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Frequency of Periodic Motion

Frequency of Periodic Motion

The number of cycles completed per second. Frequency is the reciprocal of period. Note: The unit representing cycles per second is hertz.

 

 

See also

Periodic motion, simple harmonic motion, frequency of a periodic function

Key Formula

f=1Tf = \frac{1}{T}
Where:
  • ff = Frequency, measured in hertz (Hz)
  • TT = Period, the time for one complete cycle, measured in seconds (s)

Worked Example

Problem: A pendulum swings back and forth, completing one full swing every 0.5 seconds. What is the frequency of the pendulum's motion?
Step 1: Identify the period. The pendulum completes one full cycle in 0.5 seconds, so the period is 0.5 s.
T=0.5 sT = 0.5 \text{ s}
Step 2: Apply the frequency formula. Frequency is the reciprocal of the period.
f=1T=10.5f = \frac{1}{T} = \frac{1}{0.5}
Step 3: Calculate the result.
f=2 Hzf = 2 \text{ Hz}
Answer: The pendulum has a frequency of 2 Hz, meaning it completes 2 full cycles every second.

Another Example

Problem: A vibrating guitar string completes 440 cycles in one second. What is its frequency, and what is its period?
Step 1: Determine the frequency directly from the given information. The string completes 440 cycles per second.
f=440 Hzf = 440 \text{ Hz}
Step 2: Find the period by taking the reciprocal of the frequency.
T=1f=1440T = \frac{1}{f} = \frac{1}{440}
Step 3: Calculate the period.
T≈0.00227 sT \approx 0.00227 \text{ s}
Answer: The guitar string vibrates at 440 Hz (the musical note A), with each cycle taking approximately 0.00227 seconds.

Frequently Asked Questions

What is the difference between frequency and period?
Frequency and period are reciprocals of each other. Period (TT) tells you how long one cycle takes in seconds, while frequency (ff) tells you how many cycles fit into one second. If the period is large (slow motion), the frequency is small, and vice versa. They are related by f=1/Tf = 1/T.
What are the units of frequency?
Frequency is measured in hertz (Hz). One hertz equals one cycle per second. For very fast oscillations, you may see kilohertz (kHz = 1,000 Hz), megahertz (MHz = 1,000,000 Hz), or gigahertz (GHz = 1,000,000,000 Hz).

Frequency vs. Period

Frequency counts how many cycles occur per second, while period measures how many seconds one cycle takes. They are mathematical inverses: f=1/Tf = 1/T and T=1/fT = 1/f. A high frequency means a short period, and a low frequency means a long period. For example, a motion with a period of 0.25 s has a frequency of 4 Hz.

Why It Matters

Frequency is central to understanding sound, light, and all wave phenomena. The pitch of a sound is determined by its frequency — higher frequency means a higher-pitched note. In electronics, radio stations are identified by their broadcast frequency, and computer processor speeds are measured in gigahertz, reflecting billions of cycles per second.

Common Mistakes

Mistake: Confusing frequency and period by writing f=Tf = T instead of f=1/Tf = 1/T.
Correction: Remember that frequency and period are reciprocals, not equal. If the period is 2 seconds, the frequency is 1/2=0.51/2 = 0.5 Hz, not 2 Hz.
Mistake: Counting half-cycles instead of full cycles when determining frequency.
Correction: A complete cycle means the object returns to its starting position and direction. For a pendulum, one full cycle is a swing to the right AND back to the left, returning to where it started. Counting only the swing in one direction gives you half a cycle.

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