Gravitational Fields and Orbits

The video lesson for this topic is coming soon.

Essential Question

Why doesn’t the Moon fall down to Earth?

Earth and the Moon pull on each other with an enormous force, so why doesn’t the Moon crash into Earth? By the end of this lesson, you should be able to describe a gravitational field, calculate field strength with g = F/m, and explain how gravity keeps satellites and moons in orbit.

Gravitational Field

Mass creates a field around it
Field lines point toward the mass
Closer lines = stronger field

Earth drawn as a circle with twelve field lines around it. Every arrow points inward, toward Earth's center. Caption: field lines point toward the mass.

Just as a charge creates an electric field, a mass creates a gravitational field in the space around it. Any other mass placed in that field feels a pull. Gravitational field lines always point toward the mass, because gravity only attracts. The lines are crowded near the planet, where the field is strong, and spread out farther away, where it is weaker.

Field Strength

g = F / m

g = gravitational field strength, in newtons per kilogram (N/kg)
F = gravitational force on the object, in newtons (N)
m = mass of the object, in kilograms (kg)

The equation g = F / m in a yellow box. Below it, three cards: g is the gravitational field strength in newtons per kilogram, F is the gravitational force in newtons, and m is the mass in kilograms. Caption: at Earth's surface, g equals 9.8 N/kg, which is the same as 9.8 m/s squared.

Gravitational field strength is the force of gravity on each kilogram of mass. At Earth’s surface, every kilogram is pulled with 9.8 N, so g = 9.8 N/kg. A newton per kilogram is the same unit as a meter per second squared, which is why the field strength is also the acceleration of a falling object. This is the same idea as E = F/q, with mass in place of charge.

Weaker with Distance

g = GM/d²
M: mass of the planet
Double d: ¼ the field

The equation g = G M over d squared in a yellow box. Below it, three cards: at Earth's surface g is 9.8 N/kg, at two Earth radii from the center g is 2.5 N/kg, and at the Moon's distance, 60 Earth radii, g is 0.0027 N/kg.

Combining g = F/m with Newton’s law of gravitation gives g = GM/d², where M is the mass of the planet and d is the distance from its center. The field follows an inverse square law. At twice Earth’s radius from the center, the field is one fourth of 9.8 N/kg, about 2.5 N/kg. At the Moon’s distance, about 60 Earth radii away, it is only 0.0027 N/kg, but that small pull is still enough to hold the Moon in orbit.

Gravity Bends the Path

Gravity pulls toward the center
Velocity points sideways
Gravity is the centripetal force

A satellite on a dashed circular orbit around Earth. At the top of the orbit, a green velocity arrow v points sideways to the right, along the orbit, and a blue force arrow F sub g points down toward Earth's center.

A satellite moves sideways while gravity pulls it toward Earth’s center. The pull is always at a right angle to the motion, so it doesn’t speed the satellite up or slow it down. It only turns its direction, bending the path into a circle. In other words, gravity provides the centripetal force for every orbit, the same way the tension in a string keeps a ball moving in a circle.

Always Falling

Orbiting objects are always falling
They move sideways fast enough
Earth curves away below them

Earth with a tall mountain on top. Three paths start at the mountain peak, heading right. The slowest launch curves down and lands nearby, a faster launch lands farther around Earth, and the fastest goes all the way around in a circle, labeled orbit.

Imagine launching a ball sideways from a very tall mountain. A slow launch curves down and lands nearby. A faster launch travels farther before it lands. Because Earth is round, its surface curves away under the ball. If the launch is fast enough, about 7.9 km/s near Earth’s surface, the ball falls toward Earth just as fast as the ground curves away, so it never lands. That’s an orbit. Satellites and the Moon are always falling, but they keep missing. Astronauts float because they are falling right along with their spacecraft, not because there is no gravity.

Orbital Speed

Set F_g equal to F_c
Satellite’s mass cancels out
Closer orbit: faster speed

A three-step derivation. Gravity equals centripetal force. G M m over d squared equals m v squared over d. Down arrow to the result in a yellow box: v equals the square root of G M over d.

Setting the gravitational force equal to the centripetal force, GMm/d² = mv²/d, and solving for v gives the speed needed for a circular orbit: v = √(GM/d). The satellite’s own mass m cancels, so a tiny satellite and a huge space station at the same height move at the same speed. A smaller orbit radius means a faster speed.

Worked Examples

Try each problem on your own first, then check your work against the solution.

Example 1: Field Strength on Mars

A rover on Mars picks up a 2.5 kg rock. Mars pulls on the rock with a force of 9.3 N. What is the gravitational field strength on Mars?

A gray rock labeled 2.5 kg resting on red Martian ground, with a blue arrow pointing down labeled 9.3 N.

Knowns: m = 2.5 kg, F = 9.3 N
Equation: g = F/m
Substitute: g = 9.3 N / 2.5 kg
Answer: g = 3.7 N/kg, toward the center of Mars

That’s about 38% of Earth’s 9.8 N/kg, so you would weigh a little more than a third of your Earth weight on Mars.

Example 2: Gravity at the Space Station

The space station orbits 4.2 × 10⁵ m above Earth’s surface. Earth’s mass is 5.97 × 10²⁴ kg and its radius is 6.37 × 10⁶ m. What is g at the station?

Earth with a satellite above it. The radius R is drawn from Earth's center to the surface and the height h from the surface to the satellite, with d equals R plus h.

Knowns: h = 4.2 × 10⁵ m, M = 5.97 × 10²⁴ kg, R = 6.37 × 10⁶ m
Distance: d = R + h = 6.37 × 10⁶ m + 0.42 × 10⁶ m = 6.79 × 10⁶ m
Equation: g = GM/d²
Top: GM = (6.67 × 10⁻¹¹)(5.97 × 10²⁴) = 3.98 × 10¹⁴
Bottom: d² = (6.79 × 10⁶ m)² = 4.61 × 10¹³ m²
Answer: g = 3.98 × 10¹⁴ ÷ 4.61 × 10¹³ = 8.64 N/kg

That’s about 88% of the field at Earth’s surface. Astronauts on the station feel weightless because they are in free fall, not because gravity is missing.

Example 3: Orbital Speed

The space station orbits 6.79 × 10⁶ m from Earth’s center. Earth’s mass is 5.97 × 10²⁴ kg. How fast must the station move to stay in orbit?

The space station on a dashed circular orbit around Earth, with a red radius line from Earth's center to the orbit labeled d, and a velocity arrow labeled v equals question mark.

Knowns: d = 6.79 × 10⁶ m, M = 5.97 × 10²⁴ kg, G = 6.67 × 10⁻¹¹ N·m²/kg²
Equation: GMm/d² = mv²/d, so v = √(GM/d)
Substitute: v = √(3.98 × 10¹⁴ ÷ 6.79 × 10⁶)
Divide: 3.98 × 10¹⁴ ÷ 6.79 × 10⁶ = 5.86 × 10⁷ m²/s²
Answer: v = √(5.86 × 10⁷ m²/s²) = 7.66 × 10³ m/s

That’s about 7.7 km every second, fast enough to circle Earth in about an hour and a half.

Big Ideas

Mass makes a field; lines point toward it
g = F/m (N/kg), weaker with distance
Gravity is the centripetal force for orbits
Orbiting objects are always falling

Back to the essential question: the Moon is falling toward Earth all the time, but it also moves sideways so fast that it keeps missing. Earth’s gravitational field, which points toward Earth and weakens with distance, supplies the centripetal force that bends the Moon’s path into an orbit. Next up, Comparing Gravity and Electric Forces puts gravity side by side with the electric force to see how they are alike and how they differ.