The story of adding forces

A long time ago…

An ailing father on his deathbed tells his four sons to bring him a bundle of sticks.

He then tells them to break the bundle as a whole. They fail.

He then tells them to take separate sticks & break them. Everyone succeeds.

Moral: Stay together or the world out there will break YOU. Simple & effective!

In this post, we’ll apply the morals of this story to find simple ways to add forces in mechanics. Coming to the point,

When 2 forces are in the same direction they add up.

When they are in opposite directions, they get subtracted.

Simple & Easy. Isn’t it?

In the story, if all the 4 sons had together tried to break the bundle, they’d have succeeded.

That gives us another moral to learn:

Moral: Together a team can solve any problem. No matter how big it is.

Being an engineer involves doing rather than talking. We must be able to apply our knowledge. A knowledge or moral that we can’t apply is a waste for engineers.

But this can only be done when their direction is the same. What about the cases in which the direction differs?

We don’t have to look far for that.

Assume 2 forces which are not in the same direction.

Now if we were to add them as 2 intact individual forces. Check out the following difficulties:

Parallelogram Law of Forces: If two forces are drawn starting from a common point in magnitude & direction, their resultant is given by the diagonal starting from the same point & passing through the opposite corner of the parallelogram formed from them.

OR

Triangle Law of Forces: If forces are drawn in magnitude & direction such that tail of 2nd force starts @ the head of 1st, the line joining tail of 1st force with the head of 2nd, gives the resultant (both in magnitude & direction).

There are 2 optional methods we can use to add the forces:

  1. Analytical: Deriving & Remembering a complicated a formula.
  2. Geometric: Using geometric instruments to draw & measure with accuracy.

When the forces are more than 2, it becomes more complicated.

Polygon Law: If a set of forces are represented both in magnitude & direction as edges of a polygon drawn in order, their vector sum is represented by the closing edge of the polygon in reverse order.

Again there are the same 2 optional methods we can use.

But look how difficult it gets. Parallelogram or triangle law with analytical method would mean we’d have to find resultant of 2 forces at a time. Then add that resultant to the next & so on. Till all the forces are added.

Using geometric instruments would mean drawing the polygon with represented lines at the angles which ensure proper direction is maintained. Still, the accuracy would depend on the skills of one who draws the diagram.

Isn’t there a simple method which involves remembering a simple formula & using a calculator to fast & reliable results?

It is! Just open the bundle as in the story. Divide & you can rule.

A force is already single. “How to break it?” you say. Well, I’d say it’s more like a log which can be splintered.

Use trigonometry to split the forces into a horizontal & a vertical component each. Now all the components are either vertical or horizontal.

All the vertical components can be added to get vertical component of resultant. Similarly, all the horizontal components can be added to get horizontal component of resultant.

The resultant can of course be found using pythagoras theorem for magnitude & trigonometric tangent ratio for direction.

So here the moral of mechanics is:

Moral: Divide a complicated problem to make it simple.

That brings me to the conclusion:

  1. We must stay together in a team to be strong.
  2. We must join our strengths to counter big problems.
  3. We must divide problems into parts to make them simple.

Now that’s what I call rediscovering more meaning in the same old stories.

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