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Calculating the Period T of the Moon

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Calculating the Period (T) of the Moon’s Orbit about the
Earth
In order to calculate the Moon’s circular orbit about the Earth
there are three equations we need to apply. The first is what
was derived at the beginning of class:
This equation as we discussed gives us the
period of an object’s circular orbit.
The other two equations come from our understanding of what
the Moon is doing and what is causing the Moon to do this.
Since we know the Moon is moving in a circle, which force
equation is definitely going to be involved?
=
The other equation comes from our understanding of what is
acting as the centripetal force.
The following are values we will need to
calculate the moon’s period (T).
mass of the Moon = 7.36 × 1022 kilograms
mass of Earth = 5.9742 × 1024 kilograms
G = 6.672 × 10-11 N.m2/kg2
Mean distance of earth to moon = 3.84 x 108 m
Before we do that though, since we know that the gravitational force is
acting as the centripetal force in this case, what can we say about those
two forces?
=
=
Remember the point of this exercise is to find the period of the Moon’s orbit
about the Earth. We have the expression for the period (T):
and on the previous slide we were given the mean
distance between the Earth and the Moon, aka, the radius
of orbit. So, the only thing we need now is the value of v.
This is where the gravitational and centripetal forces come
into play.
=
We can solve the above expression for v and then substitute that into
our expression for T. Solving for v, we get:
Then substituting v
into our expression
for T we get:
Before we make our calculations, please take note of
the fact that the adjacent equation is valid for any
satellite engaged in circular motion about a planet.
calculate the period of the Moon’s orbit
mass of Earth = 5.9742 × 1024 kilograms
G = 6.672 × 10-11 N.m2/kg2
Mean distance of earth to moon = 3.84 x 108 m
T = 2368147.819 s
And of course we can find the period in days by
performing a straightforward unit conversion:
" 1hr % " 1day %
2368147.819s • $
'=
' •$
# 3,600s & # 24hr &
27.409 days