Newton’s law of universal gravitation states that every object in the universe attracts every other object with a gravitational force. The strength of that force depends on the masses of the two objects and the distance between their centers. Objects with greater mass exert stronger gravitational forces, while objects farther apart attract each other less strongly.
First published by English physicist Isaac Newton in 1687, this law provided a mathematical explanation for phenomena ranging from objects falling toward Earth to the motion of the Moon and planets. It established that the same fundamental force that pulls an apple toward the ground also helps keep planets in orbit around the Sun.
What is Newton’s law of universal gravitation?
Newton’s law of universal gravitation describes gravity as a universal attraction between objects that have mass. Unlike forces that require direct contact, such as friction or a push, gravity acts between objects separated by empty space.
Earth attracts a person standing on its surface, but the person also attracts Earth. The forces are equal in strength and opposite in direction, although Earth’s enormous mass means its resulting acceleration is far smaller than the person’s.
The law applies to all objects with mass, including planets, stars, moons, asteroids, and ordinary objects on Earth. However, the gravitational force between small everyday objects is usually so weak that it is difficult to notice without sensitive equipment.
Newton’s central insight was that gravity follows the same basic rule throughout the universe. The motion of falling objects on Earth and the motion of celestial bodies could be understood through one mathematical relationship rather than separate explanations.
Newton’s law of universal gravitation formula
The mathematical formula for Newton’s law of universal gravitation is:
F = G(m₁m₂ ÷ r²)
The equation can also be written as:
F = Gm₁m₂/r²
Each symbol represents a specific quantity:
- F is the gravitational force between the two objects, measured in newtons (N).
- G is the universal gravitational constant, approximately 6.674 × 10⁻¹¹ N·m²/kg².
- m₁ is the mass of the first object, measured in kilograms (kg).
- m₂ is the mass of the second object, measured in kilograms (kg).
- r is the distance between the centers of the two objects, measured in meters (m).
The formula reveals two essential relationships. First, gravitational force is directly proportional to the product of the two masses. Second, it is inversely proportional to the square of the distance between their centers.
The universal gravitational constant, G, determines the strength of gravity in this relationship. Its small numerical value helps explain why gravitational attraction between ordinary objects is so difficult to detect.
The equation produces the magnitude of the attractive force. The force acts along the line connecting the centers of the two objects, with each object pulled toward the other.
How mass affects gravitational force
Mass measures the amount of matter in an object and determines how strongly it participates in gravitational attraction under Newton’s law.
If the mass of one object doubles while the other mass and the distance remain unchanged, the gravitational force doubles. If both masses double, the force becomes four times as strong because the product of the masses increases by a factor of four.
For example, imagine two objects attracting each other with a gravitational force of 10 newtons. If one object’s mass doubles without changing anything else, the force becomes 20 newtons. If both objects’ masses double, the force becomes 40 newtons.
This relationship helps explain why planets and stars have significant gravitational influence. Their enormous masses produce stronger gravitational attractions than those generated by small objects.
Mass should not be confused with weight. Mass remains essentially the same regardless of location, while weight is the gravitational force acting on an object. A person has nearly the same mass on Earth and the Moon, but weighs less on the Moon because lunar gravity is weaker.
How distance affects gravitational force
Distance has a particularly important effect on gravity because the force follows an inverse-square relationship. This means gravitational force decreases in proportion to the square of the distance between two objects.
If the distance between their centers doubles, the force becomes one-fourth as strong. If the distance triples, the force becomes one-ninth as strong. If the distance increases by a factor of four, the force falls to one-sixteenth of its original strength.
For example, suppose two objects exert a gravitational force of 36 newtons at a particular distance. If they are moved twice as far apart, their gravitational force decreases to 9 newtons, assuming their masses remain unchanged.
The inverse-square relationship is especially important in astronomy. A planet experiences a stronger gravitational attraction from the Sun when it is closer to the Sun and a weaker attraction when it is farther away.
Distance in the formula means the separation between the objects’ centers of mass, not the gap between their surfaces. For spherical objects that do not overlap, this distinction makes the calculation straightforward.
Examples of Newton’s law of universal gravitation
The law becomes easier to understand when applied to familiar situations. Its formula can calculate the attraction between objects on Earth, estimate the gravitational pull between astronomical bodies, and explain why gravity changes with distance.
Example 1: Calculating the gravitational force between two objects
Imagine two objects, each with a mass of 10 kilograms, separated by 2 meters between their centers. To calculate the gravitational force, use:
F = Gm₁m₂/r²
Substitute the known values:
F = (6.674 × 10⁻¹¹ × 10 × 10) ÷ 2²
F = (6.674 × 10⁻¹¹ × 100) ÷ 4
F ≈ 1.67 × 10⁻⁹ N
The gravitational force is approximately 0.00000000167 newtons.
This force is extremely small. It illustrates why the gravitational attraction between ordinary objects is usually overwhelmed by other forces, such as friction or the support provided by a table or floor.
Example 2: Calculating the gravitational force between Earth and a person
Consider a person with a mass of 70 kilograms standing near Earth’s surface. Earth’s mass is approximately 5.97 × 10²⁴ kilograms, and Earth’s mean radius is about 6.37 × 10⁶ meters.
Using the law of universal gravitation:
F = Gm₁m₂/r²
F = (6.674 × 10⁻¹¹ × 5.97 × 10²⁴ × 70) ÷ (6.37 × 10⁶)²
F ≈ 686 N
Earth therefore exerts a gravitational force of approximately 686 newtons on the person.
Near Earth’s surface, this force corresponds to a weight of about 686 newtons. Dividing the force by the person’s mass gives an acceleration of approximately 9.8 meters per second squared, the familiar acceleration due to Earth’s gravity.
The calculation assumes Earth is approximately spherical and uses its mean radius. Local variations in gravity and the effects of Earth’s rotation are ignored.
Example 3: Understanding why gravity weakens with distance
Suppose a satellite is at a distance of 7,000 kilometers from Earth’s center. If it moves to a distance of 14,000 kilometers from Earth’s center, its distance from the center doubles.
Because gravitational force follows an inverse-square relationship, the force at the greater distance is:
F₂ = F₁/2²
F₂ = F₁/4
The gravitational force becomes one-fourth as strong.
This example also shows why altitude must be measured from Earth’s center when applying the formula. Doubling a satellite’s altitude above Earth’s surface does not necessarily double its distance from Earth’s center.
Example 4: Explaining the Moon’s orbit around Earth
Earth and the Moon attract each other through gravity. Earth’s gravitational pull continually changes the direction of the Moon’s motion, keeping it in orbit rather than allowing it to travel in a straight line through space.
The Moon also attracts Earth with an equal and opposite gravitational force. Because Earth has much greater mass, its acceleration in response to the Moon is much smaller.
The Moon’s orbit is not perfectly circular, so the distance between the two bodies changes over time. As their separation changes, the strength of their gravitational attraction changes according to Newton’s law.
Gravity does not need to be stronger than the Moon’s motion to maintain the orbit. Instead, it continually redirects the Moon’s velocity. The Moon is effectively falling toward Earth while moving sideways fast enough to keep missing the planet.
How Newton’s law explains planetary motion
Newton’s law of universal gravitation helped explain why planets follow regular paths around the Sun. The Sun contains most of the solar system’s mass, so its gravitational attraction strongly influences the motion of the planets.
A planet moving through space would travel in a straight line at constant velocity if no net force acted on it. The Sun’s gravitational force continually changes the planet’s direction, producing an orbit.
The shape and size of an orbit depend on several factors, including the planet’s speed, direction of motion, distance from the Sun, and the gravitational influence of other bodies. Under an idealized two-body model, Newton’s laws predict circular or elliptical orbits, along with other possible trajectories.
Newton’s framework also explained Kepler’s laws of planetary motion, which describe patterns in the shapes, speeds, and periods of planetary orbits. In particular, it showed that a planet’s orbital period is related to its average distance from the Sun.
The same principles apply to moons orbiting planets, artificial satellites orbiting Earth, and spacecraft traveling between celestial bodies. In practical astronomy and spaceflight, calculations often include the gravitational effects of several objects rather than treating every system as an isolated pair.
Why objects fall toward Earth
When an object is released near Earth’s surface, Earth’s gravitational attraction accelerates it downward. If air resistance is negligible, objects near the surface fall with approximately the same acceleration regardless of their mass.
This may seem surprising because Newton’s law predicts a stronger gravitational force on a more massive object. However, Newton’s second law states that acceleration equals net force divided by mass:
a = F/m
A more massive object experiences a proportionally greater gravitational force, but it also has proportionally greater resistance to changes in motion, called inertia. These effects cancel in the calculation of gravitational acceleration near Earth.
For an object near Earth’s surface, the gravitational acceleration can be approximated by:
g = GM/r²
Here, M is Earth’s mass and r is the distance from Earth’s center. Near the surface, g is approximately 9.8 m/s².
In everyday conditions, air resistance can cause objects of different shapes and masses to fall at different rates. A feather, for instance, falls more slowly than a stone in air because drag has a much greater effect on the feather’s motion. In a vacuum, where air resistance is absent, both would fall with the same gravitational acceleration at the same location.
What Newton’s law does not explain completely
Newton’s law of universal gravitation is highly effective for describing many everyday gravitational effects and predicting the motion of planets, moons, and satellites. However, it is not the most complete description of gravity known to modern physics.
In the early 20th century, Albert Einstein developed the general theory of relativity, which describes gravity in terms of the curvature of spacetime caused by mass and energy. Spacetime combines the three dimensions of space with time into a single framework.
According to general relativity, objects move along paths shaped by spacetime geometry. What Newton describes as a gravitational force can, in this broader theory, be understood through the way mass and energy influence spacetime.
Newton’s law remains an excellent approximation when gravitational fields are relatively weak, speeds are much lower than the speed of light, and extremely high precision is not required. It is widely useful in engineering, astronomy, and introductory physics because it provides accurate results for many practical calculations with relatively simple mathematics.
General relativity becomes necessary for certain situations, including precise descriptions of Mercury’s orbital precession, the behavior of matter near black holes, and the timing corrections required for satellite navigation systems.
This does not make Newton’s law incorrect for ordinary applications. Rather, it shows that Newton’s formulation is a powerful approximation within a broader physical theory.
Why Newton’s law of universal gravitation matters
Newton’s law of universal gravitation provides a common mathematical explanation for the attraction between objects throughout the universe. By connecting gravitational force to mass and distance, it explains why objects fall toward Earth, why moons orbit planets, and why planets travel around stars.
Its formula also makes quantitative predictions. Given the masses of two objects and the distance between their centers, the law allows scientists to calculate the magnitude of their gravitational attraction. Those calculations underpin many applications in physics, astronomy, and space exploration.
The central principle is straightforward: every object with mass attracts every other object with mass. Greater masses produce stronger attraction, while greater separation weakens it according to the inverse square of the distance. Together, these relationships reveal how a force that is almost imperceptible between small objects can govern the motion of planets, stars, and entire planetary systems.