To move any body against frictional force,we need to apply some force.This force shall be sufficient enough to over come the frictional force so that the body can move.Then the applied force is able to over come maximum static frictional force.Every body will have its weight and it applies some force on lower surface that is in contact.It will automatically provide the normal reaction.
Let us consider a force F is applied on the body with an angle to the horizontal and we are actually trying to pull the body.This applied force is not completely acting along the direction of motion and we need to know the part of force that is causing the motion.Hence it shall be resolved into components using vector laws.The vertical component of force acts against the weight and it reduces the effective weight.Thus normal reaction also decreases.
Horizontal component of force tries to move the body and frictional force acts against this horizontal component.If once the body starts moving,we can calculate the effective force as the difference between the applied forces horizontal component and frictional force.
By simplifying the mathematical equation further,we can get the minimum required force to just move the body against the friction as shown below.
However, if the applied force is horizontal, then resolution of force is not required and the job is much simpler.It can be written as applied force minus frictional force.Here frictional force is the product of coefficient of friction and normal reaction.
Refer the following video for detailed mathematical proof for the minimum required force to pull the body against the friction.
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