Gravitational Fields
Explore Newton's law of universal gravitation, gravitational field strength, and how gravitational potential energy works on a planetary scale.
Newton's Law of Universal Gravitation
Every object with mass attracts every other object with mass. Newton quantified this with his law of universal gravitation: the gravitational force between two masses is proportional to the product of their masses and inversely proportional to the square of the distance between their centres.
The Gravitational Force Equation
F = GMm / r2
G
6.67 × 10-11 N m2 kg-2
M, m
Masses of two objects (kg)
r
Distance between centres of mass (m)
Inverse-square law: If the distance between two masses doubles, the gravitational force becomes one-quarter as strong. If the distance triples, the force becomes one-ninth. This relationship is crucial for understanding how gravity weakens with distance.
Gravitational Field Strength
A gravitational field exists around any mass. The gravitational field strength (g) at a point in space is defined as the gravitational force per unit mass placed at that point. It tells us how strong gravity is at any location.
Field Strength Formula
g = F/m = GM/r2
Units: N kg-1 (equivalent to m s-2)
At Earth's surface: g ≈ 9.8 N kg-1
Field Direction
Gravitational field lines point radially inward toward the centre of the mass.
Field lines are closer together where the field is stronger (near the surface) and farther apart at greater distances.
Gravitational Field Around a Planet
Field lines point radially inward toward the mass. The field is a radial field that weakens with distance (1/r2).
Gravitational Potential Energy
For objects near Earth's surface, we use U = mgh. But for objects at planetary distances, we need the more general expression. The gravitational potential energy of a mass m at distance r from a mass M is always negative, reflecting that work must be done to move the mass away from M.
Gravitational Potential Energy
U = -GMm / r
The negative sign means U = 0 at infinite distance, and energy decreases (becomes more negative) as objects get closer. This is the bound state of gravitational attraction.
Escape Velocity
The minimum speed needed for an object to escape a planet's gravitational field (reach infinity with zero kinetic energy) can be found using energy conservation:
½mv2 - GMm/r = 0
vescape = √(2GM/r)
For Earth's surface: vescape ≈ 11.2 km s-1
Key Vocabulary
Gravitational Field
The region of space around a mass in which another mass experiences a gravitational force. It is a vector field pointing toward the source mass.
Field Strength (g)
The force per unit mass at a point in a gravitational field, measured in N kg-1. It equals the acceleration due to gravity at that point.
Inverse-Square Law
The relationship where gravitational force and field strength decrease in proportion to 1/r2 as distance r from the source mass increases.
Escape Velocity
The minimum launch speed needed for an object to escape the gravitational pull of a planet or star without further propulsion.
Worked Examples
Calculate the gravitational force between Earth (5.97 × 1024 kg) and the Moon (7.35 × 1022 kg) separated by 3.84 × 108 m.
Step 1: F = GMm/r2
Step 2: F = (6.67 × 10-11)(5.97 × 1024)(7.35 × 1022) / (3.84 × 108)2
Step 3: F = 2.926 × 1037 / 1.475 × 1017
Answer: F ≈ 1.98 × 1020 N
Find the gravitational field strength at an altitude of 200 km above Earth's surface. (REarth = 6.37 × 106 m)
Step 1: r = REarth + altitude = 6.37 × 106 + 2.00 × 105 = 6.57 × 106 m
Step 2: g = GM/r2 = (6.67 × 10-11 × 5.97 × 1024) / (6.57 × 106)2
Step 3: g = 3.982 × 1014 / 4.316 × 1013
Answer: g ≈ 9.2 N kg-1 (still quite strong -- gravity does not disappear in low orbit)
Calculate the escape velocity from the surface of Mars. (MMars = 6.42 × 1023 kg, RMars = 3.39 × 106 m)
Step 1: vescape = √(2GM/R)
Step 2: vescape = √(2 × 6.67 × 10-11 × 6.42 × 1023 / 3.39 × 106)
Step 3: vescape = √(2.524 × 107)
Answer: vescape ≈ 5020 m s-1 ≈ 5.0 km s-1 (less than half of Earth's escape velocity)
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
If the distance between two masses is tripled, the gravitational force between them becomes:
Question 2
The gravitational field strength g at a point in space depends on:
Question 3
Gravitational potential energy U = -GMm/r is negative. This means:
Question 4
The value of G in Newton's law of gravitation is:
Question 5
A planet has twice the mass of Earth and twice the radius. Its surface gravitational field strength compared to Earth's is:
Key Concepts Summary
- ●Newton's law of gravitation: F = GMm/r2 -- force is proportional to both masses and inversely proportional to the square of the distance.
- ●Gravitational field strength: g = GM/r2 is the force per unit mass at a point. It does not depend on the test mass.
- ●Field lines are radial and point inward, showing that gravity is always attractive.
- ●Gravitational PE: U = -GMm/r is negative, indicating a bound system. Energy must be added to escape the field.
- ●Escape velocity: vescape = √(2GM/r) is independent of the escaping object's mass.