The Gravitational Field
g = F/m, field lines, and the inverse square law
Do Now: The Moon is 384,000 km away, with nothing in between. How can Earth pull on it without touching it?
Through its gravitational field. Every mass creates a field in the space around it. Any other mass in that field feels a force.
A. What Is a Field?
B. Field Strength
Gravitational field strength (N/kg)
g = force per kilogram on a test mass. It does not depend on the test mass.
At Earth’s surface, g = 9.81 N/kg. Since 1 N/kg = 1 m/s², that’s also the acceleration of free fall.
C. Where Does g Come From?
Start with Newton’s law of gravitation and divide by the test mass:
Fg = Gm1m2r2 → g = Fgm2 = Gm1r2
m = mass making the field, r = distance from its center, G = 6.67 × 10−11 N•m²/kg²
Earth’s mass is 5.97 × 1024 kg and its radius is 6.37 × 106 m. Find g at the surface.
g = (6.67 × 10−11)(5.97 × 1024)(6.37 × 106)2
g = 9.81 N/kg ✓
Mars has a mass of 6.42 × 1023 kg and a radius of 3.39 × 106 m. Find g on Mars, and the weight of a 60.-kg student there.
g = (6.67 × 10−11)(6.42 × 1023)(3.39 × 106)2
g = 3.73 N/kg
Fg = mg = (60. kg)(3.73 N/kg) = 224 N — about 38% of the student’s weight on Earth
D. The Inverse Square Law
The Space Station orbits about 400 km above Earth, at about 6,770 km from Earth’s center. Compare g there to the surface value.
Earth’s radius is 6,370 km.
g = 9.81 × (6,3706,770)2 = 8.7 N/kg — about 89% of surface gravity!
The astronauts float because they are in free fall, not because gravity is gone.
Explore: Field Explorer
Drag the test mass around the planet. Change the planet and the mass.
1. Change the test mass. Does g change? Does Fg change?
2. Drag the mass from 1R to 2R. What happens to g?
3. Compare g at the surface of each planet.
1. g stays the same; Fg changes in proportion to the mass. g belongs to the field, not the test mass. 2. g drops to one-fourth. 3. Moon 1.62, Mars 3.73, Earth 9.81, Jupiter about 24.8 N/kg.
E. Practice
A 50.-kg astronaut stands on the Moon, where g = 1.62 N/kg. Find her weight.
Fg = (50. kg)(1.62 N/kg) = 81 N
What is g at 3 Earth radii from Earth’s center?
9.81 ÷ 32 = 1.09 N/kg
A 2.0-kg mass feels a 7.4-N gravitational force on a planet. Find g.
g = 7.4 / 2.0 = 3.7 N/kg (about the same as Mars)
A planet has twice Earth’s mass and twice Earth’s radius. Compare its surface g to Earth’s.
×2 for mass, ÷22 = ÷4 for radius → half of Earth’s: about 4.9 N/kg
Check Your Understanding
5 quick questions — pick an answer to see if you’re right.
1. Gravitational field strength is measured in
2. Doubling a test mass at the same location makes g
3. If you move to 2 Earth radii from Earth’s center, g becomes
4. Field lines around a planet point
5. Astronauts on the Space Station feel weightless because
Mapping Earth’s Gravity From Orbit
Satellite missions such as GRACE map Earth’s gravitational field. At Earth’s surface (one Earth radius, R, from the center), the field strength is about 9.8 N/kg.
What is the gravitational field strength at an altitude of one Earth radius above the surface (2R from the center)?
▶ Want a challenge? Open 6 more Regents-style questions (harder · mixed formats)
Challenge score: 0 / 6
C1. Which graph best represents gravitational field strength versus distance from Earth’s center (outside Earth)?
C2. Substitute numerical values into a mathematical representation that can be used to determine the gravitational field strength at a distance of 3R from Earth’s center. Then calculate the field strength.
C3. A student makes the following claim.
Place a check mark in either the Support or Refute box. Justify your response using a mathematical relationship.
C4. The International Space Station orbits about 6.77 × 106 m from Earth’s center, and Earth’s radius is 6.37 × 106 m. Determine the gravitational field strength at the station.
C5. GRACE detected that the field is slightly stronger above a dense mountain range. Which explanation is best?
C6. At the space station, Earth’s field is still about 8.7 N/kg, yet astronauts appear “weightless.” Explain this observation.