Question

The maximum value of $v_0$ for which the disk will roll without slipping is

• A. $\mu g\sqrt{\frac{M}{k}}$
• B. $\mu g\sqrt{\frac{M}{2k}}$
• C. $\mu g\sqrt{\frac{3M}{k}}$
• D. $\mu g\sqrt{\frac{5M}{2k}}$

Answer 1

Answer: (C)

In case of pure rolling, mechanical energy will conserved.
$\therefore \frac{1}{2}Mv^2_0+\frac{1}{2}\bigg(\frac{1}{2}MR^2\bigg)\bigg(\frac{v_0}{R}\bigg)^2=2\bigg[\frac{1}{2}kx^2max\bigg]$
$\therefore x_{max}=\sqrt{\frac{3M}{4k}}v_0$
As, $f=\frac{2kx}{3}$
$\therefore F_{max}=\mu Mg=\frac{2kx_{max}}{3}$
$=\frac{2k}{3}\sqrt\frac{3M}{4k}v_0$
$\therefore v_0=\mu g\sqrt{\frac{3M}{k}}$
$\therefore $ Correct option is (c).