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 have our own weight which we can
measure every morning on the bathroom scale. We know when one thing
is heavy and when another thing 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.
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 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.
F = G * m1 * m2 / 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, ge
.
ge = G * m Earth / (d Earth)^2
ge = 9.8 m/sec^2 = 32.2 ft/sec^2
The weight W, or
gravitational force, is then just the mass of an object times the
gravitational acceleration.
W = m * ge
An object's mass does not change from place to place, but an object's
weight does change because the gravitational acceleration ge 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:
m = Ws / ge = Wo / go
Wo = Ws * go / ge
where Ws = surface weight (250,000 pounds), Wo is the orbital
weight, and go 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.
go / ge = (d Earth)^2 / (d orbit)^2
go / ge = (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
freefall
of all objects 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.
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, ge = 9.8 m/sec^2, 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
gm is given by:
gm = G * m Moon / (d Moon)^2
gm = 1.61 m/sec^2 = 5.3 ft/sec^2
and the gravitational acceleration of
Mars
gmar is given by:
gmar = G * m Mars / (d Mars)^2
gmar = 3.68 m/sec^2 = 12.1 ft/sec^2
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.
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.
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 rocket
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.
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