Hooke's Law
A spring exerts a force proportional to how much it is stretched. This is called a Restoring Force.
\[F = -kx\]
- \(k\): Spring constant (stiffness).
- \(x\): Displacement from center.
The Period of a Spring
\[T = 2\pi \sqrt{\frac{m}{k}}\]
Worked Examples
Example 1: Finding the Period
A 2kg mass is on a spring with \(k = 50\). Find the period.
- \(T = 2\pi \sqrt{2/50} = 2\pi \sqrt{1/25} = 2\pi (1/5) = 1.25\) seconds.
Example 2: Frequency
Find the frequency of the spring in Example 1.
- \(f = 1/T = 1 / 1.25 = 0.8\) Hz (cycles per second).
The Bridge to Quantum Mechanics
Every molecule in your body—the DNA, the proteins, the water—is held together by chemical bonds that act exactly like tiny Springs. In Chapter 13, we will study the Quantum Harmonic Oscillator, which is the single most important model in all of physics. It describes how atoms in a crystal vibrate and how photons are created. The spring constant \(k\) determines the energy levels of the quantum system. Understanding springs is the key to understanding the "Solid State" of matter.