4.3.2 Why must 3 forces intersect at the same point?

If three (and only three) forces result in a body in static equilibrium, then (the lines of action of) those three forces must intersect at the same point. Why? Because of rotational equilibrium.

Let’s consider the scenario illustrated below: A rod, hinged at one end and pulled by a cord at the other, is in equilibrium. Obviously, the tension force T by the cord must act along the cord, and the weight W of the rod must act vertically downward. Can we tell the direction of the force exerted by the hinge H?

Firstly, we notice that (the lines of actions of) W and T intersect at point P. This means that W and T produces zero moment about point P.

Since the rod is in rotational equilibrium, it must be experiencing zero net moment. In addition, since the rod is also in translational equilibrium, the net moment is zero not just about the hinge, but about any point and every point, including point P.

But both W and T produce zero moment about point P. So H must also produce zero moment about point P. The is possible only if H’s line of action also passes through P. We are thus able to deduce that H is directed towards P as well.

Technically, H can be either towards or away from P. But the “away from P” option is clearly impossible for the scenario here. Because H, W and T must also sum up to zero

Video Explanation

Why Three Forces Must Intersect at the Same Point?

Demonstration

Tension, Tension and Weight

Concept Test

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