🌍 Newton's Law of Universal Gravitation

The Universal Law

Every object in the universe attracts every other object with a gravitational force. Newton's Law of Universal Gravitation describes this attraction:

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

where:

• $F_g$ = gravitational force (N)
• $G$ = universal gravitational constant = $6.67 \times 10^{-11}$ N·m²/kg²
• $m_1, m_2$ = masses of the two objects (kg)
• $r$ = distance between the centers of the masses (m)

💡 Key Insight: Gravity acts between ALL objects, but we only notice it for very massive objects (like planets) because $G$ is extremely small!

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Characteristics of Gravitational Force

1. Always Attractive

Unlike electric force (which can attract or repel), gravitational force is always attractive.

2. Action-Reaction Pairs

By Newton's 3rd Law, the force Earth exerts on you equals the force you exert on Earth:

$F_{Earth \to you} = F_{you \to Earth}$

Both have magnitude $\frac{Gm_{Earth}m_{you}}{r^2}$

3. Inverse Square Law

Force is inversely proportional to $r^2$:

• If distance doubles, force becomes $\frac{1}{4}$ as strong
• If distance triples, force becomes $\frac{1}{9}$ as strong

4. Long Range

Gravity has infinite range (never becomes exactly zero), though it becomes negligibly small at large distances.

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Gravitational Field

The gravitational field $g$ at a point is the gravitational force per unit mass:

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

where $M$ is the mass creating the field and $r$ is the distance from its center.

At Earth's Surface

$g = \frac{GM_{Earth}}{R_{Earth}^2} = 9.8 \text{ m/s}^2$

where:

• $M_{Earth} = 5.97 \times 10^{24}$ kg
• $R_{Earth} = 6.37 \times 10^6$ m

Above Earth's Surface

At height $h$ above Earth's surface:

$g_h = \frac{GM_{Earth}}{(R_{Earth} + h)^2}$

As altitude increases, $g$ decreases.

Example: At the altitude of the International Space Station (~400 km):

$g_{ISS} \approx 8.7 \text{ m/s}^2$

(About 89% of surface gravity - astronauts aren't "weightless" due to zero gravity, but due to free fall!)

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Weight vs. Mass

Mass

• Definition: Amount of matter in an object
• Property: Intrinsic to the object
• Units: kg
• Changes?: Never changes

Weight

• Definition: Gravitational force on an object: $W = mg$
• Property: Depends on location
• Units: N (newtons)
• Changes?: Yes! Different on Moon, Mars, in orbit, etc.

Example: A 70 kg person

• Mass: 70 kg (everywhere)
• Weight on Earth: $W = 70 \times 9.8 = 686$ N
• Weight on Moon: $W = 70 \times 1.6 = 112$ N (Moon's $g \approx 1.6$ m/s²)

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Gravitational Force Inside vs. Outside

Outside a Uniform Sphere

Use the distance from the center of the sphere:

$F = \frac{GMm}{r^2}$

This applies to any point outside the sphere (including at the surface).

Inside a Uniform Sphere

Only the mass inside radius $r$ contributes to the force. The shell of mass outside $r$ has zero net effect!

$F = \frac{GM_{inside}m}{r^2}$

For a uniform sphere: $M_{inside} = M\frac{r^3}{R^3}$

So: $F = \frac{GMm}{R^3}r$

The force is proportional to $r$ inside a uniform sphere (increases linearly from zero at center).

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Orbital Motion

When gravitational force provides the centripetal force, objects orbit!

Circular Orbits

Set gravitational force equal to required centripetal force:

$\frac{GMm}{r^2} = \frac{mv^2}{r}$

Simplify (mass $m$ cancels):

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

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

Orbital Speed

The orbital speed is independent of the satellite's mass and depends only on:

• The mass of the central body ($M$)
• The orbital radius ($r$)

Important: Closer orbits are faster! (Smaller $r$ means larger $v$)

Orbital Period

Using $v = \frac{2\pi r}{T}$:

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

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

Kepler's 3rd Law: $T^2 \propto r^3$

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Escape Velocity

The escape velocity is the minimum speed needed to escape a planet's gravitational pull:

$v_{esc} = \sqrt{\frac{2GM}{R}}$

For Earth: $v_{esc} \approx 11,200$ m/s (about 25,000 mph!)

Note: This is $\sqrt{2}$ times the orbital velocity at the surface.

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⚠️ Common Mistakes

Mistake 1: Forgetting r²

❌ Wrong: $F_g = \frac{Gm_1m_2}{r}$ ✅ Right: $F_g = \frac{Gm_1m_2}{r^2}$

Mistake 2: Using Surface Distance for Satellites

❌ Wrong: Using Earth's radius for a satellite 1000 km up ✅ Right: Use $r = R_{Earth} + h = 6370 + 1000 = 7370$ km

Mistake 3: Confusing Weight and Mass

❌ Wrong: "I'm weightless in space, so my mass is zero" ✅ Right: Mass is constant; weight is $mg$ where $g$ varies with location

Mistake 4: Thinking Gravity Stops

❌ Wrong: "There's no gravity in space" ✅ Right: Gravity extends infinitely (though it weakens with distance). Astronauts feel "weightless" because they're in free fall, not because gravity is absent.

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Problem-Solving Strategy

1. Identify the masses and the distance between their centers
2. Use $F_g = \frac{Gm_1m_2}{r^2}$ for gravitational force
3. For orbits, set $F_g = F_c$: $\frac{GMm}{r^2} = \frac{mv^2}{r}$
4. Remember the satellite's mass cancels in orbital calculations
5. Check if distance is from center or from surface (add radius if needed)

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Real-World Applications

GPS Satellites

• Orbit at ~20,200 km altitude
• Must account for weaker gravity (slower time due to general relativity)
• Orbital period ~12 hours

Geostationary Satellites

• Orbit at ~35,800 km altitude
• Period = 24 hours (matches Earth's rotation)
• Appear stationary above a point on equator

Tides

• Moon's gravity creates tidal bulges
• Differential force (near side vs. far side) causes tides
• Sun also contributes (spring and neap tides)

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Key Formulas Summary

| Formula | Description | |---------|-------------| | $F_g = \frac{Gm_1m_2}{r^2}$ | Gravitational force between masses | | $g = \frac{GM}{r^2}$ | Gravitational field strength | | $W = mg$ | Weight of object | | $v_{orbit} = \sqrt{\frac{GM}{r}}$ | Orbital speed | | $T = 2\pi\sqrt{\frac{r^3}{GM}}$ | Orbital period | | $v_{esc} = \sqrt{\frac{2GM}{R}}$ | Escape velocity |

Constants:

• $G = 6.67 \times 10^{-11}$ N·m²/kg²
• $M_{Earth} = 5.97 \times 10^{24}$ kg
• $R_{Earth} = 6.37 \times 10^6$ m