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Gravitation: Laws, Formulas, Mass, Weight and Orbits
Learn how every mass attracts every other mass, why objects fall toward Earth, how mass differs from weight, and how gravity keeps planets and satellites in orbit.
What Is Gravitation?
Gravitation is the universal force of attraction between any two objects that have mass. The force acts along the line joining the centres of the two objects and is always attractive.
Gravity is not limited to Earth. Earth attracts you, but you also attract Earth. The force exerted by you on Earth has the same magnitude as the force exerted by Earth on you, although Earth’s much larger mass gives it a very small acceleration.
Newton’s Universal Law of Gravitation
Newton’s law states that the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.
F = Gm1m2 / r2
F = gravitational force, G = universal gravitational constant, m1 and m2 = masses, and r = distance between their centres.
Effect of mass
If either mass increases, the gravitational force increases. If one mass doubles while the other values stay constant, the force doubles.
Effect of distance
Gravity follows an inverse-square relationship. If the distance doubles, the force becomes one-quarter as large because 22 = 4.
The gravitational constant, G
The value of the universal gravitational constant is approximately:
Its very small value shows that the gravitational attraction between ordinary-sized objects is usually weak.
In calculations, use mass in kilograms (kg), distance in metres (m), and force in newtons (N).
Acceleration Due to Gravity
When an object falls near Earth’s surface, Earth’s gravitational force gives it an acceleration called the acceleration due to gravity, represented by g. Near Earth’s surface, its approximate value is 9.8 m/s2. The standard value used for precise reference is 9.80665 m/s2.
M = mass of the planet or other attracting body, and R = distance from its centre. At the surface of a nearly spherical planet, R is approximately the planet’s radius.
This equation shows that g increases when the planet’s mass increases and decreases when the distance from the planet’s centre increases.
Mass and Weight
Mass
Mass is the amount of matter in an object. It is measured in kilograms and remains essentially the same when the object moves from Earth to the Moon.
Weight
Weight is the gravitational force acting on an object. It is measured in newtons and changes when the local value of g changes.
W = weight in newtons, m = mass in kilograms, and g = local acceleration due to gravity in m/s2.
Find the weight of a 60 kg student on Earth
Given: m = 60 kg and g = 9.8 m/s2
Formula: W = mg
Substitution: W = 60 × 9.8
Answer: W = 588 N
The student’s mass is 60 kg, while the student’s weight on Earth is approximately 588 N.
Gravitational Force Near a Planet
For an object of mass m near a spherical planet of mass M and radius R, Newton’s law gives:
Since g = GM/R2, this becomes F = mg. Near the surface, this force is the object’s weight.
Free Fall and Gravitational Acceleration
Free fall is motion in which gravity is the only significant force acting on an object. Near Earth’s surface, and when air resistance is ignored, all objects have approximately the same downward acceleration, g, regardless of their masses.
In real life, air resistance can make a feather fall more slowly than a stone. This difference is caused by the air, not by different gravitational accelerations.
- Gravity pulls objects toward the centre of the attracting body.
- In ideal free fall, mass does not change the value of the acceleration due to gravity.
- Air resistance may change the motion we observe.
Gravity and Circular Orbits
A satellite does not need a continuous engine thrust to remain in orbit. Its forward motion and Earth’s gravitational pull combine to produce a curved path. Gravity acts as the centripetal force that continually bends the satellite’s path toward Earth.
Orbital speed
For a circular orbit around a body of mass M, the ideal orbital speed is:
v = orbital speed, r = distance from the centre of the attracting body, and G and M have their usual meanings.
At a larger orbital radius, the required circular-orbit speed is lower, although the satellite takes longer to complete one orbit.
Escape speed
The minimum ideal speed needed to escape the gravitational influence of a spherical body, ignoring air resistance and rotation, is:
ve = escape speed and R = distance from the centre at the starting point.
Gravitational Potential Energy
Gravitational potential energy is energy associated with an object’s position in a gravitational field. Close to Earth’s surface, the change in gravitational potential energy is often written as:
ΔU = change in gravitational potential energy, Δh = change in height, and g is treated as approximately constant.
For objects at large distances from a planet, where g changes noticeably, the gravitational potential energy relative to infinity is:
The negative sign indicates that the object is in a bound gravitational system when the reference level at infinity is chosen as zero.
Important Symbols and Units
| Symbol | Meaning | SI unit |
|---|---|---|
| F | Gravitational force | newton (N) |
| G | Universal gravitational constant | N·m2/kg2 |
| m, m1, m2 | Masses | kilogram (kg) |
| r | Distance between centres, or orbital radius | metre (m) |
| g | Acceleration due to gravity | m/s2 |
| W | Weight | newton (N) |
| v | Speed, including orbital speed | m/s |
Common Mistakes and Misconceptions
- Confusing mass and weight: mass is measured in kilograms; weight is a force measured in newtons.
- Using distance from the surface instead of the centre: in F = Gm1m2/r2, r is the distance between the centres of mass.
- Forgetting to square r: the inverse-square relationship is r2, not r.
- Thinking heavier objects fall faster in a vacuum: without air resistance, objects near Earth have the same gravitational acceleration.
- Thinking gravity exists only on Earth: every object with mass attracts every other object with mass.
- Assuming astronauts have no gravity in orbit: astronauts and spacecraft are still affected by gravity; they are continuously falling around Earth.
How to Solve Gravitation Problems
- Write down the given quantities and identify what must be found.
- Choose the correct formula, such as F = Gm1m2/r2 or W = mg.
- Convert all measurements to SI units: kg, m and s.
- Substitute the values carefully, using brackets for powers and scientific notation.
- Check the unit and decide whether the answer is physically reasonable.
Quick Recap
- Gravitation is the attractive force between masses.
- Newton’s law is F = Gm1m2/r2.
- Gravity becomes stronger when mass increases and weaker when distance increases.
- Near Earth, the acceleration due to gravity is approximately 9.8 m/s2.
- Weight is calculated using W = mg, while mass is measured in kilograms.
- Gravity keeps planets, moons and artificial satellites in orbit.
- Always use the distance between centres and keep units consistent.