Isaac Newton (1642-1727) occupies a unique place in the history of human thought – he was both one of the greatest mathematicians and greatest scientists ever. The biographer of mathematicians, Eric Bell (Bell, 1986) said of him “his successors capable of appreciating his work have almost without exception pointed to him as the supreme intellect that the human race has produced”. For all his genius though, Newton was certainly no popularizer of science or math. His written work is very difficult to follow, for he made few concessions to those with less ability (which was everyone).

Although much of Newton’s work is accessible only to specialists, some of his results can be understood and appreciated by the rest of us. In this post I will describe one such investigation – his effort to determine how the force of gravity decreases with distance from the earth. The results were very significant in a scientific sense, and the way he carried out the work shows astonishing insight and imagination.

I will proceed through a few sections covering essential background before describing the penultimate calculations.

Universal Gravitation

It’s often said that Newton “discovered” gravity, but that is nonsense, of course. Clearly, some force emanating from the earth pulls objects downward, in the direction of the earth’s center. Newton though, realized that gravity was a universal property of matter, rather than some special property that only earth has. Every mass in the universe, said Newton, pulls on every other mass with a force proportional to the product of the two masses. Thus the Sun, planets, and the Moon all attract each other, meaning that the things outside our earth are subject to the same laws that hold for the earth.

Now, the force of gravity clearly must decrease with distance – it couldn’t be that every pair of masses attracts other masses with the same force, no matter how far away. Newton thought that the force probably dropped off as the square of the distance, but he needed evidence to support this hypothesis.

Acceleration

If the velocity of something is changing, the rate at which it is changing is called acceleration. For example, if the velocity of a car changes from 30 meters per second to 50 meters per second over a period of 5 seconds, the (average) acceleration is (50 – 30) / 5 = 4 meters per second per second.

F=MA

Exert a force on a mass, and its velocity will change (i.e., it accelerates). Newton’s Second Law of Motion says that the acceleration is equal to the force divided by the mass ( A = F / M, or F = MA). So the acceleration will be larger if the force is larger or if the mass is lower.

This law helps to understand what happens when a mass is falling. The force involved is due to gravity, and is proportional to the product of the earth’s mass times the object’s mass. So in F/M, the F goes up in exact proportion to M, cancelling the M in the denominator. Therefore, A remains the same for different masses, and that explains why the rate of fall does not depend on mass. (You may remember Galileo’s experiment where he dropped different weight objects from the Tower of Pisa to show that they hit the ground at the same time).

Finally, notice that this law suggests a way to quantify the force of gravity – just measure acceleration for a falling mass. The gravitational force and the acceleration it produces are proportional. At the surface of the earth, the gravity force is sufficient to produce an acceleration of 9.8 meters per second per second.

Circular Motion

Even Newton didn’t understand motion in a circle at first; it seemed to be fundamentally different from straight-line motion. What he came to understand was that circular motion can be considered to be a combination of two separate motions; straight-line motion and motion toward the center.

The mass in the diagram, with velocity v, would move, in the absence any external forces, from point a to b in the time ?t, and this motion would be at constant speed. Instead, the mass is at point c after time . The difference is that it has also “fallen” through the distance bc. The combination of straight line motion and inward motion is such that the mass follows a circular path.

Now, the velocity of the inner-directed motion has to increase with time, i.e. there must be acceleration. Since there is an inward directed acceleration, there must be an inward directed force. For example, if the mass is being whirled around at the end of a string, the tension in the string provides the force. If it is a planet revolving around the Sun, then gravity provides the inward directed force.

Newton derived a formula for the inward acceleration: a = V² / R. V is the velocity around the circle, and R is its radius. All high school physics courses cover the use of this formula, although none explain how it can be derived – in a future post, I will do that.

The Inverse Square Formula

Newton had a problem – he knew the strength of gravity at the earth’s surface, 6,375km from the center, and it was strong enough to make a mass accelerate at 9.8 meters per second per second. To see if the inverse square law holds, he needed to find the strength of earth’s gravity at some far distant point. His solution was to measure gravity’s strength at the Moon!

The key is the V²/R formula; the Moon orbits Earth in a (more or less) circular orbit. Newton could work out both V and R, and thus find the acceleration – that’s the measure of how strong Earth’s gravity is at the distance of the Moon.

Let’s do the numbers (in modern units, and not exactly the numbers Newton used):

The average centre-to-centre distance (R) from the Earth to the Moon is about 384000 km.

We can get V by dividing the circumference of the Moon’s orbit (2?R) by the time for one orbit, 27.3 days (we must convert 27.3 days to seconds, however):

So, acceleration = V²/R = 1022.9²/3.84×10⁸ = 0.0027 meters per second per second.

Now, the Moon is further from the center of the earth by the factor 384000/6375 = 60.2. If gravity does indeed fall off by the square of distance, then the gravity at the Moon should be 1/60.2² = 1/3624 as strong as on earth.

The actual ratio is (drum roll please) 0.0027 / 9.8 = 1/3630. Wow!

Thus encouraged, Newton went on to even more difficult problems, such as, what is the shape of a planet’s orbit if the inverse square law holds? (the answer is, an ellipse).

Postscript

My hope is that the reader will be impressed that a person mired in the 1600s could, through the power of thought alone, measure the strength of gravity at the vast distance of the Moon.

Now I should add that an implicit assumption has been that a spherical body like the earth attracts other masses as if the earth’s mass was all concentrated at the center. This is far from obvious. Remember that, according to universal gravitation, every tiny piece of the earth pulls on an object (like our falling mass) differently, depending on its distance and direction from the mass (with the distance being squared). The assumption is that summing up those infinite number of forces from all the constituent parts of the sphere will work out to be the same as if all mass is at the earth’s center.

The assumption is true, but calculus is required to prove it. It was partly for problems like this that Newton invented calculus (when he was 23 years old).

Bibliography

Bell, E. T. (1986). Men of Mathematics – The Lives and Achievements of the Great Mathematicians from Zeno to Poincare. Touchstone.

Share/Save

This entry was posted on Thursday, January 14th, 2010 at 11:20 am and is filed under Mathematics History, Physics, Thrilling Math. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.