Question

A wheel (to be considered as a ring) of mass $m$ and radius $R$ rolls without sliding on a horizontal surface with constant velocity $v$. It encounters a step of height $\frac{R}{2}$ at which it ascends without sliding. Choose the correct statement(s).

• A. The angular velocity of the ring just after it comes in contact with the step is $\frac{3v}{4R}$.
• B. The normal reaction due to the step on the wheel just after the impact is $\frac{mg}{2}-\frac{9m{v}^2}{6R}$.
• C. The normal reaction due to the step on the wheel increases as the wheel ascends.
• D. The normal reaction due to the step on the wheel stays constant as the wheel ascends.

Answer 1

Answer: (A), (B), (C)

The normal reaction due to the step on the wheel increases as the wheel ascends.


Angular momentum about $IAOR$ remains conserved. [no net external torque]

After hitting the step, point $O$ behaves as the instantaneous axis of rotation. Let ${\omega }_i$ be the initial angular velocity about centre and $\omega$ be the angular velocity about point $O$ immediately after collision.

Then,
$L_i=L_f$

$\Rightarrow I_c{\omega }_i+m{v}_i{r}_{\perp }=I_O\omega$


$\Rightarrow$$\omega =\frac{I_c{\omega }_i+m{v}_i{r}_{\perp }}{I_O}$

$\left[\begin{array}{c}{I}_c=m{R}^2,{\omega }_i=\frac{v}{R}\\ r_{\perp }=\frac{R}{2},v_i=v\\ I_O=m{R}^2+m{R}^2\\ =2m{R}^2\end{array}\right]$

$\Rightarrow$$\omega =\frac{m{R}^2\left(\frac{v}{R}\right)+mv\left(\frac{R}{2}\right)}{2m{R}^2}$

$\omega =\frac{3v}{4R}\dots \dots \left(1\right)$

Now, just after impact
$mg\cos {60}^{\circ }-N=\frac{m{v}_c^2}{R}$ [from FBD]

$\Rightarrow N=\frac{mg}{2}-\frac{m(R\omega {)}^2}{R}\dots \dots \left(2\right)[\because v_c=R\omega ]$

Using $\left(1\right)$ in $\left(2\right)$,

$N=\frac{mg}{2}-\frac{9m{v}^2}{16R}$

As the wheel ascends, $\theta$ (which was initially ${60}^{\circ }$) decreases and $v_c$ decreases [from conservation of mechanical energy]

$N=mg\cos \theta -\frac{m{v}_c^2}{R}$

$\Rightarrow$ Thus, $N$ increases.

So, options $\left(a\right),\left(b\right)$ and $\left(c\right)$ are correct.

Why this question? This is a multi-conceptual question that involves the concepts of instantaneous axis of rotation, angular momentum conservation and Newton's laws in circular motion.