Gravity — Newton's Law of Universal Gravitation

Drop two textbooks on a desk and they do not visibly drift toward each other, yet the same force that is too weak to notice between them holds planets in orbit. Newton's law of gravitation captures both extremes with one rule: any two masses pull on each other, modeled by

If one mass grows, the force grows; 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 This Formula Applies

The formula is exact for point masses. It also works for spherically symmetric bodies, such as idealized planets, when you are outside the body — under that condition the body acts as if all its mass sat at its center. If the mass distribution is irregular, one direct plug-in may not be enough.

The Steps For A Gravitation Problem

1. Check the model. Use the simple formula for point masses, or for spherically symmetric bodies when you are outside them.
2. Measure the right distance. Use the center-to-center distance $r$, not the gap between surfaces.
3. Substitute into the formula. Apply $F = G m_1 m_2 / r^2$ with consistent SI units.
4. Interpret the result. The force is always attractive and points toward the other mass.

The step that carries the most intuition is the distance. The $1/r^2$ term is the part students need to feel, not just memorize: distance matters more than most first-time readers expect. Double the distance and the force becomes $1/4$; triple it and it becomes $1/9$. That inverse-square pattern is the heart of universal gravitation.

The Whole Procedure On Two Objects

Suppose two small objects have masses $5\ \mathrm{kg}$ and $10\ \mathrm{kg}$, with centers $2.0\ \mathrm{m}$ apart. The model check passes (point masses), and the distance is already center-to-center. Substitute:

So the gravitational force is about $8.34 \times 10^{-10}\ \mathrm{N}$. Interpreting it: the force is attractive and extremely small, which is why gravity between everyday objects is hard to notice — yet the same law dominates when one mass is enormous, such as Earth or the Sun.

This is also why weight near Earth's surface is often written as $W = mg$. Weight is the gravitational force between you and Earth, and since your distance from Earth's center barely changes compared with Earth's radius, the force simplifies to that local approximation. The universal law is the more general idea underneath it.

Where Each Step Trips People Up

Step 2 errors: using the surface-to-surface gap instead of the center-to-center distance.

Step 1 errors: forgetting that the simple formula applies directly to point masses, or to spherically symmetric bodies only when you are outside them.

Step 3 errors: missing the square on $r$ and treating gravity as proportional to $1/r$ instead of $1/r^2$.

A naming trap: mixing up $G$, the universal constant, with $g$, the local gravitational field strength near Earth.

The law shows up in falling objects, satellite motion, planetary orbits, and the link between mass and weight; in many problems it connects directly to circular motion, since orbital motion needs a centripetal force. To test that the inverse-square step is solid, keep the same masses but change the distance from $2.0\ \mathrm{m}$ to $4.0\ \mathrm{m}$ and predict the result first — if the idea is clear, the new force should be one fourth of the original.