Gravitational Potential

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Potential at distance r from isolated mass M (zero at infinity)?

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1. **Definition**: Gravitational potential ($V$) at a point is the work done per unit mass in bringing a mass from infinity to that point.
2. **Significance of Sign**: Since the gravitational force is attractive, work is done by the field itself (not an external agent). This results in a negative potential.
3. **Mathematical Form**: $V = -\int_{\infty}^r g \cdot dr$.
4. **Calculation**: $V = -\int_{\infty}^r \frac{GM}{x^2} dx = -[ -\frac{GM}{x} ]_{\infty}^r = -\frac{GM}{r}$.
5. **Conclusion**: At infinity, $V = 0$. Closer to the mass, it becomes more negative, reflecting a deeper potential well.

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Gravitational Potential

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