Table of Contents
Hooke's Law Explained
Hooke's Law, formulated by Robert Hooke in 1660, states that the force exerted by a spring is directly proportional to its displacement from equilibrium, as long as the elastic limit is not exceeded. This linear relationship (F = -kx) is the foundation of spring mechanics and applies to many elastic systems beyond simple coil springs, including rubber bands, bending beams, and even atomic bonds.
The spring constant k (measured in N/m) describes the stiffness of the spring. A stiffer spring has a higher k value and requires more force for the same displacement. The negative sign indicates that the spring force always opposes the displacement (restoring force).
Formulas
Spring Constant Examples
| Spring Type | k (N/m) |
|---|---|
| Slinky | 0.5-1 |
| Ballpoint pen spring | 50-100 |
| Car suspension spring | 10,000-50,000 |
| Trampoline spring | 3,000-5,000 |
| Watch spring | 0.01-0.1 |
| Garage door spring | 5,000-15,000 |
Spring-Mass Oscillation
When a mass is attached to a spring and displaced, it oscillates back and forth in simple harmonic motion (SHM). The period T = 2π√(m/k) depends only on mass and spring constant, not on amplitude. This makes spring-mass systems ideal timekeepers and is the principle behind mechanical watches.
- Amplitude: Maximum displacement from equilibrium position.
- Period: Time for one complete oscillation cycle.
- Frequency: Number of oscillations per second (f = 1/T).
- Angular frequency: ω = 2πf = √(k/m).
Frequently Asked Questions
What happens beyond the elastic limit?
Beyond the elastic limit, the spring deforms permanently (plastic deformation) and Hooke's Law no longer applies. The force-displacement relationship becomes nonlinear, and the spring will not return to its original shape when released. This is called yielding.
How do springs in series vs. parallel combine?
Springs in series: 1/k_total = 1/k1 + 1/k2 (softer overall). Springs in parallel: k_total = k1 + k2 (stiffer overall). This is opposite to how resistors combine in electrical circuits.
What is the natural frequency of a spring-mass system?
The natural frequency is f = (1/2π)√(k/m). A 500 N/m spring with a 1 kg mass oscillates at f = (1/2π)√(500) = 3.56 Hz, completing about 3.56 cycles per second.