12.1 Newton's Law of Universal Gravitation

Gravitational Force and Newton's Law of Universal Gravitation

Every object with mass pulls on every other object with mass. Newton's Law of Universal Gravitation gives us the precise mathematical relationship governing that pull, connecting mass and distance to the strength of the gravitational force. This law unified what had been two separate ideas (things falling on Earth and planets orbiting in space) into a single framework.

Inverse square law of gravitation

The gravitational force between two objects decreases as the square of the distance between them increases:

$F \propto \frac{1}{r^2}$

What does "inverse square" actually mean? If you double the distance between two objects, the force doesn't just halve. It drops to one-quarter. Triple the distance, and the force falls to one-ninth. On a force vs. distance graph, this produces a steep hyperbolic curve that flattens out but never quite reaches zero.

The reason for this pattern is geometric. Gravitational influence spreads outward in all three dimensions, like the surface of an expanding sphere. The area of a sphere grows as $4\pi r^2$, so the force per unit area dilutes by $r^2$ as you move farther away.

• This same inverse square behavior shows up elsewhere in physics: light intensity and electrostatic force both follow $\frac{1}{r^2}$ relationships.
• The force never reaches zero, no matter how large $r$ gets. Gravity has infinite range, though it becomes negligibly small at large distances.

Calculation of gravitational force

Newton's Law of Universal Gravitation is expressed as:

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

where:

• $F$ = gravitational force (in Newtons, N)
• $G$ = gravitational constant = $6.674 \times 10^{-11} \, N \cdot m^2 / kg^2$
• $m_1$ and $m_2$ = the two masses (in kg)
• $r$ = distance between the centers of the two masses (in m)

Note that $r$ is measured center-to-center, not surface-to-surface. For a person standing on Earth, $r$ is Earth's radius (~$6.37 \times 10^6$ m), not zero.

Steps to solve a gravitational force problem:

1. Identify the two masses ($m_1$ and $m_2$) and convert to kilograms if needed.
2. Determine the center-to-center distance $r$ between them, in meters.
3. Plug values into $F = G\frac{m_1 m_2}{r^2}$.
4. Solve and check units. Your answer should come out in Newtons.

Quick example: Find the gravitational force between Earth ($5.97 \times 10^{24}$ kg) and a 70 kg person standing on the surface ($r = 6.37 \times 10^6$ m).

$F = (6.674 \times 10^{-11}) \frac{(5.97 \times 10^{24})(70)}{(6.37 \times 10^6)^2} \approx 686 \, N$

That's roughly the person's weight, which is exactly what you'd expect since weight is the gravitational force Earth exerts on you.

Experimental determination of G

Gravity is universal, but the constant $G$ is extraordinarily small. You can't just look up its value from theory; it had to be measured experimentally.

The Cavendish Experiment (1798):

Henry Cavendish designed a remarkably clever apparatus to measure the tiny gravitational attraction between lead spheres in a lab.

1. Two small lead spheres were mounted on opposite ends of a lightweight horizontal rod.
2. The rod was suspended from its center by a thin wire (a torsion fiber).
3. Two large, fixed lead spheres were placed near the small ones.
4. The gravitational pull between the large and small spheres caused the rod to twist the wire slightly.
5. By measuring the angle of twist and knowing the torsion properties of the wire, Cavendish calculated the force between the spheres.
6. With the force, the known masses, and the measured distances, he solved for $G$.

Cavendish's result was remarkably close to the modern accepted value. The experiment is often called "weighing the Earth" because once you know $G$, you can combine it with $g$ (9.8 m/s²) and Earth's radius to calculate Earth's mass.

Modern techniques (beam balance experiments, atom interferometry) have refined the measurement, but $G$ remains one of the least precisely known fundamental constants because the gravitational force between lab-sized objects is so tiny and so easily disturbed by vibrations, air currents, and other environmental factors.

Universality of gravitational law

Newton's law applies to all objects with mass, from dust grains to galaxy clusters. This was a revolutionary idea. Before Newton, people treated "earthly" physics and "heavenly" physics as completely separate domains. The law of universal gravitation showed that the same force making an apple fall also keeps the Moon in orbit.

• Predictive power: Newton's law enabled the mathematical prediction of planetary orbits. In 1846, astronomers used gravitational calculations to predict the existence and location of Neptune before anyone had seen it through a telescope.
• Limitations: At very strong gravitational fields (near black holes) or very high speeds (near the speed of light), Newton's law breaks down. Einstein's general relativity extends and corrects it in these extreme regimes. For most everyday and solar-system-scale problems, Newton's law works extremely well.
• Technological applications: GPS satellites require precise knowledge of gravitational effects on timing. Spacecraft trajectories depend entirely on gravitational calculations.
• Cosmological implications: The law underlies our understanding of how galaxies form and rotate. Observations of galactic rotation that don't match predictions from visible mass led to the hypothesis of dark matter, one of the biggest open questions in modern physics.