Gravitation – Potential and Potential Energy

Gravitational Potential and Potential Energy

Gravitational potential at a point is the amount of work needed to bring a unit mass from infinity to that point.

Gravitational potential is defined to be negative since gravity is attractive in nature. A negative potential means that no external work is needed to bring a unit mass from infinity to that point, since the gravity-producing mass would be doing the work to pull the unit mass to that point.

Mathematically,

\begin{aligned} \phi = -\frac{GM}{r} \end{aligned}

Note that potential is a scalar quantity. Hence, the potential at a point is simply the algebraic sum of the potential of the different masses at that point.

\begin{aligned} \phi_\text{sum} = -\frac{GM_1}{r_1}  - \frac{GM_2}{r_2} - ... \end{aligned}

Similarly, gravitational potential energy is defined as

Gravitational potential energy of a mass at a point is the amount of work needed to bring the mass from infinity to that point.

\begin{aligned} \text{GPE} = -\frac{GMm}{r^2} \end{aligned}

You may find this concept similar to gravitational force and field strength. Both gravitational force and potential energy invloves the product of two masses \begin{aligned} Mm \end{aligned} while field strength and potential involves just the gravity-producing mass \begin{aligned} M \end{aligned}.

Gravitational Potential vs Field Strength

It is easy to compare the relative values of potential and field strength because of the similar form of equations.

\begin{aligned} g &= -\frac{GM}{r^2} \\ \phi &= -\frac{GM}{r}\end{aligned}

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GPE = mgh

In many junior Physics text, gravitational potential energy is quoted with the formula
\begin{aligned} \text{GPE} = mgh \end{aligned}
This formula assumes that the change in height in insignificant compare to the radius of the Earth. This formula also calculates the change in the potential energy due to a change in position.

The formula \begin{aligned} F_G = \frac{GMm}{r} \end{aligned} calculates the actual amount of potential energy a mass possess due to its position. This formula does not calculate the change in potential energy. To calculate the change in potential energy,

\begin{aligned}\text{change in GPE} &= \frac{GMm}{r_1} - \frac{GMm}{r_2}\\\end{aligned}

If \begin{aligned} r_1 \approx r_2 \end{aligned} ,

\begin{aligned} \text{change in GPE} &= GMm ( \frac{1}{r_1} - \frac{1}{r_2} ) \\&= GMm (\frac{r_2-r_1}{r_1 r_2})\\&=gm(r_2-r_1)\\&=mgh\end{aligned}

From this, we have our old formula \begin{aligned} \text{GPE} = mgh \end{aligned}

Summary

  1. Gravitational potential at a point is the amount of work needed to bring a unit mass from infinity to that point.
  2. Gravitational potential energy of a mass at a point is the amount of work needed to bring the mass from infinity to that point.
  3. \begin{aligned} \phi = -\frac{GM}{r} \end{aligned}
  4. The gravitational potential has a larger magnitude than the field strength.
  5. When the change in height is small, we may use \begin{aligned} \text{change in GPE} = mgh \end{aligned}
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