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Gravitational Force Calculator

Calculate the force between two masses

Object Parameters

This calculator determines the gravitational force between two objects according to Newton's Law of Universal Gravitation. [1, 2] This fundamental law states that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. [2, 3]

  • Inverse-Square Law: Doubling the distance between two objects reduces the gravitational force to one-quarter of its original value. [4]
  • Universal Constant (G): The force is quantified by the gravitational constant, G, an empirical physical constant. [5]

F = G (m₁m₂) / r²

Enter parameters and click Calculate

About Gravitational Force Calculator

The Cosmic Glue: A Master Guide to Our Gravitational Force Calculator

From the gentle fall of an apple from a tree to the majestic dance of galaxies across billions of light-years, the universe is governed by a silent, invisible, and utterly relentless force: **gravity**. It is the cosmic glue that binds matter together, the architect of planets, stars, and solar systems, and the conductor of the grand celestial ballet.

For centuries, the motion of the heavens was a profound mystery. It took the genius of Sir Isaac Newton to finally formulate a law that not only explained why objects fall to the Earth but also why the Moon orbits the Earth and the Earth orbits the Sun. He proposed a radical idea: the same force is responsible for both. This was the birth of the Law of Universal Gravitation, a cornerstone of science that forever changed our perception of the cosmos.

Welcome to the definitive guide to this fundamental force. Our Gravitational Force Calculator is a precision tool built upon Newton's legendary equation. This comprehensive article will empower you to calculate the gravitational pull between any two objects, and more importantly, it will take you on a journey to understand the profound implications of this universal law.

What is Gravitational Force? The Universal Attraction

Newton's Law of Universal Gravitation states that every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Let's break down that profound statement:

  • It is Universal: This force isn't unique to Earth. It exists between you and your computer, between the Sun and Jupiter, and between two dust particles in a distant nebula. Anything with mass exerts a gravitational pull.
  • It is Always Attractive: Gravity always pulls objects together; it never pushes them apart.
  • It Depends on Mass: The more massive the objects, the stronger the gravitational force between them. This is why we feel the Earth's pull so strongly but are completely unaware of the tiny gravitational pull between ourselves and a nearby car.
  • It Weakens with Distance (Rapidly!): This is the most crucial and perhaps least intuitive part. The force is not just weaker with distance; it's weaker with the *square* of the distance. This is known as an **inverse-square law**.

The Inverse-Square Law in Action

Imagine two objects a certain distance apart. If you double the distance between them, the gravitational force doesn't become half as strong—it becomes one-quarter (1/2²) as strong. If you triple the distance, the force becomes one-ninth (1/3²) as strong. This rapid drop-off is why the Sun's immense gravity can hold Pluto in its orbit billions of miles away, yet we don't feel its pull directly on our bodies. The Earth is so much closer that its pull completely dominates.

Newton's Grand Equation: The Heart of the Calculator

Newton encapsulated his entire law into a single, beautiful, and powerful equation that our calculator uses to perform its calculations.

F = G * (m₁ * m₂) / r²

Let's dissect each component of this cosmic formula:

F: Gravitational Force

This is the final result—the magnitude of the attractive force between the two objects. It is measured in Newtons (N). The force is mutual; object 1 pulls on object 2 with the exact same force that object 2 pulls on object 1, just in the opposite direction (Newton's Third Law).

m₁ and m₂: The Masses

These are the masses of the two objects in question, measured in kilograms (kg). The formula shows that the force is directly proportional to the product of these masses. Doubling one mass doubles the force. Doubling both masses quadruples the force.

r: Distance Between Centers

This is the distance between the centers of mass of the two objects, measured in meters (m). For spherical objects like planets, this is simply the distance between their geometric centers. This is the term that gets squared, representing the inverse-square law.

G: The Gravitational Constant

'G' is one of the most important numbers in all of physics. It is the Universal Gravitational Constant. It is a scaling factor that converts the relationship between mass and distance into the precise force in Newtons. Its value is incredibly small, which explains why gravity is the weakest of the four fundamental forces.

G ≈ 6.67430 × 10⁻¹¹ N·m²/kg²

How to Use the Gravitational Force Calculator

Step 1: Enter the Mass of Object 1 (m₁)

Input the mass of the first object in kilograms (kg). For astronomical calculations, you may need to use scientific notation (e.g., the mass of the Earth is approximately 5.972e24 kg).

Step 2: Enter the Mass of Object 2 (m₂)

Input the mass of the second object, also in kilograms.

Step 3: Enter the Distance (r)

Input the distance between the centers of the two objects in meters (m). Be careful with units! If a distance is given in kilometers, you must multiply by 1,000 to convert it to meters before entering it.

Step 4: Calculate and Interpret

The calculator will instantly apply Newton's formula and provide the gravitational force in Newtons. This allows you to explore the relationships. What happens if you halve the distance? The force should quadruple. What if you double one of the masses? The force should double.

Worked Examples: From People to Planets

Example 1: The Force Between Two People

Calculate the gravitational force between a 70 kg person and an 80 kg person standing 1 meter apart.

  • • Inputs: m₁ = 70 kg, m₂ = 80 kg, r = 1 m.
  • • Formula: F = G * (m₁ * m₂) / r²
  • • Calculation: F = (6.674e-11) * (70 * 80) / 1² => F = (6.674e-11) * 5600 ≈ 3.74 × 10⁻⁷ N
  • • Result: The force is 0.000000374 Newtons. This is an incredibly tiny, utterly negligible force, which is why we don't feel ourselves being pulled towards other people. It highlights just how weak gravity is on a human scale.

Example 2: The Earth and the Moon

Calculate the gravitational force that holds the Moon in orbit around the Earth.

  • • Inputs:
    • Mass of Earth (m₁): 5.972 × 10²⁴ kg
    • Mass of Moon (m₂): 7.342 × 10²² kg
    • Distance (r): 384,400 km = 3.844 × 10⁸ m
  • • Formula: F = G * (m₁ * m₂) / r²
  • • Calculation: F = (6.674e-11) * (5.972e24 * 7.342e22) / (3.844e8)²
  • • Result: F ≈ 1.98 × 10²⁰ N
  • • Interpretation: This is an unimaginably colossal force—nearly 200 quintillion Newtons! This is the centripetal force that constantly pulls the Moon towards the Earth, preventing it from flying off into space.

From Universal Law to g = 9.81 m/s²

Where does the familiar g = 9.81 m/s² come from? It's derived directly from Newton's Universal Law! We can calculate the force of gravity on a person (let's say you, with mass m_you) standing on the Earth's surface. This force is what we call your weight.

Weight = G * (m_earth * m_you) / r_earth²

But we also know from Newton's Second Law that Force = mass * acceleration, so Weight = m_you * g. If we set these two expressions for weight equal to each other:

m_you * g = G * (m_earth * m_you) / r_earth²

Your mass, m_you, appears on both sides and cancels out! We are left with a formula for g itself:

g = (G * m_earth) / r_earth²

Plugging in the values for G, the mass of the Earth, and the radius of the Earth gives a value of approximately 9.81 m/s². This shows that the local acceleration due to gravity is just a specific application of the universal law.

Frequently Asked Questions (FAQ)

Q: Is Newton's Law of Gravitation completely correct?

For almost all practical purposes, yes. It can predict planetary orbits with incredible accuracy. However, Albert Einstein's Theory of General Relativity provides a more complete and accurate description. Einstein showed that gravity is not a force in the traditional sense, but a curvature of spacetime caused by mass and energy. Newton's law is an excellent approximation of General Relativity in all but the most extreme gravitational fields (like near a black hole).

Q: Why is gravity so much weaker than other forces like magnetism?

This is one of the deepest mysteries in physics, often called the "hierarchy problem." The weakness is reflected in the tiny value of the Gravitational Constant, G. A small refrigerator magnet can overcome the gravitational pull of the entire Earth to lift a paperclip. Physicists have many theories as to why this is, but a definitive answer remains elusive.

Calculate the Attraction of the Cosmos

Newton's Law of Universal Gravitation is one of the greatest intellectual achievements in human history. It connected the terrestrial with the celestial and revealed that the universe operates on a set of predictable, mathematical laws.

Our calculator is your portal to exploring this profound law. Use it to feel the scale of the cosmos, to understand the forces that shape our solar system, and to appreciate the invisible bonds that connect every piece of matter in the universe.

Frequently Asked Questions

What is gravitational force?
Gravitational force is the attractive force between any two objects with mass. It's described by Newton's Law of Universal Gravitation.
How is gravitational force calculated?
F = G(m₁m₂)/r², where F is force, G is gravitational constant (6.674×10⁻¹¹ N·m²/kg²), m₁ and m₂ are masses, and r is distance between centers.
What units are used in gravitational force calculations?
Standard SI units: force in newtons (N), mass in kilograms (kg), distance in meters (m). G's units are N·m²/kg².
Why is the gravitational constant (G) so small?
G's small value reflects gravity's weakness compared to other fundamental forces. It's the proportionality constant in Newton's law.
How does distance affect gravitational force?
Force decreases with the square of distance (inverse square law). Double the distance = ¼ the force; triple = 1/9 the force, etc.
What is the gravitational force between two 1kg masses 1m apart?
F = (6.674×10⁻¹¹ × 1 × 1)/1² = 6.674×10⁻¹¹ N (extremely small - shows why we don't notice gravity between everyday objects).
How do I calculate gravitational force on Earth's surface?
Use F = mg, where g = 9.81 m/s² (average). This is a simplified version where Earth's mass and radius are incorporated into g.
What's the difference between g and G?
G is the universal gravitational constant. g is local gravitational acceleration (9.81 m/s² on Earth), derived from G and Earth's properties.
How does mass affect gravitational force?
Force is directly proportional to each mass. Double one mass = double the force. Double both masses = quadruple the force.
What is the gravitational force inside a planet?
At radius r inside a uniform sphere: F = G(M_inside)m/r², where M_inside is the mass within radius r (decreases toward center).
How do I calculate orbital velocity using gravity?
Set gravitational force equal to centripetal force: v = √(GM/r), where M is central body's mass, r is orbital radius.
What is the gravitational force at the center of Earth?
Zero - all mass pulls equally in all directions, canceling out. This assumes Earth is perfectly spherical and uniform (which it isn't exactly).
How does altitude affect gravitational force?
Force decreases with (R+h)², where R is Earth's radius, h is altitude. At 2R (Earth radius up): force is ¼ surface value.
What is the gravitational force between Earth and Moon?
F ≈ (6.674×10⁻¹¹ × 5.97×10²⁴ × 7.34×10²²)/(3.84×10⁸)² ≈ 1.98×10²⁰ N (enormous, but balanced by Moon's orbital motion).
How do I calculate gravitational potential energy?
U = -GMm/r (negative because force is attractive). Near Earth's surface: U ≈ mgh (h << Earth radius).
What is the gravitational force in Einstein's relativity?
General Relativity replaces force with spacetime curvature. Newton's law is an excellent approximation for weak fields like our solar system.
How does gravity compare between planets?
Surface gravity g = GM/R². Jupiter has 2.5× Earth's g despite being 318× more massive (it's also much larger - R² term dominates).
What is the gravitational force between Earth and Sun?
F ≈ (6.674×10⁻¹¹ × 1.99×10³⁰ × 5.97×10²⁴)/(1.5×10¹¹)² ≈ 3.54×10²² N (keeps Earth in orbit).
How do I calculate escape velocity using gravity?
v_escape = √(2GM/R), derived by setting kinetic energy equal to gravitational potential energy.
What is the gravitational force at a Lagrange point?
At L1-L3 points, gravitational forces from two bodies balance the centripetal force needed for orbital motion (complex calculation).
How does gravity work in a binary star system?
Both stars orbit their common center of mass (barycenter). Gravitational force is F = G(m₁m₂)/r² where r is separation distance.
What is the gravitational force in a black hole?
At event horizon, escape velocity = c. Within, gravity dominates all forces. Newton's law breaks down; need General Relativity.
How do I calculate tidal forces from gravity?
Tidal force ≈ 2GMmR/d³, where R is object's radius, d is distance to gravitating body. Difference in gravity across object's length.
What is the gravitational force in a hollow sphere?
Inside: zero (shell theorem). Outside: behaves as if all mass were concentrated at center (Newton's law applies normally).
How does gravity vary with latitude on Earth?
About 0.5% difference from pole (9.83 m/s²) to equator (9.78 m/s²) due to Earth's rotation (centrifugal effect) and oblateness.
What is the gravitational force between protons?
F ≈ (6.674×10⁻¹¹ × 1.67×10⁻²⁷ × 1.67×10⁻²⁷)/(1×10⁻¹⁵)² ≈ 1.86×10⁻³⁴ N (negligible compared to electromagnetic force between them).
How do I calculate gravitational time dilation?
Δt₀ = Δt√(1 - 2GM/rc²). Strong gravity slows time. GPS satellites must account for this relativistic effect.
What is the gravitational force in a galaxy?
For stars orbiting galactic center: F = GM(r)m/r², where M(r) is mass within star's orbit. Dark matter affects observed rotation curves.
How does gravity work in quantum mechanics?
No complete quantum theory of gravity yet. At small scales, gravity is negligible compared to other forces. String theory attempts unification.
What is the gravitational force between galaxies?
F = G(m₁m₂)/r² applies, but cosmological expansion dominates at large scales. Local groups of galaxies remain gravitationally bound.
How do I calculate gravitational lensing?
Light bending angle α = 4GM/rc². Requires General Relativity - Newtonian gravity predicts half the correct value.
What is the gravitational force in the early universe?
At extremely high densities, quantum gravity effects would dominate. Current physics can describe back to ~10⁻⁴³s after Big Bang.
How does gravity vary in the solar system?
Sun's gravity dominates (99.86% solar system mass). At Earth's orbit: F = 0.006 N/kg (vs 9.81 N/kg at Earth's surface).
What is the gravitational force in a neutron star?
Surface gravity ~10¹¹× Earth's. Intense gravity compresses matter to nuclear densities. Relativistic effects crucial.
How do I calculate gravitational waves?
Requires solving Einstein's field equations for accelerating masses. Amplitude h ≈ 2G/c⁴ × (E_kinetic/r). LIGO detects h ~10⁻²¹.
What is the gravitational force in a wormhole?
Theoretical construct - would require negative energy to stabilize. Gravity behavior depends on specific hypothetical model.
How does gravity work in higher dimensions?
String theories suggest gravity may appear weak because it 'leaks' into extra dimensions. Force law would change at very small scales.
What is the gravitational force at the edge of the universe?
No edge in standard cosmology. On cosmological scales, gravity competes with dark energy driving accelerated expansion.
How do I calculate gravitational redshift?
z = Δλ/λ ≈ GM/rc². Photons lose energy climbing out of gravity wells, shifting to longer (redder) wavelengths.
What is the gravitational force in a vacuum?
Same as in matter - gravity acts on mass/energy regardless of intervening material (though matter can shield gravitational effects).
How does gravity behave near cosmic strings?
Theoretical 1D defects - would cause conical spacetime with unusual gravitational lensing effects (double images of background objects).
What is the gravitational force of dark matter?
Follows F = Gm₁m₂/r² like normal matter, but dark matter only interacts gravitationally (no electromagnetic effects).
How do I calculate gravitational effects in a galaxy cluster?
Treat as N-body problem with dark matter included. Typically requires numerical simulations due to complexity.
What is the gravitational force in a Bose-Einstein condensate?
Extremely weak - only relevant for very large condensates. Typically dominated by quantum effects at these scales.
How does gravity work in the holographic principle?
Speculative theory suggesting gravity in a volume may be described by physics on its boundary. Exact relationship still being explored.

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