〰️Introduction to Oscillations
Periodic Motion
Periodic motion is motion that repeats at regular time intervals. Simple Harmonic Motion (SHM) is the most important type of periodic motion.
SHM occurs when the restoring force is proportional to the displacement from equilibrium and directed back toward equilibrium:
F_restore = −kx
This negative sign is crucial: the force always acts opposite to displacement, pulling/pushing the object back toward the center. This is what makes the motion oscillate.
Examples: mass on a spring, pendulum (small angles), tuning forks, atoms in a crystal.
Key Vocabulary
Amplitude (A): maximum displacement from equilibrium. Larger amplitude = larger oscillations.
Period (T): time for one complete oscillation (s). Doesn't depend on amplitude (for ideal SHM)!
Frequency (f): oscillations per second (Hz = 1/s). f = 1/T.
Angular frequency (ω): ω = 2πf = 2π/T (rad/s)
The position of an object in SHM can be described by: x(t) = A cos(2πt/T) = A cos(ωt)
The velocity and acceleration follow: v(t) = −Aω sin(ωt) a(t) = −Aω² cos(ωt) = −ω²x
Think About It
If you double the amplitude of a mass-spring oscillation, how does the period change? What about the maximum speed?
📐 Key Equations
Simple Harmonic Motion fundamentals
x(t) = A\cos(ω t)T = (1)/(f) = (2π)/(ω)v_max = Aωa_max = Aω²⚠️ Common Mistakes
Misconception: Thinking that a larger amplitude means a longer period.
✓ Correct thinking: For ideal SHM, the period is completely independent of amplitude. A bigger swing covers more distance but at proportionally higher speed — the time cancels out.
Why: T = 2π/ω depends only on the system's physical properties (m and k for springs, L and g for pendulums), never on how far you pull it.
Misconception: Confusing frequency (f, in Hz) with angular frequency (ω, in rad/s).
✓ Correct thinking: They are related by ω = 2πf. Use f when counting oscillations per second; use ω in the position/velocity/acceleration equations x(t) = A cos(ωt).
Why: The factor of 2π comes from converting one full cycle (360° or 2π radians) into the angular measure that appears in the cosine function.
Misconception: Assuming the object is always at maximum speed at t = 0.
✓ Correct thinking: At t = 0, x(t) = A cos(0) = A, meaning the object starts at maximum displacement with zero velocity. Maximum speed occurs when x = 0 (equilibrium).
Why: v(t) = −Aω sin(ωt). At t = 0, sin(0) = 0 so v = 0. Maximum speed occurs when sin(ωt) = ±1, i.e., when x = 0.
📝 Practice Problems
Try these problems. Check your answer when ready.
An oscillator has a period of 0.4 s. What is its frequency and angular frequency?
An object in SHM has amplitude 0.3 m and angular frequency 5 rad/s. What is its maximum speed?
A mass oscillates with x(t) = 0.2 cos(4t) meters. Find: (a) amplitude, (b) angular frequency, (c) period, (d) maximum acceleration.
If the amplitude of an SHM oscillator is doubled, how does the maximum speed change? How does the period change?
An object in SHM has position x(t) = 0.5 cos(3πt) m. At what times during the first period does the object first reach x = 0.25 m?
x = A\cos(ω t)Finished reading through this lesson?