Gravity

by Dr. Eric Wegryn


Sir Isaac Newton (1643-1727) is considered by many people to be the greatest scientist of all time. He made numerous important discoveries, including proof that the attractive force of gravity is universal—that is, the force that makes the Moon orbit Earth is the same force which causes apples (and other objects) to fall to the ground.

For those of you who love math, Newton expressed the force of gravity in one of the most important equations of all time:
    F = (G * m1 * m2) ÷ r2
This means that the force of gravity, F, between any two bodies is directly proportional to each of their masses (m1 and m2), and inversely proportional to the distance between them—center-to-center—squared (r2).

But Newton wasn't the first to bring reason to the physics of moving bodies. Galileo made famous the Law of Falling Bodies. He showed that all bodies fall with the same constant acceleration. A freely falling body will speed up 9.8 meters per second every second that it falls (when air resistance is negligible). Galileo also did not accept the traditional Aristotelian belief that rest was the natural state for all objects, and motion had to be sustained (except for the heavenly spheres, which obeyed different laws of nature). Galileo understood the property of inertia, but it took Newton to express the Laws of Motion clearly for posterity.

Newton's Three Laws of Motion can be stated in different ways; here is one way.
    1. A body at rest will remain at rest, and one in motion will remain in uniform motion, unless a net force acts upon it from something else. Uniform motion means no acceleration, i.e. no speeding up, slowing down, or changing direction.

    2. A body will change its motion (accelerate) proportional to the net force acting on it, and inversely proportional to its mass. Using symbols, this is expressed as F = m * a.

    3. For each force imposed on a body, there is always an equal and opposite reaction force imposed upon the source. That is, forces act between two objects in opposite directions.

There are several things to notice about Newton's Laws of Motion:
    1. The First Law is simply a special case of the Second Law, i.e. if the force F is zero, then acceleration will be zero, no matter what the mass. Newton knew that the First Law was redundant and not really necessary, but he stated it anyway to directly refute the prevailing Aristotelian belief that bodies in motion 'naturally' want to come to rest.

    2. The Second Law provides one way to define and measure the property we call mass: putting a force on an object and measuring its acceleration. The resistance to change in motion is called inertia, so the m in the formula F = m * a is inertial mass.

    3. In the Third Law (action-reaction), the equal but opposite forces act upon different bodies. This law leads to the important Laws of Conservation of Momentum, and Conservation of Angular Momentum.

Newton showed that the force of gravity between any two bodies is directly proportional to each of their masses, and inversely proportional to the distance between them squared:
    Law of Universal Gravity: F = (G * m1 * m2) ÷ r2
The constant of proportionality, G, is called the universal gravitational constant. It is one of the fundamental, unchanging constants of nature, and should not be confused with g, the acceleration due to gravity (see Galileo above), which depends on the planet's size and mass. The value of G was first measured by Cavendish in 1798, and is 6.67 x 10-11 N m2/kg2.

Newton's Law of Universal Gravity provides a completely independent way of defining and measuring mass. In this case, mass is not resistance to change in motion (inertia), but the property that causes bodies to be attracted, gravitational mass. Nobody totally understands why these two seemingly independent properties should give the same numbers for a body's mass, but they do (this is Einstein's Principle of Equivalence).

What this means is that even though the m in F=ma is inertial mass, and the m in F=GMm/r2 is gravitational mass, we can assume that they are the same. Then, by setting these two equations equal, we can see what it would be like to stand on different planets.

Here's how: Standing on Earth's surface, you have a certain weight. Weight the force of gravity between you and Earth. (Actually, what you feel is the reaction force opposing this gravitational force.) So w = m (GME/rE2), where m is your mass (in kg). But, from Newton's 2nd Law, weight is also the force that causes the acceleration of falling bodies, g, i.e. w = m g.

As a final honor to Sir Isaac Newton, the metric unit of force (and weight) was named after him. One newton (N) is the force that will cause 1 kilogram to accelerate 1 meter per second per second. If you do not know your metric weight in newtons, you can convert from pounds to kilograms (a kilogram weighs about 2.2 pounds), then to newtons using w=mg.