A classical mechanics problem by Kartik Sharma

This is a tricky one.

When is minimum force required? Yes, it is not given that force is applied horizontally.

Therefore, we have to find at what angle force must be applied so that we get a minimum force.

$\text{F - force applied}$

$\mu - \text{coefficient of friction}$

$\text{N - normal force}$

$\text{m - mass of the block}$

$\alpha - \text{angle force makes with the horizontal}$

For no slipping,

$\displaystyle F\cos(\alpha) \le \mu N$

$\displaystyle F\sin(\alpha) + N = mg$

Therefore,

$\displaystyle F \le \frac{mg}{cos(\alpha) + \mu \sin(\alpha)}$

$\displaystyle F \le \frac{mg}{\sqrt{1 + \mu^2}}$

Therefore, to just move, this is the minimum force.

Hence,

$\mu = \frac{F^2}{W^2} - 1 = 0.75$