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Newton’s Second Law

From my education and experience, I would venture to say that the vast majority of engineering mechanics can have its origins traced back to Newton’s Second Law. It is amazing how much this relatively simple little law can be developed and tell you a great deal about how the world works. In a previous post on kinematics, we discussed the relationship between position, velocity, and acceleration. There we said that those relationships could be developed independent of the forces that created the various motions. Well the study of kinetics is the study of the forces which cause motion and how that all works and it of course is based on Newton’s Second Law. The form I would like to state Newton’s Second Law in is this:

The summation symbol is to show that it is the net force that creates motion or the sum of the forces. For instance if I have a ball on the table and I push in one direction with a certain force and also simultaneously push in the exact opposite direction with the same force, the ball will not move. (It may deform, but no motion.) Now I applied two forces, but nothing happened. That is what is meant by the net force or sum of forces. There must be an unbalanced total force to create motion. The differential on the right-hand side is the rate of change of linear momentum with respect to time. Now I know that some, not well acquainted with calculus may say that this doesn’t look that simple, but I mean it only has one term on the left and one on the right. A equals B. That’s it. The motion of almost all things (at least all the everyday things you see. Some variation lies with subatomic particle motion, and things moving at close to the speed of light, but that’s another topic.) Is summed up in this?! Amazing!

Now if we make a few modifications we can manipulate the equation into the form that most people are familiar with. Now linear momentum is the product of the mass and velocity of a body.

Well if we assume that the mass of the body is constant and does not change with time as the body moves, and realizing that the time rate of change of velocity is the definition of the instantaneous acceleration, you have:

This is typical form you see. Finally we will divide by the mass to change it to this form:

What you can understand from this is that as the sum of the unbalanced forces increases, the acceleration increases, and as the mass of a body increases, the acceleration decreases. This is of course all of our real world experience. The harder something is pushed, the more it speeds up. The bigger it is, the harder it is to speed up.

Already you can see that we can extrapolate a lot of understanding from this little relationship. I plan to build on this in future posts to show some of its application to the different areas of engineering.

Also, here are a few links to some additional helps. First is a physics lecture on Newton’s Laws provided by MIT OCW. Second is a interactive graphical demonstration of Newton’s Second Law provided by Wolfram Demonstrations. Finally, a Khan Academy video.