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Period Calculator

Calculate the period of oscillation or rotation

Input Parameters

From FrequencyFrom Wavelength
Hz

The period (T) of a wave is the time it takes to complete one full cycle or oscillation. [1, 5, 8, 12] It is a fundamental property of periodic motion and is measured in units of time, such as seconds (s). [1, 7]

  • From Frequency (f): The period is the reciprocal of the frequency. [1, 3, 6, 14] A high frequency corresponds to a short period, and vice-versa. [9, 10, 11]
  • From Wavelength (λ): The period can be found by dividing the wavelength by the wave speed. [21, 22]

T = 1 / f

T = λ / v

Enter parameters and click Calculate

About Period Calculator

The Rhythm of Motion: The Ultimate Guide to Our Period Calculator

The universe is filled with cycles, rhythms, and repetitions. The Earth completes one rotation on its axis, giving us a day. A clock's pendulum swings back and forth with reassuring regularity. A plucked guitar string vibrates to create a steady musical note. Each of these repeating events can be characterized by a fundamental quantity that describes the duration of a single, complete cycle. This quantity is known as the **Period**.

The period is the cornerstone of understanding periodic motion. It is the answer to the question, "How long does one full cycle take?" While its counterpart, frequency, tells us "how often" something happens, the period gives us a tangible measure of time—the fundamental building block of any oscillation or rotation.

Welcome to the definitive guide to this essential concept. Our Period Calculator is a versatile tool designed to help you navigate the simple but powerful relationships between period, frequency, and the properties of oscillating systems like pendulums and springs. This comprehensive article will not only guide you through the calculator's use but will also take you on a journey through the physics of periodic motion, revealing the simple math that governs the timing of our world.

What is the Period? Measuring a Single Cycle

In physics, the **Period (T)** is defined as the time required to complete one full cycle of a repeating event. It is a measure of the duration of an oscillation or rotation.

Consider a spinning carousel. The time it takes for the carousel to make one complete 360-degree rotation is its period. For a bouncing spring, the time it takes to go from its highest point, down to its lowest, and back up to its highest point again is its period.

The standard unit of the period in the International System of Units (SI) is the **second (s)**.

The Fundamental Relationships: The Heart of the Calculator

Our calculator is built on the simple, elegant, and powerful relationships that connect the period to other key properties of oscillations and waves.

1. Period and Frequency: An Inverse Relationship

The period is most famously linked to **Frequency (f)**. They are two sides of the same coin, describing the same motion from different perspectives.

- **Period (T)** asks: "How many seconds does one cycle take?"
- **Frequency (f)** asks: "How many cycles happen per second?"

They are perfect inverses of each other:

T = 1 / f   and   f = 1 / T

An event with a long period has a low frequency. An event with a short period has a high frequency. This is the most basic way to determine the period of any repeating phenomenon.

2. Period of a Simple Pendulum

For a simple pendulum (a mass suspended by a light string), the period of its swing (for small angles) depends on only two factors: the length of the string and the strength of gravity.

T = 2π * √(L / g)
  • L is the **length** of the pendulum in meters (m).
  • g is the **acceleration due to gravity** (≈ 9.81 m/s² on Earth).

Remarkably, the period of a simple pendulum does *not* depend on the mass of the bob or the amplitude of the swing (as long as the swing is small). A heavier bob on the same string will have the same period as a lighter one.

3. Period of a Mass-Spring System

For a mass attached to an ideal spring and oscillating horizontally or vertically, the period of its oscillation depends on the mass and the stiffness of the spring.

T = 2π * √(m / k)
  • m is the **mass** of the oscillating object in kilograms (kg).
  • k is the **spring constant** in Newtons per meter (N/m), which measures the spring's stiffness.

A larger mass will oscillate more slowly (longer period), while a stiffer spring will cause the mass to oscillate more rapidly (shorter period).

How to Use the Period Calculator

1. Choose the Calculation Method

Select the system you are analyzing:

  • From Frequency: The most general method.
  • Simple Pendulum: For a swinging mass on a string.
  • Mass-Spring System: For an oscillating spring.

2. Enter the Known Variables

- For **Frequency**, enter the frequency f in Hertz (Hz).
- For a **Simple Pendulum**, enter the length L in meters and optionally adjust the gravitational acceleration g.
- For a **Mass-Spring System**, enter the mass m in kilograms and the spring constant k in N/m.

3. View the Result

The calculator will instantly compute the period T in seconds.

Worked Examples: The Timing of Oscillations

Example 1: A Grandfather Clock

The pendulum of a grandfather clock is designed to have a period of exactly 2 seconds (1 second to swing left, 1 second to swing right). What is the required length of this pendulum on Earth?

  • • Knowns: T = 2 s, g = 9.81 m/s².
  • • Goal: Find Length (L).
  • • Formula: Start with T = 2π * √(L / g) and rearrange to solve for L.
    T² = (2π)² * (L / g) => L = g * T² / (2π)².
  • • Calculation: L = 9.81 * (2)² / (4π²) ≈ 9.81 * 4 / 39.48 ≈ 0.994 meters.
  • • Result: The pendulum needs to be almost exactly one meter long. This is why grandfather clocks are so tall!

Example 2: A Car's Suspension

A corner of a 1600 kg car is supported by a spring that has a spring constant of 50,000 N/m. If this corner of the car hits a bump and starts to oscillate, what is the period of the oscillation? (Assume the corner's effective mass is 1/4 of the total, so 400 kg).

  • • Inputs: m = 400 kg, k = 50,000 N/m.
  • • Formula: T = 2π * √(m / k)
  • • Calculation: T = 2π * √(400 / 50000) = 2π * √(0.008) ≈ 2π * 0.0894 ≈ 0.56 seconds.
  • • Result: The car's suspension will naturally oscillate with a period of about 0.56 seconds. Shock absorbers are added to "damp" this oscillation quickly so the car doesn't keep bouncing.

Frequently Asked Questions (FAQ)

Q: Does the amplitude (how far something swings or bounces) affect the period?

For an ideal simple pendulum and an ideal mass-spring system, no. The period is independent of the amplitude. This property, known as isochronism, is what makes these systems excellent timekeepers. A pendulum with a large swing takes the same time to complete a cycle as one with a small swing. In reality, for very large amplitudes, this breaks down slightly, but for most practical purposes, it holds true.

Q: How would I measure the period of a real-world object?

The most accurate way is to time multiple cycles. For example, use a stopwatch to measure the time it takes for a swing to complete 10 full cycles. Then, divide the total time by 10. This minimizes the human error associated with starting and stopping the watch for a single, short cycle.

Q: How is the period of a wave related to its wavelength?

They are not directly related. The period (T) and frequency (f) describe the *temporal* behavior of a wave (what happens at one point over time). The wavelength (λ) describes the *spatial* behavior (a snapshot of the wave at one instant in time). They are linked through the wave speed v by the universal wave equation v = fλ. You can substitute f = 1/T to get v = λ/T, or T = λ/v.

The Master Clock of Physics

The period is a simple yet profound concept that forms the basis of timekeeping and the analysis of all periodic systems. It allows us to predict the behavior of pendulums, the oscillations of structures, the rhythm of waves, and the orbits of planets.

Use our Period Calculator to explore these fundamental relationships. Design a pendulum for a specific period, analyze the oscillation of a spring, or simply convert between frequency and period to build your intuition. By understanding the timing of a single cycle, you gain the key to understanding the rhythm of the entire system.

Frequently Asked Questions

What is period in physics?
Period (T) is the time taken for one complete cycle of oscillation or rotation in a periodic motion.
How do I calculate the period of a simple pendulum?
For a simple pendulum: T = 2π√(L/g), where L is length and g is gravitational acceleration (9.81 m/s²).
What's the period formula for a mass-spring system?
For a mass-spring system: T = 2π√(m/k), where m is mass and k is spring constant.
How do I calculate orbital period?
Kepler's Third Law gives orbital period: T² = (4π²r³)/(GM), where r is orbital radius and M is central mass.
What units should I use for period calculations?
Use consistent SI units: seconds (s) for time, meters (m) for length, kilograms (kg) for mass.
Does pendulum mass affect its period?
For simple pendulums, no - period depends only on length and g. For physical pendulums, mass distribution matters.
How does amplitude affect period?
For simple harmonic motion (small angles/displacements), period is amplitude-independent. For large amplitudes, period increases slightly.
What's the period of a circular orbit?
T = 2π√(r³/GM) where r is orbit radius, G is gravitational constant, M is central mass.
How do I calculate rotational period?
Rotational period T = 2π/ω, where ω is angular velocity in rad/s.
What's the difference between period and frequency?
Period (T) is time per cycle, frequency (f) is cycles per time. They're reciprocals: f = 1/T.
How does string length affect pendulum period?
Period increases with the square root of length: T ∝ √L. Double length → period increases by √2 (≈1.414) times.
What's the period of Earth's rotation?
Earth's sidereal rotation period (relative to stars) is about 23h 56m 4s (86164 seconds).
How do I calculate period from angular frequency?
T = 2π/ω, where ω is angular frequency in radians/second.
What's the period of a vertical spring-mass system?
Same as horizontal: T = 2π√(m/k). Gravity affects equilibrium position but not period.
How does gravity affect pendulum period?
Period increases as gravity decreases: T ∝ 1/√g. On Moon (g/6), period would be √6 ≈ 2.45 times longer.
What's the period of a physical pendulum?
T = 2π√(I/mgd), where I is moment of inertia about pivot, d is distance from pivot to center of mass.
How do I calculate period from frequency?
T = 1/f. Example: 50 Hz → T = 1/50 = 0.02 seconds.
What's the period of a torsional pendulum?
T = 2π√(I/κ), where I is moment of inertia and κ is torsion constant of the wire.
How does mass affect spring period?
Period increases with mass: T ∝ √m. Quadruple mass → double period.
What's the period of a conical pendulum?
T = 2π√(Lcosθ/g), where L is string length and θ is angle from vertical.
How do I calculate period from RPM?
T = 60/RPM (converts revolutions per minute to seconds per revolution).
What's the period of a compound pendulum?
Same as physical pendulum: T = 2π√(I/mgd), accounting for object's mass distribution.
How does spring constant affect period?
Period decreases as spring constant increases: T ∝ 1/√k. Stiffer springs oscillate faster.
What's the period of a satellite in low Earth orbit?
About 90 minutes (5400s) for typical LEO altitudes (200-2000km).
How do I calculate period from wave speed and wavelength?
T = λ/v, where λ is wavelength and v is wave speed.
What's the period of a bifilar pendulum?
T = 2π√(d²/(2gL)), where d is separation between suspension points, L is suspension length.
How does temperature affect pendulum period?
Temperature changes affect length (thermal expansion) which changes period: T ∝ √L.
What's the period of a circular hoop swinging as a pendulum?
T = 2π√(2r/g), where r is hoop radius (special case of physical pendulum).
How do I calculate period for coupled oscillators?
Depends on coupling - normal modes have periods T = 2π/ω where ω depends on masses and coupling strength.
What's the period of a Foucault pendulum?
Same as simple pendulum (T = 2π√(L/g)) but plane of oscillation rotates due to Earth's rotation.
How does air resistance affect period?
For damped oscillations, period increases slightly: T ≈ T₀(1 + (b/2mω₀)²) where b is damping coefficient.
What's the period of a double pendulum?
Chaotic system with no simple formula - depends on initial conditions and exhibits complex motion.
How do I calculate period for a liquid in a U-tube?
T = 2π√(L/2g), where L is total length of liquid column (acts like a pendulum).
What's the period of a bar pendulum?
T = 2π√(2L/3g) for uniform rod of length L pivoted at one end.
How does amplitude affect physical pendulum period?
For small angles, period is amplitude-independent. For large angles, correction factors apply.
What's the period of a mercury pendulum?
Same as simple pendulum if bob is treated as point mass, but thermal effects may cause variations.
How do I calculate period for a spring in series?
For springs with constants k₁ and k₂ in series: T = 2π√(m(k₁+k₂)/(k₁k₂)).
What's the period of a Wilberforce pendulum?
Combines torsional and vertical oscillations with two characteristic periods that interact.
How does latitude affect pendulum period?
Effective g changes with latitude (Earth's rotation) causing small period variations (≈0.3% equator to pole).
What's the period of a magnetic dipole in a field?
T = 2π√(I/μB), where I is moment of inertia, μ is dipole moment, B is field strength.
How do I calculate period for a spring in parallel?
For parallel springs: T = 2π√(m/(k₁+k₂)). Combined spring constant is sum of individual constants.
What's the period of a charged particle in magnetic field?
Cyclotron period: T = 2πm/(qB), where m is mass, q is charge, B is magnetic field strength.
How does altitude affect pendulum period?
Higher altitude → lower g → longer period (T ∝ 1/√g). At 10km altitude, period increases ≈0.16%.
What's the period of a floating cylinder bobbing in water?
T = 2π√(h/g), where h is submerged length at equilibrium (similar to simple pendulum).
How do I calculate period for a driven harmonic oscillator?
At resonance, period matches natural period T₀ = 2π/ω₀. Off-resonance, system oscillates at driving frequency.
What's the period of an LC circuit?
T = 2π√(LC), where L is inductance and C is capacitance (electrical analogue of mass-spring system).
How does non-uniform gravity affect pendulum period?
For very long pendulums where g varies significantly with height, period calculations require integration.
What's the period of a pulsating star?
Depends on stellar properties - Cepheid variables have periods (1-70 days) related to luminosity.
How do I calculate period for small oscillations about equilibrium?
For any potential well: T ≈ 2π√(m/k_eff) where k_eff is second derivative of potential at minimum.
What's the period of a quantum harmonic oscillator?
Energy levels are quantized but classical period formula T = 2π/ω still applies to expectation values.
How does relativistic speed affect orbital period?
At relativistic speeds, observed period changes due to time dilation (T_obs = γT_proper where γ is Lorentz factor).
What's the period of a binary star system?
T² = (4π²a³)/(G(m₁+m₂)) where a is semi-major axis, m₁ and m₂ are stellar masses (Kepler's Third Law).

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