NEWTON’S SECOND LAW
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Kepler said that the planets revolve around the sun in ellipses with the sun at one focus and that they followed certain laws involving their velocities and periods. Newton was working on the law of gravitation under the assumption that gravity was the force that governed the motion of the planets. So the problem for Newton was to show that motion under a gravitational force would have to be the same as the motion Kepler described.
He found the relation between force and motion in what is called Newton’s Second Law of Motion. It takes the form of an equation that allows the path followed by an object along with the velocity of the object to be calculated if the force and the starting conditions are known. Here is the equation:
Force equals mass times acceleration,
Which is written in symbolic form as
F = ma.
The remarkable thing about this is that it works for any motion under any force that we know of as long as the conditions do not stray too far away from ordinary conditions normally encountered by humans. It is true that corrections are needed for objects moving extremely fast and for objects confined in a very small region of space such as in an atom. These corrections are supplied by Einstein’s Relativity Theory and Quantum Theory, which we will not worry about just now. These theories are not needed except under fairly extreme conditions that Newton could not have known about. Otherwise Newton’s Laws of Motion are very general. They describe any kind of motion that people could conceive of during Newton’s time.
If you (or Newton) wish to calculate the motion of a planet, first you need the starting conditions. That means you need to know where the planet starts and how fast it is moving in the beginning. Suppose it is located one astronomical unit (AU) from the sun and is not moving at all. One AU is the average distance from the earth to the sun.
DOES THIS
Of course, in that case, Newton’s Laws predict just what you would expect. The planet does not follow an ellipse at all; it drops right into the sun in a straight line. It would take a couple of months to get there. This, however, is not the usual starting condition for a planet.
Usually the planet would start out a certain distance from the sun moving nearly at right angles to the direction to the sun.
Small initial velocity to the side
In this case, the planet starts to fall toward the sun along a curve.
As
long as the initial sideways velocity is large enough, it will miss the sun and
whip around the back of it. Eventually
it returns to the starting point.
Newton’s mathematical work on this problem showed that the curve it follows is
indeed an ellipse with the sun at one focus just as Kepler said. Not only that, but since it was really
falling toward the sun, then it gathered speed as it fell. That is why its speed was the greatest as it
swung around at its closest distance form the sun. Then, as it rose back to the starting point, it lost speed just
as anything does while rising in a gravitational field. Newton was able to show that the changes in
speed would exactly satisfy Kepler’s second and third laws about the velocity
and period of the planet.
Newton’s Laws also show that the orbit could be a circle provided that the starting velocity is exactly right for the distance from the sun. Most planets have orbits that are close to circles but with deviations significant enough to bedevil all of the modelers up to the time of Kepler.
If the starting velocity becomes greater, the top of the ellipse moves farther away from the sun. There is such a thing as a velocity large enough to be called the escape velocity. If the planet starts out with such a high velocity, then it will no longer follow an ellipse. Anything following an ellipse will, of course, not escape because an ellipse is a closed curve. That means it comes back to the starting point. If an object is moving exactly with escape velocity, it follows an open curve called a parabola.
It
never comes back.
If it starts to the right of the sun moving up, then it
follows the upper (solid) part of the parabola. The dotted portion is just there to show that there is more
to the parabola, at least in an abstract mathematical sense.
If it starts out moving faster than escape velocity, then it follows another open curve called a hyperbola. That looks a little bit like the curve pictured above, but the distant portions on the left tend to straighten out to a greater degree.
So Newton could show that several paths were possible, straight line, ellipse, circle, parabola, and hyperbola, depending on the starting conditions. The real planets follow ellipses because their speeds are right for ellipses. It would take a very special speed to produce a circle, and of course, any planet with enough speed for a parabola or a hyperbola would have left the solar system long ago. Nothing is at rest in the solar system. If anything ever was, it fell into the sun long ago.
In effect, Newton showed why the planets follow ellipses and obey the other two Kepler Laws. Such motion is made necessary by the basic laws of motion that govern all motion in the universe. These basic laws were not just conjured up from someone’s brain and accepted without any testing as Plato’s idea of perfection was. Newton’s Laws take a specific mathematical form, which can be used to predict motion in all sorts of circumstances. Then these predictions can be (and are) tested by experiment. With the exceptions involving high speeds and so forth noted above, the results have always agreed with the predictions of the laws. So we are pretty sure that they represent something fundamental.
Acceleration, force, and mass all play important roles in the laws of motion. If you went through the slide show on the subject, you found out that uniform motion means motion at a constant speed in a straight line and that acceleration is any deviation from uniform motion. Acceleration can be measured and represented by a number such as 4.0 meters per second per second. This number means that the velocity is changing at a rate of 4.0 meters per second each second, and it is usually written as 4.0 meters per second squared (4.0 m/s2), which, in an algebraic sense, means the same thing. The larger this number for acceleration is then the more violent is the deviation from uniform motion.
It is harder for a large mass to accelerate than for a small mass. If you applied a force of 60 units (newtons, for example) to a mass it produces a certain acceleration. If the mass is one kilogram, then the acceleration must be 60 m/s2, according to Newton’s Second Law:
F = ma
becomes 60 = 1 x a
So a must be 60.
But if the same force is applied to 2 kilograms of mass, then the acceleration can only be 30 m/s2. That is because
F = ma
takes the form 60 = 2 x 30.
If the mass is 30 kilograms, the acceleration can only be 2 m/s2. So the larger masses always tend to hinder the acceleration. In other words, a large mass cannot change its motion very easily. This property of not being able to change your motion is called inertia, so a large mass has more inertia. In general, mass and inertia always go together.
There is a slide show at this link that illustrates the connection between force, mass, and acceleration. Look at the show and then come back here. If two objects have the same mass, then a larger force will produce a larger acceleration. But if one object has twice the mass of another, then the same force will produce half the acceleration on the large mass.
Here is what Newton’s Second Law is good for: if someone gives you force and mass, you can find the acceleration that goes with them. If you know the right mathematics, then this acceleration along with the starting conditions can be used to predict the motion at any time thereafter. That is what Newton did with the planets, among many other things, and that is what has been happening ever since.