Newton's Laws of Motion

You may have detected a cause and effect relationship among the elements of motion we have been studying under the heading of kinematics . Consider a particle at rest in our reference frame. It can't undergo a displacement, that is change position without getting a velocity. It can't change its velocity from zero without getting an acceleration. Acceleration seems to be at the root of motion as far as kinematics is concerned. Now we need to peel back the next layer and look at the cause of acceleration. That takes us out of the realm of kinematics into the next level of mechanics, called dynamics .

A
fundamental concept in dynamics is the dynamical system. Since
dynamics is the study of motion *and* forces , things that move or things
under the influence of forces or both are called dynamical
systems. For example a satellite in orbit around the Earth is a
dynamical system. So also might a beating heart be considered a
dynamical system.

So where did the word "system" come from? What makes an object a system? In general a system is considered to be a thing composed of more than one part. Now I do not want to make too big a deal about this one word "system" because I am not sure all that much thought went into choosing this word back in the early days, but in fact we can not study the motion of a single isolated point. Think about it. Things only move relative to other things, like an observer for example. So if there is motion there is a system. Likewise no force exists for a single isolated particle so again the word system seems to apply.

By applying the laws of nature to a dynamical system, we may determine its future behavior. That is really what science is all about. Predicting the future. We are in this business so as to predict the future early enough and accurately enough to profit from our knowledge.

The technique of predicting the future of a dynamical system by application of the laws of nature which govern its change as time passes is called "mathematical modeling". The way it works is this. We discover somehow the laws relating the physical quantities of the dynamical system to time. We express those laws in mathematical terms. The resulting equations may involve the derivative of some of the variables with respect to time. Then we solve the equations using the things we know to calculate the things we don't know.

This is called modeling because the mathematical functions derived from the laws of nature display the behavior of the observable quantities of the actual system. That means that we can avoid the time and expense involved in building an actual model of the dynamical system, assuming that it is even possible to build such a model.

Newton's first law of motion, sometimes called the law of inertia, was actually adopted by Newton from the work of Galileo. It states that a particle's velocity will not change unless a force is applied to the particle. This means that if a "body" (a collection of particles having appreciable mass) is at rest it remains at rest unless a force is applied. If a body is moving with some velocity, it will continue to move with the same velocity unless a force is applied. The first law is really just a qualitative statement about the persistence of motion.

Newton's second law gets into a quantitative relationship
between forces and motion. For this we need to revisit the rate
of change idea. We defined velocity as the rate of change of
position, a small displacement divided by the corresponding
change in time, as *v* =
D*r* / Dt. The velocity
vector itself may change over time so it too has a rate of
change. That rate of change of velocity is called
"acceleration". The acceleration of a body is the
change in velocity divided by the corresponding change in time,
as *a* = D*v* / Dt. We have no single word analogous to
displacement, describing the change in velocity.

The second law states that the acceleration of a body is
proportional to the force on it. This is consistent with our
experience that the harder we push on a moveable body, the
quicker its speed changes. The second law goes on to state that
the constant of proportionality between the force and the
acceleration is the "mass" of the body. In the form of
an equation the second law reads *F*=m**a*, where
*F* is the force vector, m is the scalar mass, and *
a* is the acceleration vector. The mass may be considered the
property of a body which determines its resistance to changing
its velocity.

We are using the kilogram as the unit of mass, as stated earlier. The unit of length is the meter, so displacement is in meters. Velocity is calculated as displacement divided by time so its units are meters per second. Acceleration is calculated as velocity divided by time so its units are meters per second per second, or meters per second squared. The force sufficient to accelerate one kilogram by one meter per second squared is called one Newton, in honor of the old gentleman himself, seen at the right.

You may have observed that the first law is contained in the second as the special case where force and therefore acceleration are zero.

Newton's third law addresses the nature of forces. The implicit assumption is that a force is simply a manifestation of the interaction between a pair of bodies. You might say there can not be a pushee without a pusher. The third law states that the force resulting from the interaction of two bodies acts with equal magnitude on both of them and in opposite directions. For every action, there is an equal and opposite reaction.

These three laws of nature credited to Newton are not all there are but they are enough to allow us to get started in building and analyzing mathematical models of some dynamical systems.

Suppose for example that I toss a ball straight up from the surface of the Earth. Once it leaves my hand let's assume the only force acting on it is gravity. The Vertical Ball Toss display shows that situation. I could not resist adding a bit of reality to this system by making the ball bouncy. Notice that it looses some height on every bounce. Later we will relate that to some properties of the ball and the surface on which it is bouncing.

Suppose in tossing the ball I fail to toss it straight up. That means that the initial velocity vector has not only a vertical component but also a horizontal one. In our discussion of vectors we showed how a vector may be resolved into perpendicular vector components.

Now that we have introduced force into the picture, I am going to go over again here an important idea about the independence of motion in different dimensions. One of the things implicit in Newton's second law is that the change in velocity, acceleration and force all point in the same direction. This is evident from the fact that multiplying a vector by a scalar does not change its direction. Acceleration is force divided by the scalar, mass, so acceleration is "collinear" with force, meaning that the vectors lie in the same direction. The change in velocity is acceleration multiplied by the scalar, time, so the change in velocity is collinear with the force. Notice that it is only the change in velocity which lies in line with the force, necessarily the velocity itself.

Since force and acceleration are collinear, a force vector which is vertical can have no effect on horizontal motion. Likewise a horizontal force can not affect vertical motion. This means that the motion of a particle in two or three dimensions can be studied as two or three independent one dimensional problems where the dimensions are chosen along axes which are mutually perpendicular. For example in our thrown ball situation one of the axes would be chosen vertical and the other horizontal with distance increasing in the direction of the horizontal component of the initial velocity.

Next we apply Newton's laws to each of the independent motions separately. In the vertical direction the ball moves exactly as when tossed straight up. In the horizontal direction gravity has no effect on the ball's motion since it acts perpendicular to that path. It is Newton's first law which applies in this direction. Since the ball experiences no horizontal force it moves with uniform velocity horizontally until it returns to the ground. The Combined Ball Toss display shows the effect of having both horizontal and vertical components of initial velocity.

These ball toss displays present the output of the model in the most physical but least useful manner. What you see is an animation of the motion of a ball tossed with a certain initial velocity. Except for the ability to precisely set the initial velocity, this model brings us no closer to predicting the future than actually tossing a ball would do. To get a better grip on the data generated by the model, we could make a video tape of the moving ball, run it one frame at a time and measure the ball's position in each frame. Then we could tabulate the results and plot the table on a graph. In fact that is the way dynamics data was handled in the old days.

Now that mathematical modeling on a computer is possible, we can go directly to a more useful presentation. As a first approach we can capture the motion by recording the ball's flight, plotting its trajectory so that we can see where it has been as well as where it is. That is the sort of display we saw in the Projectile Motion example. A tossed ball and an artillery shell should have some things in common.

A still more useful representation of the model output would allow you to answer directly questions like, "What is the maximum height reached by a projectile?", "At what time is the maximum height reached?", "When does the projectile hit the ground... At what point?" and so on. This information is contained in plots of vertical and horizontal position vs time as illustrated on the next display. On such plots you can just place the cursor to get the required value. If more precision is required than you can get from the graph, you will need a table of values.

There is a sort of a recipe for solving problems involving Newton's laws as applied to individual particles . Rember we are still dealing with particles, even though we have moved on from kinematics. However messy a problem may seem to be, frequently we can boil it down to the movement of one or more particles under the influence of various forces. The key to solving these problems is to use a trick called the " free body diagram" to keep track of the forces applied.

Even if there are multiple particles involved, isolate them one at a time and carefully identify the forces which apply to the one on which you are working. Then choose the orientation of your reference frame so as to simplify your work as much as possible. Next draw a diagram showing the particle with each force vector attached to it. Then resolve each of the force vectors into its components along the reference frame dimensions. Add up the forces in the x, y, and z(if applicable) directions. Then to get the acceleration along each dimension divide the appropriate force by the mass. Once the accelerations are known, apply the rules we developed in kinematics, or use calculus, or do mathematical modeling on a computer, to get the velocities and positions as functions of time.

Play around with the Free Body Diagram Tool to get some idea what it can do for you. Detailed instruction are included with the display.

So far we have exercised Newton's first and second law. The third law deals with action and reaction. What this law really means is that forces always occur in pairs. In Newtonian terms there is no such thing as an isolated force. The action and reaction force pair also always acts on different bodies. If they acted on the same body, there would be no motion because the net force on everything would be zero. For any body you can imagine which is subject to some force, there is another body subject to the reactive force. This business of action and reaction does not imply a cause and effect. Either force could be considered the action force, the other is then the reaction.

Correctly identifying the action/reaction force pairs and associating each force with the correct object is sometimes a stumbling block. Imagine a book lying on a table standing on the surface of the Earth. There is a table-book force, a book-table force, a table-Earth force, an Earth-table force, an Earth-book force and a book-Earth force. The force of the Earth on the book and the equal but opposite force of the table on the book are not an action/reaction pair. They both act on the same body. The force of the Earth on the book and the force of the book on the table are not an action/reaction pair. They are both in the same direction. The force of the Earth on the book and the force of the Earth on the table are not an action/reaction pair. They involve three different bodies.

Consider the ball toss situation, If I toss a ball straight up with a certain force, my hand exerts the force on the ball and the ball exerts an equal and opposite force on my hand. The force on my hand from the ball is transmitted through the structure of my body to my feet which are planted firmly on a very large ball called the Earth. So the ball is accelerated in one direction and the person-Earth body is accelerated in the opposite direction. The reason we don't notice the movement of the Earth is the huge difference in mass between it and the ball.

Let's consider a situation where the reaction force produces a more noticeable effect. A friend of mine once tried to use a six gauge goose gun to propel a modified harpoon at giant blue fin tuna. A few moments of quiet reflection on Newton's laws could have saved him considerable discomfort. The goose gun had a mass of about 8KG. The harpoon had a mass of about 2KG. The burning of the charge of gun powder, about a quarter of a cup of it, produced equal force on the projectile and the gun. The acceleration produced by a force is inversely proportional to the mass of the body, so the velocity of the gun toward the shooter only built up one fourth as fast as the velocity of the harpoon toward the fish. Even so the collision between the gun and the shooter was hard to ignore.

Perhaps a more common example of the third law is the collision between two billiard balls. During the time the balls are in contact, they are subject to equal and opposite forces. The situation in this little two body system from shortly before to shortly after the collision is shown in the Billiards display.

In the Billiards display, the operation of Newton's third law is seen in the action and reaction of the two balls. Perhaps later we will expand the capability of this display to see some more interesting effects. You will notice that the magnitude of the average force between the two balls during their contact time is displayed at the end of the run. The directions of the action and reaction forces are shown by the yellow and green vectors, which are not to scale by the way. The assumptions used in calculating this average force is that the balls each have a mass of 0.25kg and they remain in contact for 1 millisecond. We will deal with this sort of impulse force in more detail later.

In the next section of the course, I will go into some of the details of setting up and solving the kind of equations which make mathematical modeling possible.