Spring-Mass System
A spring-mass system consists of a mass attached to an ideal spring that obeys Hooke's Law, where the restoring force is proportional to the displacement from equilibrium. When displaced and released, the mass oscillates in simple harmonic motion with a period that depends on the mass and the spring constant, but not on amplitude. This system is the canonical model for oscillatory behavior in physics and engineering.
Key Formula
T = 2π × √(m / k), F = −kx (Hooke's Law)
LaTeX: T = 2\pi\sqrt{\frac{m}{k}}, \quad F = -kx
| Symbol | Meaning | Unit |
|---|---|---|
| T | Period of oscillation | s |
| m | Mass attached to spring | kg |
| k | Spring constant (stiffness) | N/m |
| F | Restoring force | N |
| x | Displacement from equilibrium | m |
Worked Example
Problem
A mass of 0.50 kg is attached to a spring with spring constant k = 200 N/m. It is displaced 0.08 m from equilibrium and released. Find the period, frequency, and maximum speed.
Solution
Step 1: Period — T = 2π√(m/k) = 2π√(0.50/200) = 2π√(0.0025) = 2π × 0.05 = 0.314 s. Step 2: Frequency — f = 1/T = 1/0.314 ≈ 3.18 Hz. Step 3: Angular frequency — ω = √(k/m) = √(200/0.50) = √400 = 20 rad/s. Step 4: Maximum speed — vmax = Aω = 0.08 × 20 = 1.6 m/s.
Answer
T ≈ 0.314 s, f ≈ 3.18 Hz, vmax = 1.6 m/s.
Effect of mass and spring constant on the period of a spring-mass system
| Mass m (kg) | Spring Constant k (N/m) | Period T (s) | Frequency f (Hz) | Stiffer or Heavier? |
|---|---|---|---|---|
| 0.50 | 200 | 0.314 | 3.18 | Reference |
| 1.00 | 200 | 0.444 | 2.25 | Heavier mass |
| 0.50 | 800 | 0.157 | 6.37 | Stiffer spring |
| 2.00 | 200 | 0.628 | 1.59 | Much heavier |
| 0.50 | 50 | 0.628 | 1.59 | Softer spring |
| 0.25 | 400 | 0.157 | 6.37 | Light + stiff |
Interactive Tools
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Related Terms
Simple Harmonic Motion
Simple Harmonic Motion (SHM) is a type of periodic oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and always directed toward it. The motion follows a sinusoidal pattern over time, characterized by constant amplitude, frequency, and period in the absence of damping. SHM is the basis for understanding pendulums, springs, sound waves, and alternating electric circuits.
Mechanical Resonance
Mechanical resonance occurs when an oscillating system is driven at its natural frequency, causing the amplitude of oscillation to grow dramatically — theoretically without bound in an undamped system. Every mechanical system has one or more natural frequencies at which it vibrates freely after being disturbed. Resonance is critical in engineering design, as it can cause catastrophic structural failure (Tacoma Narrows Bridge, 1940) or be harnessed usefully in musical instruments, clocks, and sensors.
Pendulum
A pendulum is a mass (called a bob) suspended from a fixed point by a string or rod that oscillates back and forth under the influence of gravity. For small angular displacements (less than about 15°), a simple pendulum exhibits simple harmonic motion, and its period depends only on its length and the local gravitational acceleration, not on mass or amplitude. Pendulums have historically been used in clocks and are fundamental to understanding oscillatory systems.
Named after Robert Hooke (1635–1703), who formulated the spring law "ut tensio, sic vis" (as the extension, so the force) in 1676, published in 1678 as "De Potentia Restitutiva." The word "spring" derives from Old English "springan" (to leap or burst forth).