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Harmonic Motion and Trigonometric Models

Trigonometry25 minDifficulty4 out of 10

You should know: trigonometric graphs

Overview

Simple harmonic motion (SHM) is the repetitive back-and-forth motion of a system — a mass on a spring, a pendulum swinging through small angles, a vibrating guitar string — whose displacement over time is exactly a sinusoidal function. The position is modeled as x(t) = A cos(ωt + φ), where A is the amplitude (maximum displacement), ω is the angular frequency (radians per unit time), and φ is the phase constant that fixes where in the cycle the motion starts. This model arises directly from Newton's second law applied to a restoring force proportional to displacement (F = −kx), and the same trigonometric machinery used to analyze the graphs of sine and cosine — period, amplitude, phase shift — describes velocity, acceleration, energy, and frequency of the oscillating system.

Intuition

SHM is just the graph of cosine reinterpreted: instead of x being a horizontal axis variable, it is time t, and the vertical output is a physical displacement instead of an abstract y-value. Everything learned about graphing y = A cos(Bx+C) transfers directly — amplitude A is the maximum swing of the oscillator, angular frequency ω plays the role of B and compresses or stretches the cycle in time, and phase φ shifts where the clock 'starts' relative to the peak. The restoring-force condition a(t) = −ω²x(t) is what guarantees the motion is exactly sinusoidal in the first place: whenever acceleration is proportional to negative displacement, the solution to that differential equation is a sum of sine and cosine at angular frequency ω, which combines into the single shifted cosine A cos(ωt+φ).

Interactive Graph

x(t) = A cos(ωt + φ) — simple harmonic motion; drag sliders to change amplitude, angular frequency, and phase

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Formal Definition

Definition

For simple harmonic motion with amplitude A, angular frequency ω, and phase constant φ:

x(t)=Acos⁡(ωt+φ)x(t) = A\cos(\omega t + \varphi)
Position
v(t)=dxdt=−Aωsin⁡(ωt+φ)v(t) = \frac{dx}{dt} = -A\omega\sin(\omega t + \varphi)
Velocity
a(t)=d2xdt2=−Aω2cos⁡(ωt+φ)=−ω2x(t)a(t) = \frac{d^2x}{dt^2} = -A\omega^2\cos(\omega t + \varphi) = -\omega^2 x(t)
Acceleration (restoring)
T=2πω,f=1T=ω2πT = \frac{2\pi}{\omega}, \qquad f = \frac{1}{T} = \frac{\omega}{2\pi}
Period and frequency

Worked Examples

  1. Substitute t = π/6 into x(t) = 5cos(2t).

    x ⁣(π6)=5cos⁡ ⁣(2⋅π6)=5cos⁡ ⁣(π3)=5⋅12=2.5x\!\left(\frac{\pi}{6}\right) = 5\cos\!\left(2\cdot\frac{\pi}{6}\right) = 5\cos\!\left(\frac{\pi}{3}\right) = 5\cdot\frac{1}{2} = 2.5

  2. Period T = 2π/ω with ω = 2.

    T=2π2=π secondsT = \frac{2\pi}{2} = \pi \text{ seconds}

Answer: x(π/6) = 2.5 cm; period T = π seconds ≈ 3.1416 s.

Practice Problems

Difficulty 3/10

For x(t) = 5cos(2t), find the maximum speed of the oscillator.

Difficulty 4/10

A pendulum's angular displacement is modeled by θ(t) = 0.2cos(3t + π/4) radians. Find the phase constant, angular frequency, and the period of oscillation.

Difficulty 6/10

A 0.1 kg mass on a spring with spring constant k = 40 N/m oscillates with amplitude 0.1 m. Using ω = √(k/m), find the angular frequency and the ordinary frequency (in Hz) of oscillation.

Quiz

In x(t) = A cos(ωt + φ), the parameter ω controls:
The defining differential-equation condition for simple harmonic motion is:
For x(t) = 5cos(2t), the maximum speed of the oscillator is:

Summary

  • Simple harmonic motion is modeled by x(t) = A cos(ωt+φ), directly analogous to graphing y = A cos(Bx+C).
  • Velocity and acceleration follow by differentiating; acceleration satisfies a(t) = −ω²x(t), the restoring-force condition that produces sinusoidal motion.
  • Period T = 2π/ω and frequency f = ω/(2π) translate the graphing parameters into physical quantities for springs, pendulums, and waves.

References

  1. WebsiteWikipedia — Simple harmonic motion