Class 11 > Unit # 05: Work, Energy & Power > Absolute Gravitational Potential Energy (AGPE)

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Gravitation & Work - Absolute Gravitational Potential Energy

Q. Define absolute gravitational potential energy and derive expression for Absolute Gravitational Potential Energy.

Definition

"The absolute gravitational potential energy of a body at a given position is defined as the work done by an external agent in bringing the body from infinity to that point against the gravitational force, without acceleration."

Figure: Dividing large radial displacement into $n$ small intervals to compute gravitational work.

Derivation

For ordinary heights near the Earth's surface, the gravitational force $F = mg$ is assumed constant. However, for large displacements (such as space flights or planetary distances), the gravitational force decreases with distance according to Newton's Law of Gravitation: $$F = \frac{G M_e m}{r^2}$$ Since the force is variable, the simple formula $W = F \cdot S$ cannot be applied directly.

To overcome this, we divide the large distance from point $A$ ($r_A$) to point $B$ ($r_B$) into a large number $n$ of small displacement intervals of equal width $\Delta r$ (where $\Delta r = r_2 - r_1$).

The magnitude of gravitational force at the two ends of the first interval are:

$F_1 = \frac{G M_e m}{r_1^2} \quad \text{and} \quad F_2 = \frac{G M_e m}{r_2^2}$

The average force $F$ acting throughout the first interval can be approximated using the geometric mean of the distances:

$F = \sqrt{F_1 F_2} = \frac{G M_e m}{r_1 r_2}$

Since $\Delta r = r_2 - r_1$, we can write $r_1 r_2 \approx r^2$ where $r$ is the midpoint distance. The work done in moving the body through the first interval $\Delta r$ is:

$W_1 = F \Delta r = \frac{G M_e m}{r_1 r_2} (r_2 - r_1) = G M_e m \left(\frac{r_2 - r_1}{r_1 r_2}\right) = G M_e m \left(\frac{1}{r_1} - \frac{1}{r_2}\right)$

Similarly, the work done across subsequent intervals is:

• Second interval: $W_2 = G M_e m \left(\frac{1}{r_2} - \frac{1}{r_3}\right)$
• $n$-th interval: $W_n = G M_e m \left(\frac{1}{r_{n-1}} - \frac{1}{r_n}\right)$

Summing up all the work done from initial position $r_A$ to final position $r_B$:

$W_{\text{total}} = W_1 + W_2 + \dots + W_n = G M_e m \left(\frac{1}{r_A} - \frac{1}{r_B}\right)$

Reference Point at Infinity

If the reference point $B$ is chosen at infinity ($r_B = \infty$), the potential energy at infinity is taken as zero ($\frac{1}{r_B} = 0$).

Therefore, the absolute potential energy $U$ at a distance $r$ from the center of the Earth is:

$U = -\frac{G M_e m}{r}$

When the body lies on the surface of the Earth ($r = R_e$), the Absolute Gravitational Potential Energy is given by:

$\text{A.G.P.E.} = -\frac{G M_e m}{R_e}$

Where $R_e$ is the radius of the Earth, $M_e$ is the mass of the Earth, $m$ is the mass of the body, and $G$ is the gravitational constant. The negative sign indicates that the gravitational field is an attractive force.

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