Forces in Equilibrium

Forces in Equilibrium

If two equal and opposite forces pull at a ball, they cancel out each other. We say that the forces are at equilibrium. If the ball is at rest before the forces are applied, then it would just remain at rest. If the ball is already moving, then it would continue moving at the same velocity.

Suppose there are three forces acting on the ball. Suppose these forces lie in the same plane, but not along the same line. Is it possible that they cancel each other exactly to give a zero resultant force? With only two forces, we know this happens when they are equal and opposite. With three forces, how do we tell whether they are in equilibrium?

In order to understand this, we must first know how to find the resultant of two forces that are in different directions, such as A and B. Suppose that force A is 4 N and force B is 3 N. Consider the case when only A is acting. Suppose that after a short time, the ball moves 4 cm in the direction of A. Next, consider the case when only B is acting. After the same time, the distance moved in the dierction of B must be 3 cm. This is because the distance is proportional to the acceleration, which is in turn proportional to the force.

Finally, consider the case when both forces A and B are acting. Then after the same time, the distance moved must be 4 cm in direction A and 3 cm in direction B. The only way this can happen is if it moves in the direction of the diagonal D of the parallelogram formed by forces A and B. Alternatively, we can draw the relation between A, B and D as a triangle. D is then the resultant of forces A and B. Its direction is given by the side D of the triangle.

If forces A, B and C above are in equilibrium, the force D must be at equilbrium with C, since D is the combined effect of A and B. This means that D must be equal and opposite to C.

So if we reverse the arrow of D in the above triangle, D would become C.

In this way, we find a simple way to tell if the three forces are in equilibrium: They must form a closed triangle. We call this a vector triangle. To clarify how this works, consider how we should draw this triangle. Start with the original sketch for the forces A, B and C.

Suppose we draw these with the correct directions, and the lengths of the arrows to scale with the magnitudes of the forces. Then imagine moving these arrows around without changing any of the directions. Try joining the tail of one to the head of the other, and repeat until all three are joined head to tail. If they are not in equilibrium, the triangle would not be closed. That is, there would be a gap at one corner, where the head and tail of two of the arrows would not meet.

If they are in equilibrium, the triangle would be closed. The head of every arrow would touch the tail of another arrow.

This is the condition for equilibrium of three forces - they must form a closed triangle. This result is very useful. If we know two of the forces, we can find the third force that is needed for equilibrium - by doing a scale drawing for example.

The Physics Notes