KAHIBARO
Table of Contents
A Universal Number in Gravitation
The gravitational constant, written as $G$, is the proportionality constant in Newton's law of gravitation. It tells us how strong the gravitational attraction is between two masses.
Newton's law of gravitation is
$$
F = G \frac{m_1 m_2}{r^2}
$$
where $F$ is the gravitational force, $m_1$ and $m_2$ are the two masses, and $r$ is the distance between their centers.
Without $G$, the formula would not give the correct size of the force. The constant sets the scale of gravity in nature.
The gravitational constant is
$$
G \approx 6.674 \times 10^{-11}\ \text{N m}^2/\text{kg}^2
$$
This is a universal constant, the same everywhere in classical physics.
Why It Is So Small
The value of $G$ is very small. This means gravity is a very weak force unless at least one of the masses is very large. That is why small everyday objects do not noticeably attract each other, while planets, moons, and stars do.
For example, two books on a table do attract one another gravitationally, but the force is far too tiny to notice without very sensitive equipment.
Units of the Gravitational Constant
We can find the units of $G$ from Newton's law of gravitation:
$$
G = \frac{F r^2}{m_1 m_2}
$$
Using SI units,
$$
[F] = \text{N}, \quad [r] = \text{m}, \quad [m] = \text{kg}
$$
so the units of $G$ are
$$
[G] = \text{N m}^2/\text{kg}^2
$$
Since $1\ \text{N} = 1\ \text{kg m/s}^2$, we can also write
$$
[G] = \text{m}^3 \text{kg}^{-1}\text{s}^{-2}
$$
Both unit forms are correct.
Difference Between $G$ and $g$
Beginners often confuse $G$ with $g$, but they are different quantities.
| Symbol | Meaning | Typical Value | Same everywhere? |
|---|---|---|---|
| $G$ | Gravitational constant | $6.674 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$ | Yes |
| $g$ | Gravitational field strength or free fall acceleration near a planet | About $9.8\ \text{m/s}^2$ near Earth | No |
$G$ is a universal constant of nature. By contrast, $g$ depends on the nearby astronomical body and on distance from its center.
Do not confuse
$$
G \neq g
$$
$G$ is universal, while $g$ depends on location.
Measuring the Gravitational Constant
The gravitational constant is difficult to measure because gravity between ordinary objects is extremely weak. A famous method was developed by Henry Cavendish using a torsion balance.
In a torsion balance, small masses are attached to a light rod suspended by a thin wire. Larger nearby masses attract the smaller ones, twisting the wire slightly. By measuring the twist, the gravitational force can be found, and then $G$ can be calculated.
This experiment was historically important because it allowed scientists to determine the strength of gravity quantitatively.
Role of $G$ in Physics
The gravitational constant appears whenever Newtonian gravity is written mathematically. It is essential for calculating gravitational forces between masses and for connecting mass to gravitational effects.
It also appears in many later formulas in gravitation, astronomy, and modern physics.
In Newtonian gravitation, the strength of the force is directly proportional to $G$.
If $G$ were larger, gravity would be stronger.
If $G$ were smaller, gravity would be weaker.
A Simple Numerical Example
Suppose two masses of $1\ \text{kg}$ are separated by $1\ \text{m}$. Then
$$
F = G \frac{(1)(1)}{1^2} = G
$$
So the force is
$$
F \approx 6.674 \times 10^{-11}\ \text{N}
$$
This is an extremely small force, which helps explain why gravity between small objects is hard to observe directly.
Summary
The gravitational constant $G$ is the universal constant that sets the strength of gravity in Newton's law of gravitation. Its value is very small, which is why gravity is weak for everyday objects. Its SI units are $\text{N m}^2/\text{kg}^2$, or equivalently $\text{m}^3\text{kg}^{-1}\text{s}^{-2}$. It must not be confused with $g$, the local gravitational acceleration near a planet.