Gravitation

One long-standing goal of physics is to understand the gravitational force — the force that holds you to Earth, the Moon to Earth, and Earth to the Sun. Its reach does not stop there. It binds the hundreds of billions of stars in our Milky Way, holds together the Local Group of galaxies (including Andromeda, \(2.3 \times 10^{6}\) light-years away), draws the Local Supercluster toward the Great Attractor, and tries to hold together the entire expanding universe. The same force, taken to an extreme, creates black holes: the collapsed remnants of massive stars whose surface gravity is so strong that not even light escapes. Our understanding of all of it begins with one elegant law written by Isaac Newton.

In 1665, the 23-year-old Newton made one of the great unifications in physics: the force that holds the Moon in orbit is the same force that makes an apple fall. He went further — every body in the universe attracts every other body. The familiar tug of Earth simply overwhelms the feeble attraction between everyday objects (you and a nearby apple attract each other with less than the weight of a dust speck).

The two forces of an attracting pair are equal and opposite — a third-law pair — so we speak of the gravitational force between two particles. No object placed between them can shield one from the other. And although the law is written for particles, Newton rescued the apple–Earth case with a theorem of remarkable economy:

To find the net force on one particle from many, we use the principle of superposition: compute each pairwise force separately, then add them as vectors. There is no shielding and no interference — each pull acts as if the others were not there.

Treating Earth as a uniform sphere of mass \(M\), the force on a particle of mass \(m\) at distance \(r\) from the center is \(F = GMm/r^{2}\). Setting \(F = ma_{g}\) gives the gravitational acceleration:

The measured free-fall acceleration \(g\) differs slightly from \(a_{g}\) for three reasons: Earth's mass is not uniformly distributed, Earth is not a perfect sphere (it bulges at the equator), and Earth rotates. The rotation effect is the cleanest to quantify — a crate on a scale at the equator needs a small inward net force for its circular motion, so the scale reads slightly less than the true gravitational pull:

The shell theorem has a striking companion for the interior: a uniform shell exerts no net gravitational force on a particle inside it. The pulls from all parts of the shell cancel exactly. So for a particle at radius \(r\) inside a uniform Earth, only the mass within radius \(r\) matters — the outer shells contribute nothing.

Writing the enclosed mass as \(M_{\text{ins}} = \rho\,\tfrac{4}{3}\pi r^{3}\) and substituting into \(F = GmM_{\text{ins}}/r^{2}\), the \(r^{2}\) in the denominator is overpowered, and the force grows linearly with depth toward the center:

Near Earth's surface we used \(U = mgy\) with constant \(g\). For two particles separated by an arbitrary distance \(r\), we choose the zero of potential energy at infinite separation. Because gravity does positive work as the particles approach, \(U\) is negative for every finite separation:

Gravity is conservative, so the work it does — and hence \(\Delta U\) — is independent of the path taken. Differentiating \(U(r)\) recovers the force, \(F = -dU/dr = -GMm/r^{2}\), neatly closing the loop back to Newton's law. One of the most useful consequences is the escape speed: set the total energy at the surface to zero (kinetic plus potential), since a projectile that just barely escapes arrives at infinity with neither.

Working from Tycho Brahe's painstaking naked-eye observations, Johannes Kepler distilled three empirical laws of planetary motion. Newton later showed they all follow from his law of gravitation. They hold equally for planets around the Sun and for satellites around any massive central body.

The near-constant final column is Kepler's third law made visible — one number, the same across the solar system, set entirely by the Sun's mass.

As a satellite orbits, its kinetic and potential energies trade back and forth, but its mechanical energy stays constant. For a circular orbit, applying Newton's second law (\(GMm/r^{2} = mv^{2}/r\)) gives the kinetic energy, which combines with the potential energy into a clean result:

Einstein's route to general relativity began with a single thought: a person in free fall does not feel their own weight. This is the principle of equivalence — gravitation and acceleration are locally indistinguishable. A physicist sealed in a box cannot tell whether the box rests on Earth or accelerates through deep space at \(9.8\,\mathrm{m/s^{2}}\); the scale reads the same and a dropped object falls the same way.

From there Einstein recast gravity not as a force but as the curvature of spacetime produced by mass. Two ships sailing due south from the equator draw together not because a force pulls them, but because Earth's surface is curved; likewise two apples dropped side by side converge because mass curves the space around Earth. Light feels this curvature too — passing a galaxy or black hole, its path bends (gravitational lensing), sometimes wrapping a distant quasar's image into a luminous Einstein ring. Whether gravity is best described as curved spacetime, a force, or the exchange of hypothetical gravitons remains, at the cosmological and quantum extremes, not fully understood.

Force problems reward careful vector bookkeeping and the inverse-square law; energy problems reward conservation of \(K + U\) with \(U = -GMm/r\); orbital problems hinge on Kepler's third law and \(E = -GMm/2a\). Useful constants: \(G = 6.67\times10^{-11}\,\mathrm{N\cdot m^{2}/kg^{2}}\), \(M_{E} = 5.98\times10^{24}\,\mathrm{kg}\), \(R_{E} = 6.37\times10^{6}\,\mathrm{m}\).