Question

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

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

Answer 1

Answer: (C)

In case of pure rolling, mechanical energy will conserved.
$∴\frac{1}{2}M{v}_0^2+\frac{1}{2}\left(\frac{1}{2}M{R}^2\right)(\frac{v_0}{R}{)}^2=2[\frac{1}{2}k{x}^{2}max]$
$∴x_{max}=\sqrt{\frac{3M}{4k}}{v}_0$
As, $f=\frac{2kx}{3}$
$∴F_{max}=μMg=\frac{2k{x}_{max}}{3}$
$=\frac{2k}{3}\sqrt{\frac{3M}{4k}}{v}_0$
$∴v_0=μg\sqrt{\frac{3M}{k}}$
$∴$ Correct option is (c).