energy of orbit

Question

Total mechanical energy of mass m in circular orbit radius r around M?

RankGuru · Accepted Answer

1. **Potential Energy (U)**: $U = -\frac{GMm}{r}$.
2. **Kinetic Energy (K)**: From orbital mechanics, $\frac{mv^2}{r} = \frac{GMm}{r^2} \implies mv^2 = \frac{GMm}{r}$.
   - Thus, $K = \frac{1}{2} mv^2 = \frac{GMm}{2r}$.
3. **Total Mechanical Energy (E)**: $E = K + U$.
   - $E = \frac{GMm}{2r} - \frac{GMm}{r} = -\frac{GMm}{2r}$.
4. **Important Relations**: 
   - $E = -K$
   - $U = 2E$

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