# Weight Equation

## Weight

**Weight** is the force generated by the gravitational attraction of one object on another object. The equation which describes the weight of an object is the same equation whether we are studying airplanes, rockets, or rocks. Weight is fundamentally different from the aerodynamic forces, **lift and drag**, and the thrust force. Aerodynamic forces and thrust are **mechanical forces** and the object has to be in physical contact with the gas which generates the force. The gravitational force is a **field force**; the source of the force does not have to be in physical contact with the object.

## Nature of Gravity

The nature of the gravitational force has been studied by scientists for many years and is still being investigated by theoretical physicists. For an object the size of a rocket flying near the Earth, the descriptions given three hundred years ago by Sir Isaac Newton work quite well. Newton published his theory of gravitation with his laws of motion in 1686. The gravitational force, **F**, between two particles equals a universal constant, **G**, times the product of the mass of the particles, **m _{1} and m_{2}**, divided by the square of the distance,

**d**, between the particles.

\(\LARGE F=G\frac{m_{1}m_{2}}{d^{2}}\)

If you have a lot of particles acting on a single particle, you have to add up the contribution of all the individual particles. For objects near the Earth, the sum of the mass of all the particles is simply the mass of the Earth and the distance is then measured from the center of the Earth. On the surface of the Earth the distance is about 4000 miles. Scientists have combined the universal gravitational constant, the mass of the Earth, and the square of the radius of the Earth to form the gravitational acceleration,** g _{e}**. On the surface of the Earth, it’s value is 9.8 meters per square second or 32.2 feet per square second.

\(\LARGE g_{e}=G\frac{m_{earth}}{d_{earth}^{2}}\)

The weight **W**, or gravitational force, is then just the mass of an object times the gravitational acceleration.

\(\LARGE W=mg\)

## Mass and Weight

The gravitational constant **g** depends on the mass of the planet and on the radius of the planet. So an object has a different value of the weight force on the Earth, Moon, and Mars because each planet has a different mass and a different radius. The mass of the object remains the same on these three bodies, but the weight of the object changes. Roughly speaking, the weight on the Moon is 1/6 of the weight on Earth and the weight on Mars is 1/3 of the weight on Earth.

## Altitude and Weight

Since the gravitational constant **g _{e}** depends on the square of the distance from the center of the Earth, the weight of an object decreases with altitude. Let’s do a test problem to see how much the weight of a model rocket changes with altitude. If a model can reach 35000 feet (about 7 miles) the distance to the center of the Earth is about 4007 miles. We can calculate the ratio of the gravitational constant to the value at the surface of the Earth as the square of (4000/4007) which equals .9965. If the rocket weighs 100 pounds on the surface of the Earth, it weighs 99.65 pounds at 35000 feet; it has lost .35 pounds, a very small amount compared to 100 pounds.

### Example

Let’s do another problem and compute the weight of the Space Shuttle in low Earth orbit. On the ground, the orbiter weighs about 250,000 pounds. In orbit, the shuttle is about 200 miles above the surface of the Earth. As before, the gravitational constant ratio is the square of (4000/4200) which equals .907. On orbit, the shuttle weighs 250,000 * .907 = 226,757 pounds. **Notice: the weight is not zero. The shuttle is not weightless in orbit**. “Weightlessness” is caused by the speed of the shuttle in orbit. The shuttle is pulled towards the Earth because of gravity. But the high orbital speed, tangent to the surface of the Earth, causes the fall towards the surface to be exactly matched by the curvature of the Earth away from the shuttle. In essence, the shuttle is constantly falling all around the Earth.