Physics - Rotation Question with Solution | TestHub

As we know, $\overrightarrow{J}=∆\overrightarrow{P}$

$\Rightarrow J_0=MV ... \left(\mathrm{i}\right)$

About center of the disk, we can also write

$$\begin{gathered} J_0\times h=I_{CM}\omega \\ \Rightarrow J_0\times h=\left[\frac{M}{2}\left(a^2+b^2\right)\right]\omega ... \left(ii\right) \end{gathered}$$

If $h=h_m\Rightarrow \omega =\frac{V}{b}\left(\text{pure rolling}\right)$

From equation(ii), we can write

$\Rightarrow J_0\times h_m=\left(\frac{M}{2}\right)\left(a^2+b^2\right)\frac{V}{b} ... \left(\mathrm{iii}\right)$

By dividing equation $\left(\mathrm{i}\right)$ with the equation $\left(\mathrm{iii}\right)$, we get

$\Rightarrow h_m=\frac{a^2+b^2}{2b}$

Now, considering all cases:

If $a\to 0\Rightarrow h_m=\frac{b}{2}$

If $a\to b\Rightarrow h_m=b$

If $h=h_m\Rightarrow \omega$ is independent of $a$(As we get $\omega =\frac{V}{b}$)

If $\mu =0 \text{and} h=0\Rightarrow$Wheel will slide without rolling.