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PhDorBust
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Satellite of mass m is moving with velocity v in a circular orbit of radius r about mass M.
Find the orbital angular momentum.
Know
v = \sqrt{\frac{GM}{r}}
Orbital angular momentum of a system is defined as the angular momentum of the center of mass of the system.
Let the origin be at mass M.
r_{cm} = \frac{m}{M + m} r
v_{cm} = \frac{v}{r} \frac{m}{M + m} r = \frac{vm}{M+m}
l = r_{cm} \times p_{cm} = (M + m) r_{cm} v_{cm} = \frac{m^2 rv}{M+m} = \frac{m^2}{m+M} \sqrt{GMr}
Correct answer in text is m \sqrt{GMr}
Find the orbital angular momentum.
Know
v = \sqrt{\frac{GM}{r}}
Orbital angular momentum of a system is defined as the angular momentum of the center of mass of the system.
Let the origin be at mass M.
r_{cm} = \frac{m}{M + m} r
v_{cm} = \frac{v}{r} \frac{m}{M + m} r = \frac{vm}{M+m}
l = r_{cm} \times p_{cm} = (M + m) r_{cm} v_{cm} = \frac{m^2 rv}{M+m} = \frac{m^2}{m+M} \sqrt{GMr}
Correct answer in text is m \sqrt{GMr}