Question

Two bodies of mass m and 4m are attached with string as shown in the figure. The body of mass m hanging from a string of length l is executing oscillations of angular amplitude ${\theta }_0$ while the other body is at rest. The minimum coefficient of friction between the mass 4m and the horizontal surface should be

• A. $\left(\frac{2-cos{\theta }_0}{3}\right)$
• B. $2co{s}^2\left(\frac{{\theta }_0}{2}\right)$
• C. $\left(\frac{1-cos{\theta }_0}{3}\right)$
• D. $\left(\frac{3-2cos{\theta }_0}{4}\right)$

Answer 1

The correct option is D $\left(\frac{3-2cos{\theta }_0}{4}\right)$

$\begin{array}{c}Forthebob\\ mg\left(l-l\cos {\theta }_0\right)=\frac{1}{2}m{v}^2\\ v^2=2gl\left(1-\cos {\theta }_0\right)\\ T-mg=\frac{m{v}^2}{l}\\ T-mg=2mg\left(1-\cos {\theta }_0\right)\\ andT=\mu \cdot 4mg\\ \mu =\frac{3mg-2mg\cos {\theta }_0}{4m}\\ =\frac{3-2\cos {\theta }_0}{4}\end{array}$

Hence,

option $\left(D\right)$ is correct answer.