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.
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, 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 gravitational acceleration, ge
. On the surface of the Earth, it's value is 9.8 meters per
square second or 32.2 feet per square second.
ge = G * m earth / (d earth)^2
The weight W, or
gravitational force, is then just the mass of an object times the
gravitational acceleration.
W = m * g
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.
Since the gravitational constant ge 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.
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.
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