Gravity — Newton's Law of Universal Gravitation

Gravity means any two masses pull on each other. In Newtonian physics, that pull is modeled by

$$F = G \frac{m_1 m_2}{r^2}$$

If one mass gets larger, the force gets larger. If the distance doubles, the force becomes one fourth as large. Here $m_1$ and $m_2$ are the masses, $r$ is the center-to-center distance, and $G \approx 6.67 \times 10^{-11}\ \mathrm{N \cdot m^2/kg^2}$ is the gravitational constant.

When Newton's law of gravity applies

This formula is exact for point masses. It also works for spherically symmetric bodies, such as idealized planets, if you are outside the body.

Under that condition, the body behaves as if all its mass were concentrated at its center. If the mass distribution is irregular, one direct plug-in formula may not be enough.

What the inverse-square term means

The $1/r^2$ term is the part students usually need to feel, not just memorize. Distance matters more than many first-time readers expect.

If the distance doubles, the force becomes $1/4$ of the original value. If the distance triples, it becomes $1/9$. That inverse-square pattern is the key intuition behind Newton's law of universal gravitation.

Worked example: gravitational force between two objects

Suppose two small objects have masses $5\ \mathrm{kg}$ and $10\ \mathrm{kg}$, and their centers are $2.0\ \mathrm{m}$ apart.

Start with

$$F = G \frac{m_1 m_2}{r^2}$$

Substitute the values:

$$F = (6.67 \times 10^{-11}) \frac{(5)(10)}{(2.0)^2}$$ $$F = (6.67 \times 10^{-11}) \frac{50}{4} = (6.67 \times 10^{-11})(12.5)$$ $$F \approx 8.34 \times 10^{-10}\ \mathrm{N}$$

So the gravitational force is about $8.34 \times 10^{-10}\ \mathrm{N}$.

That force is extremely small. This is why gravity between everyday objects is hard to notice, even though the same law becomes dominant when one of the masses is enormous, such as Earth or the Sun.

Why weight is often written as $W = mg$

Near Earth's surface, weight is the gravitational force between you and Earth. Because your distance from Earth's center changes only a little compared with Earth's radius, the force is often written as

$$W = mg$$

This is a useful local approximation. Newton's universal gravitation law is the more general idea underneath it.

Common mistakes with gravity formulas

• Using the surface-to-surface gap instead of the center-to-center distance.
• Forgetting that the simple formula applies directly to point masses, or to spherically symmetric bodies when you are outside them.
• Missing the square on $r$ and treating gravity as proportional to $1/r$ instead of $1/r^2$.
• Mixing up $G$, the universal constant, with $g$, the local gravitational field strength near Earth.

Where Newton's law of gravitation is used

Newton's law of universal gravitation is used to model falling objects, satellite motion, planetary orbits, and the link between mass and weight. In many introductory problems, it also connects directly to circular motion because orbital motion needs a centripetal force.

The law is especially useful because it ties ordinary gravity on Earth to motion in space with one framework.

Try a similar gravity problem

Keep the same masses, but change the distance from $2.0\ \mathrm{m}$ to $4.0\ \mathrm{m}$. Predict the result before calculating it. If the inverse-square idea is clear, the new force should be one fourth of the original. If you want to check your setup on another gravity question, GPAI Solver can walk through a similar problem step by step.