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Gravitation

Section 01
Interactive Simulation
Gravitation Simulator — SciSim
Ready
v_orb
—
m/s
g(r)
—
m/s²
T
—
s
v_esc
—
m/s
F_g
—
N
r
—
×10⁶m
Controls
Parameters
Central Mass M6.00×10²⁴kg
Orbital Radius r7.00×10⁶m
Section 02
The Idea, Step by Step

Let go of a ball and it drops. Toss it sideways and it still curves down to the ground. Now imagine throwing it so fast that, as it falls, the round Earth curves away beneath it just as quickly — it never lands. That is an orbit, and it is the same pull that drops the ball. The Moon is doing exactly this: forever falling toward Earth, forever missing it.

The deep idea is that every mass tugs on every other mass. Two things decide how hard the tug is: how much stuff each one has (its mass) and how far apart they are. More mass means a stronger pull; more distance means a weaker one — and distance matters a lot, because the pull fades with the square of the separation. Double the distance and the force drops to a quarter.

Putting numbers on it

Newton wrote this as $F = G\dfrac{m_1 m_2}{r^2}$, where $G$ is a tiny universal constant and $r$ is the centre-to-centre distance. The constant is small ($G\approx6.67\times10^{-11}$), which is why you feel no pull toward a friend standing next to you. Try the everyday case of a 1 kg book resting on Earth: with $M_\oplus=5.97\times10^{24}\,$kg and $r=R_\oplus=6.37\times10^6\,$m, the force comes out to about $9.8\,$N — that number is just the book's weight. Gravity only feels strong because Earth is enormous.

From force to orbits

For motion, it is cleaner to talk about the gravitational field $g=\dfrac{GM}{r^2}$, the pull per kilogram, which points straight toward the central mass. Setting gravity equal to the centripetal requirement of a circle, $\dfrac{GMm}{r^2}=\dfrac{mv^2}{r}$, the satellite mass cancels and you get the orbital speed $v_{\text{orb}}=\sqrt{GM/r}$. From there the period follows as $T=2\pi\sqrt{r^3/GM}$ — this is Kepler's third law, $T^2\propto r^3$. And the speed needed to break free entirely is $v_{\text{esc}}=\sqrt{2GM/r}=\sqrt{2}\,v_{\text{orb}}$. In the sim above, the Central Mass M slider sets $M$ and the Orbital Radius r slider sets $r$; every readout ($v_{\text{orb}}$, $g$, $T$, $v_{\text{esc}}$, $F_g$) updates from these two numbers alone.

Try this in the sim above

Push the orbital radius $r$ outward and watch $v_{\text{orb}}$ drop while the period $T$ grows steeply — that is $T^2\propto r^3$ in action. Then crank the central mass $M$ up and see every speed climb, since heavier centres pull harder. Finally compare $v_{\text{esc}}$ to $v_{\text{orb}}$ at any setting: escape always sits a factor of $\sqrt{2}\approx1.41$ above orbital speed, no matter where you put the satellite.

Section 03
Equations & Derivation
Newton's Law of Universal Gravitation
$$F_g = G\frac{m_1 m_2}{r^2}$$
Gravitational Field
$$g = \frac{GM}{r^2},\quad PE_g = -\frac{Gm_1 m_2}{r}$$
Orbital Mechanics
$$v_{\text{orb}}=\sqrt{\frac{GM}{r}},\quad T=2\pi\sqrt{\frac{r^3}{GM}},\quad v_{\text{esc}}=\sqrt{\frac{2GM}{r}}$$

Symbol Definitions

SymbolQuantitySI Unit
$G$Gravitational constant = 6.674×10⁻¹¹N m² kg⁻²
$M$Mass of central bodykg
$m$Mass of test bodykg
$r$Centre-to-centre distancem
$g$Gravitational field strengthN kg⁻¹
$T$Orbital periods
$v_{\text{esc}}$Escape velocitym s⁻¹
1
Newton's Law. Every mass attracts every other mass. Force $F_g=Gm_1m_2/r^2$ is always attractive, acts along the line joining centres, and obeys superposition.
2
Gravitational field. $\vec{g}=\vec{F}/m=-GM\hat{r}/r^2$. At Earth's surface: $g=GM_E/R_E^2\approx9.81$ m/s². Increases inside uniform sphere.
3
Orbital speed. Set $F_g=F_c$: $GMm/r^2=mv^2/r$, so $v_{\text{orb}}=\sqrt{GM/r}$. Faster at lower orbit.
4
Kepler's 3rd Law. $T=2\pi r/v_{\text{orb}}=2\pi\sqrt{r^3/GM}$, so $T^2\propto r^3$.
5
Escape velocity. Set $KE=|PE|$: $\tfrac{1}{2}mv^2=GMm/r$, giving $v_{\text{esc}}=\sqrt{2GM/r}=\sqrt{2}v_{\text{orb}}$.
Ref: Halliday, Resnick & Walker, 10th Ed., Ch. 13; Serway & Jewett, 8th Ed., Ch. 13; Kleppner & Kolenkow, 2nd Ed., Ch. 8.
Section 04
Frequently Asked Questions
Yes — gravity is by far the weakest of the four fundamental forces. The gravitational force between two protons is roughly $10^{36}$ times weaker than the electromagnetic force between them. Gravity dominates at astronomical scales only because it is always attractive (unlike EM) and acts over infinite range, so its effect accumulates for large masses.
Key takeaway: Gravity is the weakest fundamental force but dominates at cosmic scales because it is always attractive and has infinite range.
Above the surface: $g=GM/r^2$ decreases as $r$ increases. Inside a uniform sphere (shell theorem): only the mass within radius $r$ contributes, so $g=(4/3)\pi G\rho r$, which increases linearly with $r$ from zero at the centre to $g_s$ at the surface.
Key takeaway: Above surface: $g\propto 1/r^2$; inside uniform sphere: $g\propto r$.
GPS satellites require relativistic corrections to gravitational time dilation. Tidal forces (differential gravity) cause ocean tides and the gradual recession of the Moon. Gravitational lensing lets astronomers map dark matter. Spacecraft use gravitational slingshots to gain speed. Black holes, neutron stars, and the expansion of the universe are all gravitational phenomena.
Key takeaway: From GPS precision to black holes, gravity governs all large-scale structure in the universe.
It does fall — constantly. But it moves horizontally fast enough that the Earth's surface curves away beneath it at the same rate it falls. At ~7.9 km/s (LEO), the projectile falls ~5 m every second and the Earth curves away ~5 m per second. "Orbiting" is just free-fall with enough horizontal speed to miss the ground.
Key takeaway: An orbit is perpetual free-fall with enough horizontal velocity to keep missing the surface.
No — gravity at the ISS altitude (~400 km) is about 88% of surface gravity. You feel weightless because you and the spacecraft are both in free fall together — neither exerts a normal force on the other. True weightlessness (zero gravity) only exists infinitely far from all mass.
Key takeaway: Weightlessness in orbit is due to free fall, not zero gravity.
An orbital simulation with adjustable central mass $M$, orbital radius, and initial conditions. You can observe Kepler's 3rd Law: change the orbital radius and watch how the period changes. The escape velocity mode shows the minimum speed needed to escape the gravitational well.
Key takeaway: The simulation lets you explore orbital mechanics, Kepler's laws, and escape velocity interactively.
Resources: Khan Academy; HyperPhysics; MIT OCW.
Section 05
Common Misconceptions
❌ Gravity decreases linearly with height above Earth.
✅ Gravity decreases as $1/r^2$, not linearly. At altitude $h$: $g(h)=g_0/(1+h/R_E)^2$. At 400 km (ISS), $g\approx8.7$ m/s² — only ~11% less than at the surface. The linear approximation is valid only very near the surface.
📖 HRW 10th Ed., §13-1: Gravitation and the Principle of Superposition.
❌ Objects in orbit are beyond Earth's gravity.
✅ The ISS experiences about 88% of surface gravity. "Zero gravity" is a misnomer; astronauts feel weightless because they are in free fall. Gravity extends to infinity (it just weakens as $1/r^2$).
📖 HRW 10th Ed., §13-4: Gravitation Inside Earth.
❌ A heavier satellite requires more fuel to maintain orbit.
✅ For a given orbit radius, orbital speed $v=\sqrt{GM/r}$ depends only on the central mass and radius — not on the satellite mass. A heavier satellite requires more fuel only to change orbit (because more work must be done to accelerate it), not to maintain a circular orbit.
📖 HRW 10th Ed., §13-5: Gravitational Potential Energy.
❌ Kepler's Laws are independent of Newton's Laws.
✅ Kepler's three laws can be derived from Newton's Law of Gravitation and Newton's Laws of Motion. They are not independent empirical laws — they are consequences of the $1/r^2$ force law. This derivation was one of Newton's greatest achievements.
📖 HRW 10th Ed., §13-6: Planets and Satellites — Kepler's Laws.
Misconception research: Halloun & Hestenes (1985); Arons — A Guide to Introductory Physics Teaching, Ch. 8.