Question

A block of mass M is kept on a rough horizontal surface. The coefficient of static friction between the block and the surface is µ. The block is to be pulled by applying a force to it. What minimum force is needed to slide the block? In which direction should this force act?

Answer 1

Let the Force applied on the Block be F and it makes an angle of θ with the Horizontal.

∴ N = Mg - F Sinθ

Now, FCosθ = μN

⇒ FCosθ = μMg - FμSinθ

⇒ F = μMg/(Cosθ + μSinθ)

Now, For Force to be Minimum, the denominator must be maximum.

By the concept of the Trigonometric Ratios, Maximum value of the Expression is  

⇒ r = √(1² + μ²)

∴ F = μMg/√(1² + μ²)

Hope it helps.