How Gravitational Force Works: Newton's Law and the Constant Behind It

Every object with mass pulls on every other object with mass. You, the chair you are sitting in, the planet under your feet, and a star ten thousand light-years away are all tugging on each other right now. Most of those pulls are so faint that no instrument in your house could detect them. A few are strong enough to keep a Moon in orbit for four and a half billion years. The single equation that covers both extremes is Newton's law of universal gravitation, and once you can read it, a surprising amount of physics falls into place.

The one formula you need

Newton's law says the gravitational force between two objects is

F = G · m1 · m2 / r²

Here F is the attractive force in newtons, m1 and m2 are the two masses in kilograms, r is the distance between their centres in metres, and G is the gravitational constant. The whole sentence reads: the pull is proportional to each mass and inversely proportional to the square of the distance.

Notice there is no "which object is heavier" term. Gravity is mutual and exactly equal in both directions. The Earth pulls a falling apple with some force, and the apple pulls the Earth back with the same force. The apple accelerates dramatically while the Earth barely twitches, but that is a consequence of their wildly different masses, not of the force being uneven. The force itself is symmetric.

The smallest important number in physics

The constant G ties the formula to real-world units. Its measured value is roughly 6.674 × 10⁻¹¹ N·m²/kg² (the CODATA figure is 6.674 30e-11). Written out, that is 0.000 000 000 066 7. It is one of the tiniest constants in all of physics, and that smallness is the whole story of why gravity feels weak.

Plug in two one-kilogram masses one metre apart and the force is exactly G: about 67 trillionths of a newton. You could not feel that with the most sensitive fingertip. Gravity only becomes a force to reckon with when at least one of the masses is astronomical, because the m1 · m2 product finally grows large enough to drag that microscopic G up to something you can stand on.

This is also why G was so hard to pin down. Newton wrote the law in 1687 but never measured G; Henry Cavendish did it in 1798 with a delicate torsion balance, weighing the faint attraction between lead spheres. It remains one of the least precisely known constants in science, because you cannot shield an experiment from the gravity of everything else nearby.

A worked example, start to finish

Let me grind through a real one rather than wave at it. Two 1000 kg cars are parked 10 metres apart in a quiet lot. What is the gravitational force between them?

Drop the numbers into F = G · m1 · m2 / r²:

Numerator: G · m1 · m2 = 6.674e-11 × 1000 × 1000 = 6.674e-5
Denominator: r² = 10² = 100
Force: 6.674e-5 / 100 = 6.674e-7 N, or about 0.67 millionths of a newton

That is comparable to the weight of a single grain of fine sand. Two tonnes of steel a few car-lengths apart, and the mutual pull would not budge a feather. I ran this through the gravitational force calculator to confirm my arithmetic, and what I appreciate is that it lays out G·m1·m2 and the division by r² as separate on-screen steps, so I could see exactly where a slip would have crept in if I had fat-fingered an exponent. For homework where you have to show your working, that step list is worth more than the final number.

The inverse-square law, and why it matters

The r² in the denominator is doing quiet, heavy lifting. Because distance is squared, the force does not just shrink as objects move apart, it shrinks fast.

Double the distance, and you divide by 2² = 4. The force drops to a quarter.
Triple the distance, and you divide by 3² = 9. The force drops to a ninth.
Move ten times further out, and the pull is a hundredth of what it was.

This single fact explains the shape of the solar system. The Sun's grip on Mercury, close in, is ferocious; its grip on Neptune, thirty times further out, is roughly nine hundred times weaker. It is also why a satellite boosted to twice its orbital radius feels only a quarter of the pull it started with. If you want to see it rather than take my word for it, type any pair of masses into the calculator with some distance r, note the force, then double r and recompute. The number quarters exactly, every time.

Why the apple falls but the Moon does not

Here is the puzzle Newton famously cracked: if gravity pulls the apple straight down to the ground, why does it not yank the Moon down too? The Moon is enormous and the Earth is right there.

The answer is that the Moon is falling. Gravity is pulling it toward Earth every single second, exactly as it pulls the apple. The difference is that the Moon is also moving sideways, very fast, about a kilometre per second. So as gravity tugs it inward, its sideways motion carries it forward, and the surface of the Earth curves away beneath it at just the right rate. The Moon perpetually falls toward Earth and perpetually misses. That is what an orbit is: a continuous fall that never lands.

Run the numbers and the same formula that governs the apple holds the Moon. With Earth at 5.972e24 kg, the Moon at 7.35e22 kg, and a centre-to-centre distance of 3.84e8 m, F = G·m1·m2/r² gives about 1.98e20 N. That titanic force is what bends the Moon's straight-line momentum into a closed loop. No new physics, no separate "orbital force", just the apple's equation scaled up by twenty-six orders of magnitude.

Getting the units right

The biggest source of wrong answers is not the algebra, it is the units. G is defined for kilograms, metres, and newtons, so everything must be in SI before it touches the formula. Feed grams or kilometres straight in and your answer lands off by factors of a thousand or a million. Two other traps catch students often: squaring only one factor of r instead of the full r², and measuring distance between surfaces rather than between centres of mass. For a person on Earth, r is Earth's radius (about 6.37e6 m), not your height above the ground.

If unit conversion is where you keep stumbling, a dedicated unit converter makes the tonnes-to-kilograms and kilometres-to-metres steps painless before you ever reach the gravity calculation. And once you have the force, gravity rarely travels alone in a physics problem, so a scientific calculator is handy for the energy and velocity follow-ups that usually come next.

Gravity is the weakest of the four fundamental forces by an absurd margin, yet it is the one that shapes planets, stars, and galaxies, because unlike the others it never cancels out and reaches across any distance. One short equation, one tiny constant, and the whole architecture of the night sky follows. That is a good trade.

Made by Toolora · Updated 2026-06-13