A pendulum with a 2.5-second period needs to be about 1.55 meters long on Earth.
Simple Harmonic Motion Formula & Guide
T = 2π √(L / g)
Period of a simple pendulum
Where T is the period, L is the pendulum length, and g is the gravitational acceleration (typically 9.81 m/s² on Earth).
T = 2π √(m / k)
Period of a spring-mass system
Where T is the period, m is the mass, and k is the spring constant (stiffness).
L = gT² / 4π²
Pendulum length from period
Rearranged from T = 2π√(L/g). Use this to find the required pendulum length for a desired period.
f = 1/T · ω = 2πf = 2π/T
Frequency and angular frequency
The frequency (f) is the number of oscillations per second (Hz). The angular frequency (ω) is the rate of change of the angular displacement (rad/s).
Key Concepts
📌 What is Simple Harmonic Motion?
Simple Harmonic Motion (SHM) is a type of oscillatory motion where the restoring force is proportional to the displacement from equilibrium. Examples include pendulums (for small angles), spring-mass systems, and vibrating strings. The motion is periodic and sinusoidal.
📌 Period vs. Frequency
The period (T) is the time taken for one complete oscillation, measured in seconds. Frequency (f) is the number of oscillations per second, measured in Hertz (Hz). They are inversely related: f = 1/T.
📌 Small Angle Approximation
The pendulum formula T = 2π√(L/g) assumes small angular displacements (typically < 15°). For larger angles, the period becomes slightly longer, and a more complex formula involving elliptic integrals is required.
📌 Spring Constant
The spring constant (k) measures the stiffness of a spring. A stiffer spring (higher k) produces a shorter period. The units are Newtons per meter (N/m). For a spring-mass system, period is independent of gravitational acceleration.
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Pendulum Period
Calculate the period of a simple pendulum using T = 2π√(L/g). Supports length in meters, centimeters, and feet with selectable gravity values for Earth, Moon, or Mars.
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Spring-Mass Period
Calculate the oscillation period of a spring-mass system using T = 2π√(m/k). Supports mass in kg, g, or lb and spring constant in N/m or N/cm.
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Pendulum Length
Determine the required pendulum length for a given period using L = gT²/4π². Output in meters, centimeters, or feet with selectable gravity.
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Step-by-Step Solutions
Every calculation comes with a detailed step-by-step breakdown showing the formula, substitution, and final result with units.
⚠️ Important Note: The pendulum formulas assume small-angle oscillations (typically < 15° from vertical). For large-amplitude swings, the period will be slightly longer than calculated. Real-world effects such as air resistance, friction, and non-ideal spring behavior may also affect results.
Frequently Asked Questions
What is Simple Harmonic Motion (SHM)?
Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. Mathematically, this is expressed as F = -kx, where F is the restoring force, k is the force constant, and x is the displacement. Common examples include pendulums (for small angles), masses on springs, and vibrating strings. SHM produces sinusoidal oscillations with a constant frequency independent of amplitude.
How does pendulum length affect the period?
The period of a pendulum is proportional to the square root of its length: T ∝ √L. This means doubling the length increases the period by a factor of √2 ≈ 1.41. A longer pendulum swings more slowly. Interestingly, the period of a pendulum does not depend on the mass of the bob — a heavier bob swings at the same rate as a lighter one for the same length.
What is the difference between period and frequency?
Period (T) is the time taken for one complete cycle of oscillation, measured in seconds. Frequency (f) is the number of cycles per second, measured in Hertz (Hz). They are inversely related: f = 1/T and T = 1/f. For example, a pendulum with a period of 0.5 seconds has a frequency of 2 Hz — it completes 2 oscillations every second. Angular frequency (ω) is related by ω = 2πf = 2π/T, measured in radians per second.
Does gravity affect the spring-mass period?
No, gravity does NOT affect the period of a spring-mass system. The formula T = 2π√(m/k) contains no gravity term. A mass on a spring oscillates with the same period regardless of whether it's on Earth, the Moon, or Mars. This is because the restoring force comes from the spring itself, not from gravity. In contrast, a pendulum's period does depend on gravity because gravity provides the restoring torque.
What happens if the pendulum angle is large?
For small angles (typically less than 15°), the approximation sin(θ) ≈ θ holds, and the period formula T = 2π√(L/g) is accurate. For larger angles, the period becomes longer than the formula predicts. The exact period requires solving an elliptic integral of the first kind. For example, at 30° amplitude, the actual period is about 1.017 times the small-angle period (about 1.7% longer). At 60°, it's about 1.073 times longer (7.3% increase). Our calculator assumes the small-angle approximation.
How is Simple Harmonic Motion used in real life?
SHM appears in many real-world applications: pendulum clocks use SHM for timekeeping; suspension systems in vehicles use springs that oscillate; seismometers detect earthquakes using spring-mass systems; guitar strings vibrate in SHM to produce sound; alternating current (AC) electricity follows sinusoidal SHM patterns; molecules in solids vibrate in SHM (phonons); and tuning forks produce pure tones through SHM vibrations.