Problem 2B-01: Lever

A lever is shown in Fig. 1. A rigid massless rod AB is supported at point O. The length of AO is a, and the length of BO is b. A weight of mass m is placed at point A. A vertically downward force of magnitude F is applied at point B to lift the weight. What is the minimum value of F needed to lift the weight?

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Solution:

Let the rod forms an angle θ with the horizontal line as shown in Fig. 1. When the weight at point A is touching the ground, the vertically downward gravitational pull of magnitude mg is cancelled out by the vertically upward resistance of equal magnitude from the ground acting on point A so there is no torque created abound point O. However, once the weight is lifted up from the ground, even by a minuscule amount, the resistance force from the ground disappears, and the gravitational pull on the weight will generate a toque of the magnitude τA to rotate the rod clockwise around point O. The downward force at the end of point B creates a torque of the magnitude τB that will try to rotate the rod counter-clockwise around point O. Thus the minimum F required is to satisfy the equation τA = τB. According to Eq. (2) of Topic 2B-01, we have

τA = mg·a·cosθ ,

and

τB = mg·b·cosθ .

Thus τA = τB leads to

F = mg·(a/b) .

To make the use of lever meaningful we must have a < b so that a smaller force F can overcome a larger gravitational pull mg.