The Newtonian Constant of Gravitation, denoted by the symbol G, is an empirical physical constant that quantifies the strength of the gravitational force between two objects. It is a fundamental constant that appears in Newton's law of universal gravitation and also in Einstein's theory of general relativity. Its value represents the force in newtons between two one-kilogram masses that are one meter apart.
The first implicit measurement was conducted in 1798 by Henry Cavendish using a torsion balance. Because gravity is the weakest of the four fundamental forces, the value of G is notoriously difficult to measure with high precision, making it one of the least precisely known fundamental constants.
The Newtonian Constant of Gravitation, G, is a fundamental physical constant that defines the strength of the gravitational interaction. Its properties establish it as a universal scalar quantity essential for calculating gravitational forces between objects with mass.
| Property | Details |
|---|---|
| Scalar/Vector Nature | G is a scalar quantity. It possesses magnitude but has no associated direction. |
| SI Units | Cubic meters per kilogram per second squared (m^3 kg^-1 s^-2). |
| Magnitude | The accepted CODATA (2018) value is approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2. It is one of the most challenging fundamental constants to measure with high precision. |
| Dimensional Formula | [M]^-1 [L]^3 [T]^-2, where M represents mass, L represents length, and T represents time. |
| Universality | G is considered a universal constant, meaning its value is believed to be the same at all locations in the universe and constant throughout time. |
| Symbol | Quantity | SI Unit | Description |
|---|---|---|---|
| G | Newtonian Constant of Gravitation | m³ kg⁻¹ s⁻² | The universal constant of gravitation. |
| F | Gravitational Force | N | The attractive force between two masses. |
| m₁, m₂, M | Mass | kg | A measure of the amount of matter in an object. |
| r, a | Distance / Radius / Semi-major axis | m | The separation between the centers of mass. |
| U | Gravitational Potential Energy | J | Energy stored in a system due to its gravitational configuration. |
| g | Gravitational Field Strength | m/s² | The force per unit mass at a point in a gravitational field. |
| v | Orbital Velocity | m/s | The speed of an object in a stable orbit. |
| v_escape | Escape Velocity | m/s | The minimum speed to escape a body's gravitational pull. |
| T | Orbital Period | s | The time taken for one complete orbit. |
| r_s | Schwarzschild Radius | m | The radius of the event horizon of a non-rotating black hole. |
| c | Speed of Light in Vacuum | m/s | A universal physical constant. |
| Rμν, R, gμν, Tμν | Tensors in General Relativity | Varies | Components of the Einstein Field Equations describing spacetime curvature and mass-energy distribution. |
The Gravitational Constant G is not derived from first principles; it is an empirical constant determined through careful experimentation. The conceptual basis for its measurement comes from rearranging Newton's Law of Universal Gravitation.
To determine G, one must measure all four quantities on the right side of the equation:
By substituting the measured values of F, r, m₁, and m₂ into the rearranged formula, a value for G can be calculated. Modern experiments use highly sophisticated versions of this method to refine its value.
The Newtonian Constant of Gravitation is considered a fundamental and universal constant in physics. As such, it does not have different types, variants, or special cases; its value is assumed to be constant regardless of the physical context, location, or time.
| Type / Case | Description | When to Use |
|---|
Space Exploration and Orbital Mechanics: G is essential for calculating the trajectories of spacecraft, satellites, and probes. It's used to determine orbital velocities, periods, and the parameters for gravity-assist maneuvers to travel between planets.
Cosmology and Astrophysics: The constant is fundamental to models of stellar evolution, galaxy formation, and the expansion of the universe. It helps determine the mass of planets, stars, and galaxies, and is used to calculate phenomena like the Schwarzschild radius of black holes.
Geophysics and Geodesy: G is used to model Earth's gravitational field, which is crucial for GPS technology (which must account for gravitational time dilation). Variations in the local gravitational field, measured by gravimeters, can indicate the presence of dense ore deposits or subsurface cavities for mining and oil exploration.
Climate Science: Satellites like GRACE (Gravity Recovery and Climate Experiment) measure tiny changes in Earth's gravitational field to track the movement of water and the melting of ice sheets and glaciers, providing critical data for climate models.
Ocean Tides: The gravitational pull from the Moon and, to a lesser extent, the Sun, creates the daily rise and fall of ocean tides. The strength of this interaction is dictated by the value of G, the masses of the bodies, and their distance.
Planetary Orbits: The Sun's immense gravity, governed by the constant G, holds all the planets of our solar system in stable, predictable orbits. Without this precise gravitational balance, the planets would either fly off into space or spiral into the Sun.
Formation of Celestial Bodies: On cosmic timescales, gravity is the architect of the universe. The constant G determines the rate at which clouds of gas and dust collapse under their own gravity to form stars and planets.
Feeling Weight: The sensation of weight is a direct consequence of the gravitational force between your body's mass and the mass of the Earth. This force is calculated using G, your mass, the Earth's mass, and the Earth's radius.
The dimensions of G can be derived from Newton's law of gravitation, \(F = G m_1 m_2 / r^2\). Rearranging for G gives \(G = F r^2 / (m_1 m_2)\).
In terms of fundamental dimensions of Mass (M), Length (L), and Time (T):
\([G] = \frac{[\text{Force}] [\text{Length}]^2}{[\text{Mass}]^2} = \frac{(M L T^{-2}) (L^2)}{M^2} = M^{-1} L^3 T^{-2}\)
In SI base units, this corresponds to \(\text{m}^3 \text{kg}^{-1} \text{s}^{-2}\).
| Unit System | G Value | Units | Common Usage |
|---|---|---|---|
| SI (Standard) | 6.67430 × 10⁻¹¹ | m³ kg⁻¹ s⁻² | Scientific research |
| CGS | 6.67430 × 10⁻⁸ | cm³ g⁻¹ s⁻² | Older literature |
| Gaussian Units | 6.67430 × 10⁻⁸ | dyne cm² g⁻² | Theoretical physics |
| Astronomical Units | 1.327 × 10²⁰ | m³ s⁻² M⊙⁻¹ | Solar system dynamics (as GM) |
| Planck Units | 1 | dimensionless | Quantum gravity |
| Geometric Units | 1 | dimensionless | General relativity |
The Newtonian Constant of Gravitation, denoted by G, is a fundamental physical constant that quantifies the strength of the gravitational force between two objects. Its value is approximately 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻². It acts as the constant of proportionality in Newton's law of universal gravitation, relating the gravitational force to the masses of the objects and the distance between them.
The accepted value for G is approximately 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻². These units are derived from Newton's law of universal gravitation (F = G * (m₁m₂/r²)) to ensure the resulting force (F) is expressed in Newtons (kg·m/s²). It is also commonly written with the equivalent units of N·m²/kg².
The constant G is used whenever calculating the gravitational force between two objects using Newton's law of universal gravitation. It is essential for determining the orbital mechanics of satellites and planets, the gravitational attraction between stars in a galaxy, and the mass of celestial bodies. It is a key component in both classical mechanics and Einstein's general relativity.
This is a very common point of confusion. G is a universal constant, meaning its value (≈ 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻²) is the same everywhere in the universe. In contrast, g is the local acceleration due to gravity on a specific celestial body, which depends on the body's mass and radius (on Earth's surface, g ≈ 9.81 m/s²).
In space exploration, G is critical for calculating the precise trajectories of satellites, probes, and spacecraft. Engineers use it to determine the required orbital velocities to keep satellites in orbit and to plan gravity-assist maneuvers, where a spacecraft uses a planet's gravity to alter its path and speed. It is fundamental to navigating our solar system.
While G is defined in Newtonian physics, it is also a fundamental constant in Einstein's theory of general relativity, appearing in the Einstein field equations. In this context, G relates the curvature of spacetime to the distribution of mass and energy. Its role is central to modern astrophysics and cosmology for modeling phenomena like black holes and the expansion of the universe.