Topic 2.6: Gravitational Force

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Before We Start: Last Week's Recap

Last week in the elevator lab, your scale reading changed even though your mass never changed. If your mass didn't change, what did?

Today's New Concepts

AP CED Alignment: Unit 2, Topic 2.6

Newton's Law of Universal Gravitation

• Any two objects with mass attract each other gravitationally
• $F_g = \dfrac{Gm_1m_2}{r^2}$, where $G = 6.67\times10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$ and $r$ is the distance between the objects' centers of mass
• Always attractive, always acts along the line connecting the two centers of mass, and can be treated as acting at each object's center of mass

Worked Example: Two objects a distance $r$ apart experience gravitational force $F_g$. If the distance triples and one mass doubles: $F_g' = \dfrac{G(2m_1)m_2}{(3r)^2} = \dfrac{2}{9}F_g$ — the distance change dominates because it's squared.

The Gravitational Field and Weight

• The gravitational field describes how strongly a mass $M$ would pull on ANYTHING placed at distance $r$: $g = \dfrac{F_g}{m} = \dfrac{GM}{r^2}$
• Near Earth's surface, $g\approx9.8\text{ m/s}^2$ (sometimes rounded to $g\approx10\text{ N/kg}$ for quick estimates)
• Weight is just the gravitational force near a large body: $F_g = mg$ — same mass, different weight on different worlds

Worked Example: A planet has 3× Earth's mass and 3× Earth's radius: $g_{planet} = \dfrac{G(3M_E)}{(3R_E)^2} = \dfrac{1}{3}g_E$, so an object weighing $W_E$ on Earth weighs $\dfrac{1}{3}W_E$ there.

Apparent Weight vs. True Weight

• Apparent weight = what a scale reads = the normal force $F_N$
• When accelerating, $F_N \ne mg$ — this is exactly the elevator lab's $F_N = m(g\pm a)$
• "Weightless" means zero apparent weight (zero normal force), NOT zero gravity — an object in free fall (gravity is the only force) reads zero on a scale even though gravity is fully acting on it

Worked Example: A 60 kg person accelerates upward at 2.0 m/s² in an elevator: $F_N = m(g+a) = (60)(9.8+2.0) = 708\text{ N}$ — more than their true weight of 588 N.

Keep This Sheet!

These sheets build into your semester study guide. Keep them in order in a binder or folder — you'll want to flip back through them before quizzes, unit tests, and when AP exam review starts in the spring.