Gravitational force is the attractive force that exists between any two objects with mass. Newton’s law of universal gravitation provides a mathematical way to calculate this force using the masses of the objects and the distance between their centers. The law explains why objects fall toward Earth, why the Moon orbits our planet, and why planets move around the Sun.
To calculate gravitational force, use Newton’s universal law of gravitation:
F = G(m₁ × m₂) / r²
In this equation:
- 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 equation shows that gravitational force increases when either object’s mass increases and decreases as the distance between the objects grows. Because distance is squared in the denominator, doubling the distance reduces the gravitational force to one-fourth of its original value.
Understanding Newton’s law of universal gravitation
In the late 17th century, English physicist Isaac Newton developed the law of universal gravitation to describe the attraction between objects throughout the universe.
The central idea is that every object with mass attracts every other object with mass. This attraction does not depend on whether the objects are on Earth, in space, or part of a distant planetary system.
Newton’s law connected phenomena that might initially seem unrelated. The force that makes an apple fall toward the ground is the same fundamental gravitational interaction that keeps the Moon in orbit around Earth and holds planets in their paths around the Sun.
The law is called universal because it applies to all objects with mass. However, the strength of gravitational attraction varies depending on the objects’ masses and the distance separating them.
For everyday objects, gravity is usually too weak to notice between individual items. Earth’s enormous mass, however, produces a gravitational attraction strong enough to keep people, buildings, oceans, and the atmosphere near its surface.
How to calculate gravitational force step by step
Calculating gravitational force involves identifying the two masses, determining the distance between their centers, substituting those values into Newton’s equation, and solving.
Consider two objects with the following properties:
- First object: 5 kg
- Second object: 10 kg
- Distance between their centers: 2 meters
Start with Newton’s universal law of gravitation:
F = G(m₁ × m₂) / r²
Substitute the known values:
F = (6.674 × 10⁻¹¹ × 5 × 10) / 2²
First, multiply the two masses:
5 × 10 = 50 kg²
Next, square the distance:
2² = 4 m²
Substitute these results:
F = (6.674 × 10⁻¹¹ × 50) / 4
Multiply the gravitational constant by the product of the masses:
6.674 × 10⁻¹¹ × 50 = 3.337 × 10⁻⁹
Then divide by 4:
F = 8.3425 × 10⁻¹⁰ N
The gravitational force between the two objects is approximately 8.34 × 10⁻¹⁰ newtons.
This extremely small value illustrates why the gravitational attraction between ordinary objects is difficult to detect without sensitive instruments.
Why mass and distance affect gravitational force
Newton’s equation reveals two important relationships: gravitational force is directly proportional to the product of the masses and inversely proportional to the square of the distance between their centers.
Understanding these relationships makes it easier to predict how gravitational force changes when an object’s mass or position changes.
The effect of mass
Gravitational force increases in proportion to the product of the two masses.
If one object’s mass doubles while the other object’s 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.
This relationship explains why massive astronomical objects, such as stars and planets, produce much stronger gravitational effects than small objects.
The effect of distance
Distance has a squared effect on gravitational force. This means that increasing the separation between two objects causes their gravitational attraction to decrease rapidly.
For example:
- If the distance doubles, gravitational force becomes one-fourth as strong.
- If the distance triples, gravitational force becomes one-ninth as strong.
- If the distance increases by four times, gravitational force becomes one-sixteenth as strong.
This relationship is known as the inverse-square law.
It helps explain why Earth’s gravitational influence weakens with increasing distance from its center. Satellites in higher orbits experience less gravitational acceleration than objects near Earth’s surface, although Earth’s gravity remains significant far beyond the atmosphere.
Calculating Earth’s gravitational force on an object
One of the most familiar applications of Newton’s law is calculating the gravitational force Earth exerts on an object.
For an object near Earth’s surface, its weight is the gravitational force acting on it. Newton’s universal gravitation equation can calculate this force using Earth’s mass and the distance between the object’s center and Earth’s center.
The equation is:
F = G(M × m) / r²
Here, M represents Earth’s mass, m represents the object’s mass, and r represents the distance from Earth’s center to the object’s center.
Earth’s mass is approximately 5.97 × 10²⁴ kilograms, and its average radius is approximately 6.37 × 10⁶ meters.
Consider an object with a mass of 10 kilograms resting near Earth’s surface.
Substitute the values:
F = (6.674 × 10⁻¹¹ × 5.97 × 10²⁴ × 10) / (6.37 × 10⁶)²
Solving the equation gives a gravitational force of approximately:
F = 98 N
Therefore, Earth pulls the 10-kilogram object downward with a force of approximately 98 newtons.
This result is consistent with the familiar weight equation:
W = mg
In this equation, W is weight, m is mass, and g is the acceleration due to gravity. Near Earth’s surface, g is approximately 9.8 meters per second squared.
For the example:
W = 10 × 9.8 = 98 N
The two equations describe the same gravitational force in this situation. The universal gravitation equation explains the force using Earth’s mass and the distance from its center, while the weight equation provides a simpler method for calculations near Earth’s surface.
The difference between mass, weight, and gravitational force
Mass, weight, and gravitational force are closely related concepts, but they do not mean the same thing.
Mass measures the amount of matter in an object and is expressed in kilograms. An object’s mass does not change simply because it moves from one planet to another.
Weight is the gravitational force acting on an object. It is measured in newtons and depends on the strength of the local gravitational field.
Gravitational force is the mutual attraction between any two objects with mass. Weight is a particular example of gravitational force: the attraction between an object and the astronomical body influencing it.
For example, a person with a mass of 70 kilograms has approximately the same mass on Earth and the Moon. However, that person’s weight is much smaller on the Moon because lunar surface gravity is approximately one-sixth of Earth’s surface gravity.
This distinction is important when using Newton’s law because the equation calculates force, not mass or weight directly.
Why gravitational force acts toward the center of an object
For spherical objects such as Earth, the Moon, and the Sun, Newton’s law allows the gravitational attraction to be treated as though the object’s entire mass were concentrated at its center, provided the object has a spherically symmetric mass distribution and the calculation concerns points outside it.
This is why calculations involving planets generally use the distance between their centers rather than the distance between their surfaces.
For example, when calculating Earth’s gravitational pull on an object near the surface, the relevant distance is approximately Earth’s radius, not zero. The object’s distance from Earth’s center includes the distance from the surface to its center.
The direction of gravitational force is also important. Gravity acts along the line connecting the centers of the two objects, pulling each object toward the other.
According to Newton’s third law of motion, the two objects exert equal and opposite gravitational forces on one another.
Earth pulls an apple downward, while the apple pulls Earth upward with an equal gravitational force. Earth barely moves in response because its mass is enormously greater than the apple’s mass.
Common mistakes when calculating gravitational force
Although Newton’s gravitational equation is straightforward, several mistakes can produce incorrect answers.
Using the wrong distance: The distance in the equation must be measured between the centers of the two objects, not between their surfaces. For spherical objects, this distinction is essential.
Forgetting to square the distance: The denominator contains r², not simply r. Forgetting the square changes the calculated force significantly.
Mixing measurement units: The universal gravitational constant is expressed in units that require masses in kilograms and distances in meters. Using grams or kilometers without converting them can lead to incorrect results.
Confusing mass with weight: Mass is measured in kilograms, while gravitational force is measured in newtons. Substituting a weight value where a mass is required produces an incorrect calculation.
Using an incorrect gravitational constant: For ordinary calculations, use G ≈ 6.674 × 10⁻¹¹ N·m²/kg². The constant is universal, although its measured value is known with finite precision.
Checking the units and reviewing each mathematical step can help prevent these errors.
Applications of Newton’s law of gravitation
Newton’s law of universal gravitation is important in understanding and predicting the motion of objects throughout the solar system and beyond.
Astronomers use gravitational principles to study planetary motion, estimate the masses of celestial objects, and understand how stars and planets influence nearby bodies.
The law also helps explain satellite orbits. A satellite remains in orbit because Earth’s gravitational attraction continually changes the direction of its velocity. Rather than moving in a straight line, the satellite follows a curved path around Earth.
The Moon’s orbit is another familiar example. Earth’s gravitational attraction keeps the Moon in orbit, while the Moon’s gravity contributes to ocean tides. The Sun also influences tides, although the Moon generally has the stronger tidal effect on Earth.
Newton’s gravitational law is also useful for understanding the motion of comets, the interactions between stars, and the gravitational behavior of many other astronomical systems.
However, Newton’s law is not a complete description of gravity in every situation. Albert Einstein’s general theory of relativity provides a more comprehensive explanation, describing gravity in terms of the curvature of spacetime caused by mass and energy.
Newton’s equation remains extremely accurate and useful for most everyday situations, ordinary engineering calculations, and many applications in astronomy. Relativity becomes particularly important when dealing with extremely strong gravitational fields, very high precision, or objects moving at speeds close to the speed of light.
The key to calculating gravitational force is understanding the relationship between mass, distance, and the universal gravitational constant. By applying F = G(m₁ × m₂) / r² with consistent units, anyone can determine the gravitational attraction between two objects and better understand the physical force that shapes motion throughout the universe.