Forces toward the Center
A Central Force always points toward a fixed origin and its strength depends only on the distance \(r\). Gravity and the Electric Force are both central forces:
\[F(r) = \frac{k}{r^2}\]
These forces always conserve Angular Momentum (\(\vec{L}\)), which means the motion always stays in a flat plane.
Worked Examples
Example 1: Kepler's Laws
Using calculus on the central force equation, we can prove that planets move in ellipses and that they "sweep out equal areas in equal times." This second law is a direct consequence of the conservation of angular momentum.
The Bridge to Quantum Mechanics
The Bohr Model of the atom was the first attempt to treat the electron as a planet orbiting the nucleus under a central force. While we now know electrons aren't "little planets," the conservation of angular momentum remains a fundamental law. In Quantum Mechanics, angular momentum is quantized (\(L = \sqrt{l(l+1)}\hbar\)). This is the reason why electron orbitals have specific shapes (s, p, d, f)—they are the quantum versions of the orbits allowed by central forces.