Interactive physics simulator
Frequency
Analyze the cycle rate of periodic motion. Experiment with vibrating tuning forks, spring-mass systems, and reference circles to study how stiffness, inertia, and rotational speed govern cycles per second.
Oscillatory Frequency & Wave Rate Laboratory
Modify variables on the right controller. Enable Audio in Sound Pitch mode to hear frequency pitch shifts. Align fan and strobe speeds in Stroboscope mode to observe rotational aliasing.
Live Frequency Telemetry
- System State
- Vibrating
- Cycles Counted
- 0 cycles
- Frequency (f)
- 0.00 Hz
- Period (T)
- 0.00 s
Understanding Frequency in Periodic Motion
In physics, frequency (denoted by f or the Greek letter ν) represents the rate at which a repetitive, periodic event occurs. It is defined as the number of complete cycles, oscillations, or vibrations performed per unit of time. The SI unit of frequency is the Hertz (Hz), named after the German physicist Heinrich Hertz. One Hertz is equivalent to one complete cycle per second (1 s-1).
Frequency is directly linked to the period (T), which is the duration of one complete cycle. Since they represent reciprocal properties, they are related by the formula: f = 1/T. In systems moving in Simple Harmonic Motion (SHM), the frequency is determined by the mechanical properties of the oscillator. For instance, in a spring-mass block, frequency depends on stiffness and inertia (f0 = 1/(2π) √(k/m)), meaning stiffer springs increase the frequency, while heavier masses reduce it. In wave phenomena such as sound, frequency corresponds to our perception of pitch: high-frequency sound waves create high-pitched notes, whereas low-frequency sound waves make deep, low-pitched notes.
Key Principles
Important properties governing frequency:
- Source Property: The frequency of a wave is determined entirely by its source and does not change when the wave transitions into a different medium.
- Stiffness and Inertia: Natural frequency increases with elastic restoring stiffness (k) and decreases with mass or rotational inertia (m).
- Pitch Perception: Sound frequency directly dictates audible pitch. Sound waves above 20 kHz (ultrasound) are beyond human hearing.
- Angular Velocity relation: Rotational frequency represents Revolutions Per Second (RPS), related to angular frequency by ω = 2πf.
Formulas & Definitions
Mathematical equations defining frequency:
- Reciprocal of Period: f = 1/T. Measures cycle rate from time duration.
- Relation to Angular Speed: f = ω / 2π. Translates rad/s into cycles/s.
- Spring-Mass System natural frequency: f = (1/2π) · √(k/m). Dictated by spring stiffness k and mass m.
- Wave Equation: f = v / λ. Frequency is speed v divided by wavelength λ.
- Pendulum natural frequency: f = (1/2π) · √(g/L). Governed strictly by length and gravity.
Solved Examples
The pendulum of a wall clock completes 30 full swings (back and forth) in exactly 60 seconds. Calculate the frequency of the pendulum in Hertz (Hz) and find its period (T).
- Identify the given values: number of cycles N = 30, total time t = 60 seconds.
- Recall the frequency formula: f = N / t.
- Substitute values: f = 30 / 60 = 0.5 Hz. This means the pendulum completes 0.5 cycles per second.
- To find the period, recall that period is the reciprocal of frequency: T = 1 / f.
- Substitute the frequency: T = 1 / 0.5 = 2.0 seconds.
- The frequency of the pendulum is 0.5 Hz, and its period is 2.0 seconds.
Answer: f = 0.5 Hz, T = 2.0 s
A 0.5 kg cart is anchored to a horizontal spring of constant k = 200 N/m. Calculate (a) the angular frequency ω of its free oscillations, and (b) its natural linear frequency f0.
- Identify the values: mass m = 0.5 kg, spring constant k = 200 N/m.
- Recall the angular frequency formula for a spring-mass oscillator: ω = √(k / m).
- Substitute values: ω = √(200 / 0.5) = √400 = 20.0 rad/s.
- Recall the relation between angular frequency and linear frequency: f0 = ω / (2π).
- Substitute ω: f0 = 20.0 / (2π) = 10.0 / π ≈ 3.183 Hz.
- The angular frequency is 20.0 rad/s, and the natural linear frequency is approximately 3.183 Hz.
Answer: ω = 20.0 rad/s, f0 ≈ 3.183 Hz
A sound wave propagates in air (speed of sound v = 343 m/s) with a wavelength of λ = 0.25 meters. Calculate the frequency of this sound wave and determine if it falls within the human hearing range.
- Identify variables: speed of sound v = 343 m/s, wavelength λ = 0.25 m.
- Recall the wave speed equation: v = f · λ.
- Rearrange the formula to solve for frequency: f = v / λ.
- Substitute values: f = 343 / 0.25 = 1372 Hz (or 1.372 kHz).
- Compare this with the typical human hearing range (20 Hz to 20,000 Hz). Since 1372 Hz is within this span, it is audible to humans.
- The wave frequency is 1372 Hz, which is an audible mid-range pitch.
Answer: f = 1372 Hz (Audible)
Common Mistakes
- Swapping Period and Frequency: Thinking a frequency of 2 Hz means a period of 2 seconds. Remember that T = 1/f, so T = 0.5 s.
- Changing Medium Frequency: Believing that sound pitch increases when sound travels from air to water. The wavelength and speed change, but frequency stays constant.
- Confusing Frequency and Amplitude: Assuming a louder sound or a larger spring bounce increases the frequency. Amplitude and frequency are independent in simple harmonic motion.
- Confusing angular frequency and linear frequency: Using f and ω interchangeably without multiplying/dividing by 2π.
Practice Questions
1. If the period of an oscillating mass is cut in half, how does its frequency change?
Frequency is the reciprocal of the period (f = 1/T). If the period T is halved, the frequency f becomes 1 / (T/2) = 2/T, which is exactly double the initial frequency. The oscillator now performs twice as many cycles in the same duration of time.
2. Why does tightening a guitar string increase the frequency of its vibration?
Tightening a guitar string increases its tension. In strings, the velocity of wave propagation is directly proportional to the square root of tension (v = √(Tension/μ)). Since the wavelength of the fundamental tone is fixed by the length of the string, raising the velocity directly increases the frequency (f = v/λ), resulting in a higher pitch.
3. Explain the physical difference between linear frequency (f) and angular frequency (ω).
Linear frequency (f) measures the number of complete, 360-degree cycles performed per second, expressed in Hertz (Hz or 1/s). Angular frequency (ω) represents the rate of phase change over time measured in radians per second (rad/s). Because one full cycle corresponds to 2π radians, the angular frequency is 2π times larger than the linear frequency: ω = 2πf.
4. What parameters dictate the natural frequency of a simple pendulum? How does bob mass affect it?
The natural frequency of a simple pendulum is given by f = (1 / 2π) · √(g/L). It depends entirely on the string length (L) and local gravitational acceleration (g). Changing the mass of the bob has no effect on the frequency, because gravity acts equally on all masses, causing them to accelerate at the same rate.
FAQ
Frequently Asked Questions
What is frequency in physics?
Frequency (f) is the number of complete cycles, oscillations, or revolutions that an object or wave undergoes per unit of time.
What is the unit of frequency?
The standard SI unit of frequency is the Hertz (Hz), named after physicist Heinrich Hertz. One Hertz is equal to one cycle per second (1/s).
What is the relation between frequency and period?
Frequency and period are reciprocals: f = 1/T and T = 1/f. A shorter period corresponds to a higher frequency.
What are infrasound and ultrasound?
Infrasound refers to sound frequencies below 20 Hz, which are below the threshold of human hearing. Ultrasound refers to sound frequencies above 20,000 Hz (20 kHz), which are too high-pitched for humans to hear.
What determines the frequency of a mass-spring system?
The natural frequency is determined by the mass (m) and the stiffness of the spring (spring constant k): f = (1/2π)√(k/m). Increasing stiffness raises the frequency, while increasing mass lowers it.
What is angular frequency?
Angular frequency (ω), measured in radians per second (rad/s), represents the rate of angular rotation. It is related to linear frequency by ω = 2πf.
Does wave speed affect the frequency emitted by a source?
No. The frequency of a wave is determined solely by its source. When a wave travels from one medium to another (e.g., sound moving from air to water), its speed and wavelength change, but its frequency remains constant.