Rocket Weight

Weight

Weight is the force generated by the gravitational attraction on the rocket. We are more familiar with weight than with the other forces acting on a rocket, because each of us has our own weight which we can measure every morning on the bathroom scale. We know when one thing is heavy and when another is light. But weight, the gravitational force, is fundamentally different from the other forces acting on a rocket in flight. The aerodynamic forces, lift and drag, and the thrust force are mechanical forces. The rocket must be in physical contact with the gases which generate these forces. The gravitational force is a field force, and the rocket does not have to be in contact with the source of this force.

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, the explanation given three hundred years ago by Sir Isaac Newton is sufficient to describe the motion of the object. Newton developed his theory of gravitation when he was only 23 years old and published the theories with his laws of motion some years later. As Newton observed, the gravitational force between two objects depends on the mass of the objects and the inverse of the square of the distance between the objects. More massive objects create greater forces and the farther apart the objects are the weaker the attraction. Newton was able to express the relationship in a single weight equation. The gravitational force, F, between two particles equals a universal constant, G, times the product of the mass of the particles, m₁ and m₂, divided by the square of the distance, d, between the particles.

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

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 Earth’s gravitational acceleration, g_e.

\(\LARGE g_{e}=\frac{Gm_{\text{Earth}}}{d_{\text{Earth}}^{2}}\)

\(\LARGE g_{e}=9.8\frac{\text{m}}{\text{sec}^{2}}=32.2\frac{\text{ft}}{\text{sec}^{2}}\)

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

\(\LARGE W=mg_{e}\)

An object’s mass does not change from place to place, but an object’s weight does change because the gravitational acceleration g_e depends on the square of the distance from the center of the Earth. Let’s do a calculation and determine 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; the distance from the center of the Earth is 4200 miles. Then:

\(\LARGE m=\frac{W_{s}}{g_{e}}=\frac{W_{o}}{g_{0}}\)

\(\LARGE W_{0}=W_{s}\frac{g_{0}}{g_{e}}\)

where W_s = surface weight (250,000 pounds), W₀ is the orbital weight, and g₀ is the orbital value of the gravitational acceleration. We can calculate the ratio of the orbital gravitational acceleration to the value at the surface of the Earth as the square of Earth radius divided by the square of the orbital radius.

\(\LARGE \frac{g_{0}}{g_{e}}=\frac{d_{Earth}^{2}}{d_{orbit}^{2}}\)

\(\LARGE \frac{g_{0}}{g_{e}}=(\frac{4000}{4200})^{2}=.907\)

On orbit, the shuttle weighs 250,000 * .907 = 226,757 pounds. Notice: the weight is not zero. There is a large gravitational force acting on the Shuttle at a distance of 200 miles. The “weightlessness” experienced by astronauts on board the Shuttle is caused by the free-fall of all objects in orbit. The Shuttle is pulled towards the Earth because of gravity. 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.

Because the weight of an object depends on the mass of the object, the mass of the attracting object, and the square of the distance between them, the surface weight of an object varies from planet to planet. We have derived a gravitational acceleration for the surface of the Earth, g_e = 9.8 m/sec², based on the mass of the Earth and the radius of the Earth. There are similar gravitational accelerations for every object in the solar system which depend on the mass of the object and the radius of the object. Of particular interest for the Vision for Space Exploration, the gravitational acceleration of the Moon g_m is given by:

\(\LARGE g_{m}=\frac{Gm_{\text{Moon}}}{d_{\text{Moon}}^{2}}\)

\(\LARGE g_{e}=1.61\frac{\text{m}}{\text{sec}^{2}}=5.3\frac{\text{ft}}{\text{sec}^{2}}\)

and the gravitational acceleration of Mars g_mar is given by:

\(\LARGE g_{mar}=\frac{Gm_{\text{Mars}}}{d_{\text{Mars}}^{2}}\)

\(\LARGE g_{e}=3.68\frac{\text{m}}{\text{sec}^{2}}=12.1\frac{\text{ft}}{\text{sec}^{2}}\)

Mass and Weights of Rockets

The mass of a rocket is the same on the surface of the Earth, the Moon and Mars. But on the surface of the Moon, the weight force is approximately 1/6 the weight on Earth, and on Mars, the weight is approximately 1/3 the weight on Earth. You don’t need as much thrust to launch the same rocket from the Moon or Mars, because the weight is less on these planets.

Determining Weight

All forces are vector quantities having both a magnitude and a direction. For a rocket, weight is a force which is always directed towards the center of the Earth. The magnitude of this force depends on the mass of all of the parts of the rocket itself, plus the amount of fuel, plus any payload on board. The weight is distributed throughout the rocket, but we can often think of it as collected and acting through a single point called the center of gravity. In flight, the rocket rotates about the center of gravity, but the direction of the weight force always remains toward the center of the Earth.

The Changing Weights of Rockets

During launch the rocket burns up and exhausts its fuel, so the weight of the rocket constantly changes. For a model rocket, the change is a small percentage of the total weight, and we can determine the rocket weight as the sum of the component weights. For a full scale rocked the change is large and must be included in the equations of motion. Engineers have established several mass ratios which help to characterize the performance of a rocket with changing mass. Full scale rockets are often staged or broken into smaller rockets which are discarded during flight to increase the rocket’s performance.