Gravitation

Overview

Newton's law of universal gravitation describes the attractive force between any two masses. This force governs planetary motion, satellite orbits, and many celestial phenomena.

Newton's Law of Universal Gravitation

$F = G\frac{m_1 m_2}{r^2}$

Where:

• $G = 6.674 \times 10^{-11}$ N·m²/kg² (gravitational constant)
• $m_1, m_2$ = masses of the two objects
• $r$ = distance between centers of masses
• $F$ = magnitude of attractive force

Vector Form

$\vec{F}_{12} = -G\frac{m_1 m_2}{r^2}\hat{r}_{12}$

Force on $m_1$ due to $m_2$, pointing from $m_1$ toward $m_2$

Gravitational Field

The gravitational field due to mass $M$:

$\vec{g} = -\frac{GM}{r^2}\hat{r}$

Magnitude:

$g = \frac{GM}{r^2}$

At Earth's surface:

$g = \frac{GM_E}{R_E^2} \approx 9.8 \text{ m/s}^2$

Variation with Altitude

$g(h) = \frac{GM}{(R + h)^2} = g_0\left(\frac{R}{R+h}\right)^2$

Gravitational Potential Energy

$U = -\frac{GMm}{r}$

• Negative because we choose $U = 0$ at $r = \infty$
• Always negative (bound system)
• Increases (becomes less negative) as $r$ increases

Change in PE

$\Delta U = -GMm\left(\frac{1}{r_2} - \frac{1}{r_1}\right)$

Near Earth's Surface

For small heights $h \ll R$:

$\Delta U \approx mgh$

Escape Velocity

Minimum velocity to escape gravitational pull:

$v_{\text{escape}} = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$

For Earth:

$v_{\text{escape}} = \sqrt{2 \times 9.8 \times 6.37 \times 10^6} \approx 11.2 \text{ km/s}$

Orbital Motion

Circular Orbit Velocity

$v = \sqrt{\frac{GM}{r}}$

Orbital Period

$T = \frac{2\pi r}{v} = 2\pi\sqrt{\frac{r^3}{GM}}$

Orbital Energy

Total energy in circular orbit:

$E = KE + PE = \frac{1}{2}mv^2 - \frac{GMm}{r} = -\frac{GMm}{2r}$

(Total energy is negative for bound orbits)

Kepler's Laws

First Law (Law of Ellipses)

Planets orbit the Sun in ellipses with the Sun at one focus.

Second Law (Law of Equal Areas)

A line connecting a planet to the Sun sweeps equal areas in equal times.

$\frac{dA}{dt} = \frac{L}{2m} = \text{constant}$

Third Law (Law of Periods)

$T^2 = \frac{4\pi^2}{GM}r^3$

Or for objects orbiting the same body:

$\frac{T_1^2}{r_1^3} = \frac{T_2^2}{r_2^3}$

Geosynchronous Orbit

Orbit with period = 24 hours (stays above same point on Earth):

$r = \sqrt[3]{\frac{GMT^2}{4\pi^2}}$

For Earth:

$r \approx 42{,}200 \text{ km (35,800 km above surface)}$

Examples

Example 1: Force Between Objects

Find gravitational force between two 1000 kg masses separated by 10 m.

$F = \frac{Gm_1 m_2}{r^2} = \frac{(6.674 \times 10^{-11})(1000)(1000)}{100} = 6.67 \times 10^{-7} \text{ N}$

Example 2: Satellite Orbit

Find orbital velocity and period for a satellite at 400 km altitude.

$r = R_E + h = 6.37 \times 10^6 + 4 \times 10^5 = 6.77 \times 10^6 \text{ m}$

$v = \sqrt{\frac{GM}{r}} = \sqrt{\frac{3.99 \times 10^{14}}{6.77 \times 10^6}} = 7.67 \text{ km/s}$

$T = \frac{2\pi r}{v} = \frac{2\pi(6.77 \times 10^6)}{7670} = 5540 \text{ s} \approx 92 \text{ min}$

Example 3: Escape from Moon

Moon: $M = 7.35 \times 10^{22}$ kg, $R = 1.74 \times 10^6$ m

$v_{\text{escape}} = \sqrt{\frac{2GM}{R}} = \sqrt{\frac{2 \times 6.674 \times 10^{-11} \times 7.35 \times 10^{22}}{1.74 \times 10^6}} = 2.38 \text{ km/s}$

Example 4: Energy to Launch Satellite

Energy needed to put a 500 kg satellite in orbit at height $h = 300$ km.

$E_i = -\frac{GMm}{R_E}$

$E_f = -\frac{GMm}{2r} \text{ where } r = R_E + h$

$\Delta E = GMm\left[\frac{1}{R_E} - \frac{1}{2(R_E + h)}\right]$

$\Delta E = 3.99 \times 10^{14} \times 500 \times \left[\frac{1}{6.37 \times 10^6} - \frac{1}{2 \times 6.67 \times 10^6}\right] = 1.6 \times 10^{10} \text{ J}$

Weightlessness

In orbit, objects are in free fall—they experience apparent weightlessness because:

• Both the spacecraft and occupants fall together
• Normal force from "floor" is zero
• They are NOT outside Earth's gravity