How a Lever Works

HOW A LEVER WORKS

STATIC CASE WHERE ENERGY IS NOT BEING EXPENDED (BALANCE OF FORCES)

The rigid massless beam is balanced (m1 & m2 are not moving) in this reference frame, therefore the sum of all the forces acting on m1 and m2 are equal to zero by Newton's first law of motion. And, F3 = F1 + F2 by Newton's third law of motion. The potential energy in the system is constant and proportional to the total mass m1 + m2 and the height of the objects above the base, and the kinetic energy is equal to zero since the masses are not moving. Since the beam is rigid, the downward point forces F1 and F2 are distributed along the distances d1 and d2 respectively to produce a torsion force or torque about the pivot point. The torque for each mass can be found by integrating the force along the distance from the pivot point to each respective mass, and since the beam is balanced, these two torsional forces can be set equal to each other. 

IS EQUAL TO    

Integrating yields the result: F1*d1 = F2*d2

and

F1 = F2*d2/d1  

MOVING CASE WHERE FORCES ARE NOT EQUAL AND ENERGY IS BEING EXPENDED

 WITH MOVEMENT OF BEAM

If the lever arm is moved a distance of d1 from the left end, the amount of force it takes to move it will be:     F1 = Frictional Force x d2/d1
If energy is to be conserved, the product of force through distance on each side has to be equal. 
With a little trigonometry it can be shown that F1 = F2 x L2/L1.