define gravitational potential at a point as the work done per unit mass in bringing a small test mass from infinity to the point

Gravitational Potential

Learning Objective

Define gravitational potential at a point as the work done per unit mass in bringing a small test mass from infinity to that point.

Key Concepts

• Gravitational field and force.
• Work done by a variable force.
• Reference point at infinity where potential is taken as zero.
• Potential energy \$U = m\phi\$ where \$\phi\$ is the gravitational potential.

Derivation

Consider a test mass \$m\$ moving radially towards a point at distance \$r\$ from a mass \$M\$.

1. The gravitational force on \$m\$ is \$F = \dfrac{GMm}{r^{2}}\$ directed towards \$M\$.
2. The infinitesimal work done by the external agent in moving \$m\$ a distance \$dr\$ against the field is \$dW = -F\,dr = -\dfrac{GMm}{r^{2}}\,dr\$.
3. Integrate from \$r=\infty\$ (where \$W=0\$) to the final distance \$r\$:

\$\$

W = -\int{\infty}^{r} \frac{GMm}{r^{2}}\,dr = -GMm\left[-\frac{1}{r}\right]{\infty}^{r}= -\frac{GMm}{r}.

\$\$

The work done per unit mass is therefore

\$\$

\phi(r)=\frac{W}{m}= -\frac{GM}{r}.

\$\$

This quantity \$\phi(r)\$ is defined as the gravitational potential at distance \$r\$ from the mass \$M\$.

Important Points

• The reference point is taken at infinity, where \$\phi(\infty)=0\$.
• Potential is a scalar quantity; it can be added algebraically.
• For any point in a uniform gravitational field near Earth's surface, \$\phi = gh\$ (approximation).
• Units: joules per kilogram (J kg⁻¹).

Symbol Table

Application Example

Calculate the gravitational potential at the surface of the Earth (radius \$R{\oplus}=6.37\times10^{6}\,\text{m}\$, mass \$M{\oplus}=5.97\times10^{24}\,\text{kg}\$).

\$\$

\phi{\oplus}= -\frac{GM{\oplus}}{R_{\oplus}} \approx -\frac{6.674\times10^{-11}\times5.97\times10^{24}}{6.37\times10^{6}} \approx -6.3\times10^{7}\,\text{J kg}^{-1}.

\$\$

This negative value indicates that work must be done against the Earth's gravity to move a unit mass from the surface to infinity.

Common Misconceptions

1. Thinking that potential is always negative – it is negative only when the reference is at infinity; other reference points can give positive values.
2. Confusing potential with potential energy – potential is per unit mass, whereas potential energy depends on the mass of the object.
3. Assuming the potential is the same as the field strength – the field is the gradient of the potential, not the potential itself.

Summary

Gravitational potential \$\phi\$ at a point is defined as the work done per unit mass in bringing a test mass from infinity to that point, expressed as \$\phi = -GM/r\$. It provides a convenient scalar description of the gravitational field and forms the basis for calculating potential energy, orbital mechanics, and energy considerations in astrophysics.