1. What could scientists learn by using the Law of Superposition?
2 answers:
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A) Orbital speed:
B) Kinetic energy:
D) The orbital period is
F) The angular momentum is
G) Exponent of radial dependence:
Speed: -1/2
Kinetic energy: -1
Orbital period: 3/2
Angular momentum: 1/2
Explanation:
A)
We know that for a satellite in circular orbit around a planet of mass M, the gravitational force between the satellite and the planet is
where m is the mass of the satellite.
This force provides the centripetal force needed for the circular motion, which is
where v is the orbital speed.
Since the gravitational force provides the centripetal force, we can equate the two expressions:
And solving for v, we find
B)
The kinetic energy of an object is given by
where
m is the mass of the object
v is its speed
In this problem,
m is the mass of the satellite
is the speed of the satellite (found in part A)
Substituting, we find an expression for the kinetic energy of the satellite:
D)
The orbital speed of the satellite can be rewritten as the ratio between the distance covered during one orbit (the circumference of the orbit) divided by the period of revolution:
where
is the circumference of the orbit
T is the orbital period
We already found that the orbital speed is
Substituting into the equation,
And making T the subject,
F)
The angular momentum of an object is defined as
where
m is the mass of the object
v is its speed
r is the radius of the orbit
For the satellite here we have
m (mass of the satellite)
(orbital speed)
R (orbital radius)
Substituting,
G)
First, we rewrite the list of expressions for the different quantities that we found:
Orbital speed:
Kinetic energy:
Orbital period:
Angular momentum:
Now we observed the dependence of each quantity from R:
Orbital speed:
Kinetic energy:
Orbital period:
Angular momentum:
So the exponent of the radial dependence of each quantity is:
Speed: -1/2
Kinetic energy: -1
Orbital period: 3/2
Angular momentum: 1/2
Learn more about circular motion:
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