Blog Archives

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.

Basic Kinematic Relationships

The study of kinematics is the study of the relationships between different elements of motion like position, velocity, and acceleration. Kinematics does not consider the forces that cause these motions, only the motions themselves. In this discussion I will use the symbol s for position, v for velocity, and a for acceleration. If a body travels from one position to another position and that trip starts at one time and ends at another time, we can write the average velocity for the body as:

However realize that this is the average velocity. The velocity at any point during the trip can vary because we are only considering the beginning and ending positions and times. To find the instantaneous velocity we use the same theory but we use calculus to express the change in position with respect to time using differential elements.

In much the same way we can express the average and instantaneous acceleration of a body as follows.

Using separation of variables on the last equation we can develop an integral relationship between acceleration, velocity, and time.

This says the ending velocity is equal to the beginning velocity plus the integral of the acceleration with respect to time. If the acceleration is constant through the entire time period we get:

Next let’s do the same separation and integration to the position and velocity differential equation.

Now we could proceed and find the relationship between position and constant velocity as we did above, but I think that step is obvious and I leave that to the reader. Here I want to make a substitution and come up with a different relationship. We will substitute the above equation relating velocity and constant acceleration into the last integral as follows.

This last equation can tell us the final position of a body if we know the original position, the original velocity, the constant acceleration rate, and the time of the trip.

Finally we will develop one more kinematic formula. We start again with the differential equation relating velocity and position. Then we will use a chain rule and variable separation with integration to reach our destination.

 

Notice that this equation does not require knowledge about the time. All we need is the original velocity, the constant acceleration rate, and the change in the position to find the final velocity.

I hope you see from this discussion that kinematics is a very simple, fundamental exercise in mechanics. If we boil all of this down, we can solve any problem in general by simply applying the principles of calculus to the  two fundamental differential equations:

to formulate whatever relationship we need.