Principle of Moments
Suppose we place a uniform ruler horizontally over a triangular piece of wood. If we are very careful and put the centre exactly on top of the wooden tip, then the ruler might balance. Then suppose that I shift the ruler a small distance to te left and let go. The ruler would tilt and fall to the left. It will start with the weight W at the centre of gravity pulling it down. Since one point (P) of the ruler is still on the wood, this point would act like a pivot. As the ruler falls, it also rotates about P.
Suppose now that I tie a small piece of metal on the right hand end, and adjust the pivot P so that the ruler is again balanced. If I know the weight W of the ruler and the weight A of the metal, can I find the position of the pivot? The answer is Yes. The distances a and b of the pivot from W and A are related by the principle of moments:
The left hand side W x a is the moment of W about the pivot. The moment of a force about a pivot is defined as the product of the force and the perpendicular distance from the pivot to the line of action of the force.
That would take some time to understand. For the above ruler, it means that the moment of W about the pivot P is the force W times a certain distance. This is the perpendicular distance from the pivot to W. This means that from P, if we draw a line to W such that the line is perpendicular to W, then the distance a along this line, from P to W, is the distance we want. That is to say, the moment is W x a. Likewise, the moment of A about P would be A x b.
The weight W would rotate the ruler anti-clockwise about the pivot P, if A were not there to balance it. Likewise, the force A would tend to rotate the ruler clockwise. We say that W produces an anti-clockwise moment about P, and A gives a clockwise moment. The principle of moments, in a more general form, states that
As an example, suppose that the mass of the ruler is 40 g, and the mass of A is 10 g. What are a and b? To find the moments, we need to find the forces, which are given by the weight W = mg. However, when we substitute this into both sides of the formula above, g cancels out. So in this case, we can just take the mass as the force. Then the principle of moments give:
So the ratio a:b is 1:4. If this is a metre rule, then a would be 10 cm and b 40 cm. In this way, we have found the position of the pivot P that balances the ruler with the hanging weight at A.
I repeat the definition of moment:
It is not just the distance from the pivot to the point where the force acts but a perpendicular distance. What is the line of action anyway?
The line of action is just line that we get if we imagine extending both ends of the the arrow that represents the force into a long straight line. To see how to find the correct distance, consider a rectangular block of wood on a rough floor. I push it horizontally at a top corner. If I slowly increase my effort, there would come a point when the wood just starts to tip over. When this happens, a corner, A, on the ground would act as the pivot. The wood rotates about A. What is the moment of the force F from my hand about A?
The moment is F times a distance. What distance? Is it the distance from A to the tip of the arrow that represents F in the figure? In order to find the correct distance, we must first find the line of action. Imagine extending the arrow of F into a long line in both directions. The distance is the perpendicular distance of A from this line.
In order words, when you find the distance, draw the line and forget about the force. So the distance is h. Using the same idea, the distance for W from A is d. Applying the principle of moments, we get
Suppose the wood has height 1 m and base 50 cm, and a weight of 100 N. With what force must you push to topple it? Applying the above equation, we get
Solving, we find that F is 25 N.
Copyright 2011 by Kai Hock. All rights reserved.
Last updated: 17 March 2011