Gravitational Potential Energy

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Change in potential energy moving mass m from r₁ to r₂ (r₂ > r₁)?

RankGuru · Accepted Answer

1. **Potential Energy Formula**: $U(r) = -\frac{GMm}{r}$.
2. **Change in Energy ($\Delta U$)**: $\Delta U = U_{final} - U_{initial} = U(r_2) - U(r_1)$.
3. **Calculation**: $\Delta U = (-\frac{GMm}{r_2}) - (-\frac{GMm}{r_1}) = \frac{GMm}{r_1} - \frac{GMm}{r_2}$.
4. **Factor out GMm**: $\Delta U = GMm (\frac{1}{r_1} - \frac{1}{r_2})$.
5. **Sign Check**: If $r_2 > r_1$, then $1/r_1 > 1/r_2$, making $\Delta U$ positive. This makes sense as work must be done against gravity to increase separation.

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