Mathematics of gravity

Summary: Our understanding of gravity has changed considerably over time, such that a history of gravity is virtually a history of physics. Researchers study many different effects and conceptualizations of gravity, some of which are very far from Isaac Newton’s falling apple.

On the surface of the Earth, every object has some weight, which is simply the gravitational force that Earth exerts on it. In reality, minuscule gravitational forces are exerted on every atom of every object, the net effect of which is the same as the effect of a single force (the weight) acting at a single point, the center of gravity (CoG). If the object is sitting on a table, the downward force of gravity is balanced by the upward force provided by contact with the table, and there is no movement. Likewise, when a person holds an object like a barbell, the person must provide an upward force equal to the barbell’s weight to keep it from falling. Mathematics shows how Sir Isaac Newton’s laws of motion can explain a very complex set of observations. Scientists and mathematicians also study other conceptualizations of gravity, such as energy extraction from gravitational fields, quantum gravity, topological gravity, and supersymmetric gravity.

Properties of Gravity

Gravitational force is peculiar in that it does not depend on motion (unlike, for example, muscle forces or aerodynamic forces). The force of gravity is the same whether the object sits on a table or is allowed to fall. For an object in free fall, Newton’s second law dictates:

downward acceleration = net downward force ÷ mass, and if aerodynamic forces are small enough to be neglected, net downward force is equal to weight, so that downward acceleration = weight ÷ mass.

Another peculiarity of gravitational force is that it is directly proportional to mass. Therefore (weight ÷ mass) is the same for all objects; it is approximately 9.81 m/s² near the surface of Earth, called “acceleration due to gravity,” generally denoted by g.

Any object accelerates as it falls downward. Starting from rest (speed = 0), its speed after t seconds will be g×t. So,

Therefore, the distance traveled (d) can be calculated as d = average speed × t, which can be expressed algebraically as

This gravitational force provides a simple method for measuring a person’s visual reaction time: have the subject hold a ruler at the top and let it hang vertically. Let the subject bring his thumb and forefinger near to but not touching a known reading on the ruler, ready to grab it when it falls. At a random time, let the ruler fall. Measure the distance d it fell before it was grabbed and compute t, the reaction time, from the above equation. For d in centimeters:

When gravity is the only force, whether the object is moving up, down, or at an angle, its velocity vector changes continually but its acceleration vector remains constant (magnitude g, pointing downward). The distinction between the velocity and acceleration vectors is fundamental to dynamics. The space shuttle circling Earth has constant downward acceleration when it is not firing its rockets, though its velocity—never downward—changes direction continually. Mathematics allows one to calculate what its speed must be so that the change in direction would correspond to the known constant acceleration. This speed (about 17,500 miles per hour) then determines that the period of making a complete circle around Earth is approximately 90 minutes. Farther away from Earth, gravity is weaker, so that g is smaller. It is proportional to

where r is the distance from Earth’s center (“inverse square law”). Taking this factor into consideration, one can determine that a circular orbit at an altitude of 22,236 miles will take 24 hours to make a complete circle. This is, indeed, where communications satellites are located, so that they would seem not to be moving as seen from the rotating Earth. Similarly, the distance to the moon’s orbit can be related to its period of revolution.

The same ideas can be applied to the gravitational forces between the sun and the planets, leading to remarkably accurate descriptions of the shapes the orbits of planets can take, the change in speed as the orbit is traversed, and the relation between period of revolution and distance from the sun. All this follows from Newton’s second law and a rule of how much the gravitational force weakens with distance.

Bibliography

Buchbinder, Joseph, and Sergei Kuzenko. Ideas and Methods of Supersymmetry and Supergravity. Oxfordshire, England: Taylor & Francis, 1998.

Gamow, George. Gravity. New York: Dover Publications, 2002.

Rovelli, Carlo. Quantum Gravity. Cambridge, England: Cambridge University Press, 2007.