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Periodic Motion

Periodic Motion

Motion that repeats itself identically over and over, such as the swinging of a pendulum. If the motion can be modeled using a sinusoid it is called simple harmonic motion.

 

 

See also

Period of periodic motion, frequency of periodic motion , periodic function

Key Formula

x(t)=Asin⁡(2πf t+ϕ)x(t) = A\sin(2\pi f\, t + \phi)
Where:
  • x(t)x(t) = Position at time t
  • AA = Amplitude — the maximum displacement from the rest position
  • ff = Frequency — the number of complete cycles per second (in hertz)
  • tt = Time (in seconds)
  • ϕ\phi = Phase shift — determines where in its cycle the motion starts

Worked Example

Problem: A mass on a spring oscillates back and forth with simple harmonic motion. Its position is modeled by x(t) = 5 sin(2π · 3t), where x is in centimeters and t is in seconds. Find the amplitude, frequency, period, and position at t = 0.25 s.
Identify the amplitude: The amplitude is the coefficient in front of the sine function.
A=5 cmA = 5 \text{ cm}
Identify the frequency: The frequency appears as the coefficient of t inside the argument 2πft.
f=3 Hz (3 cycles per second)f = 3 \text{ Hz (3 cycles per second)}
Find the period: The period is the reciprocal of the frequency — the time for one complete cycle.
T=1f=13≈0.333 sT = \frac{1}{f} = \frac{1}{3} \approx 0.333 \text{ s}
Find the position at t = 0.25 s: Substitute t = 0.25 into the position function and evaluate.
x(0.25)=5sin⁡(2π⋅3⋅0.25)=5sin⁡(1.5π)=5(−1)=−5 cmx(0.25) = 5\sin(2\pi \cdot 3 \cdot 0.25) = 5\sin(1.5\pi) = 5(-1) = -5 \text{ cm}
Answer: The amplitude is 5 cm, the frequency is 3 Hz, the period is 1/3 s, and at t = 0.25 s the mass is at −5 cm (maximum displacement in the negative direction).

Frequently Asked Questions

What is the difference between periodic motion and simple harmonic motion?
Periodic motion is any motion that repeats at regular time intervals — it can follow any repeating pattern. Simple harmonic motion (SHM) is a special type of periodic motion where the restoring force is proportional to displacement, producing a sinusoidal (sine or cosine) pattern. All SHM is periodic, but not all periodic motion is SHM. For example, a heartbeat is periodic but not sinusoidal.
How do you find the period of periodic motion?
The period T is the time it takes for one complete cycle of the motion. You can measure it directly by timing one full repetition. If you know the frequency f, use T = 1/f. For instance, if something oscillates at 4 Hz, its period is 1/4 = 0.25 seconds.

Periodic motion vs. Simple harmonic motion

Periodic motion is the broader category: any motion that repeats at fixed time intervals. Simple harmonic motion is a specific subset where the motion traces a sinusoidal curve. A bouncing ball on a flat surface can be periodic (it repeats), but its position-vs-time graph is not a smooth sine wave, so it is not simple harmonic motion. A mass on an ideal spring, by contrast, moves sinusoidally and qualifies as both periodic and simple harmonic.

Why It Matters

Periodic motion is central to understanding waves, sound, light, and alternating current (AC) electricity. Engineers rely on it to design clocks, musical instruments, and electronic circuits. Recognizing that a real-world process is periodic lets you use powerful mathematical tools — especially sine and cosine functions — to predict its future behavior.

Common Mistakes

Mistake: Confusing period and frequency, or using the wrong units.
Correction: Period (T) is measured in seconds and tells you the time per cycle. Frequency (f) is measured in hertz (cycles per second). They are reciprocals: T = 1/f. Always check which one the problem asks for.
Mistake: Assuming all periodic motion is simple harmonic motion.
Correction: Simple harmonic motion is only one type of periodic motion — specifically the kind described by a sine or cosine function. Many real-world periodic motions (e.g., a square-wave signal or an irregular heartbeat) repeat regularly but are not sinusoidal.

Related Terms