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A satellite in a circular orbit of radius r has period T.
Chapter 13, Problem 62CP(choose chapter or problem)
A satellite in a circular orbit of radius \(r\) has period \(T\). A satellite in a nearby orbit with radius \(r+\Delta r\), where \(\Delta r \ll r\), has the very slightly different period \(T+\Delta T\).
a. Show that
\(\frac{\Delta T}{T}=\frac{3}{2} \frac{\Delta r}{r}\)
b. Two earth satellites are in parallel orbits with radii \(6700 \mathrm{~km}\) and \(6701 \mathrm{~km}\). One day they pass each other, \(1 \mathrm{~km}\) apart, along a line radially outward from the earth. How long will it be until they are again \(1 \mathrm{~km}\) apart?
Equation Transcription:
Text Transcription:
r
T
R+ delta r
delta r << r
T+delta T
delta T/T=3/2 delta r/r
6700 km
6701 km
1 km
Questions & Answers
QUESTION:
A satellite in a circular orbit of radius \(r\) has period \(T\). A satellite in a nearby orbit with radius \(r+\Delta r\), where \(\Delta r \ll r\), has the very slightly different period \(T+\Delta T\).
a. Show that
\(\frac{\Delta T}{T}=\frac{3}{2} \frac{\Delta r}{r}\)
b. Two earth satellites are in parallel orbits with radii \(6700 \mathrm{~km}\) and \(6701 \mathrm{~km}\). One day they pass each other, \(1 \mathrm{~km}\) apart, along a line radially outward from the earth. How long will it be until they are again \(1 \mathrm{~km}\) apart?
Equation Transcription:
Text Transcription:
r
T
R+ delta r
delta r << r
T+delta T
delta T/T=3/2 delta r/r
6700 km
6701 km
1 km
ANSWER:
Step 1 of 3
a) We need to prove that \(\frac{\Delta T}{T}=\frac{3}{2} \frac{\Delta r}{r}\).
From the Kepler's third law,
\(T^{2}=\left(\frac{4 \pi^{2}}{G M}\right) r^{3}\)
Implies period,
\(T=\sqrt{\frac{4 \pi^{2}}{G M}} r^{\frac{3}{2}}\)
Let
\(a=\sqrt{\frac{4 \pi^{2}}{G M}}\)
So the first satellite obeys \(T=a r^{\frac{3}{2}}\)
Now for the second satellite,
\(T+\Delta T=a(r+\Delta r)^{\frac{3}{2}}\)
\(T+\Delta T=a r^{\frac{3}{2}}\left(1+\frac{\Delta r}{r}\right)^{\frac{3}{2}}\)