliding on a horizontal surface R/2 at which it ascends without sliding.

• A. the angular velocity of the ring just after it comes in contact with the step is 3 v/4 R
• B. the normal reaction due to the step on the wheel just after the impact is $\frac{mg}{2}+\frac{9m{v}^2}{16R}$
• C. the normal reaction due to the step on the wheel increases as the wheel ascends
• D. the friction will be absent during the ascent.

Answer 1

Answer: (A), (C)

The correct options are
A the angular velocity of the ring just after it comes in contact with the step is 3 v/4 R
C the normal reaction due to the step on the wheel increases as the wheel ascends

$\begin{array}{c}\text{By conservation of angular momentum about point of contact:}\\ L_i=L_F\Rightarrow \left(mv\right)\frac{R}{2}+I_{cm}{\omega }_0=2m{R}^2\omega \dots \left(1\right)\\ \text{For rolling ,}{\omega }_0=\frac{v}{R}\text{:}\\ \begin{array}{cc}\text{So, eqn-(i) becomes}& \to \frac{1}{2}mvR+\frac{1}{2}m{R}^2\left(\frac{v}{R}\right)=2m{R}^{2}c\\ & \Rightarrow [\omega =\frac{3v}{4R}].\end{array}\end{array}$

$\begin{array}{c}\text{At time of contact :}\\ \begin{array}{c}\text{As it ascends,}\theta \text{decrease}\Rightarrow \cos \theta \text{increases}\\ \text{so,}N\text{increases as wheel ascents.}\end{array}\end{array}$

$\text{Option A and C are correct.}$