Last Updated: October 30, 2025
Calculate the period, frequency, and angular frequency for simple harmonic motion instantly with our advanced physics calculator supporting multiple units and real-time results. Essential for analyzing mass-spring systems, pendulums, and oscillatory motion in physics and engineering applications.
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The Simple Harmonic Motion Calculator is a specialized physics tool that calculates the period, frequency, and angular frequency for systems exhibiting simple harmonic motion (SHM). SHM describes oscillatory motion where the restoring force is directly proportional to displacement from equilibrium and directed toward the equilibrium position.
This calculator is essential for analyzing mass-spring systems, pendulums (for small angles), and any oscillatory system with a linear restoring force. It provides accurate calculations for period using mass and spring constant, enabling engineers and physicists to understand periodic motion, resonance phenomena, and oscillatory behavior across diverse applications from mechanical systems to quantum mechanics.
Period (T): Time for one complete oscillation, measured in seconds (s). For mass-spring systems, T = 2π√(m/k). Independent of amplitude (isochronous) for ideal SHM. The period determines how quickly the system oscillates.
Frequency (f): Number of oscillations per second, measured in Hertz (Hz). Related to period by f = 1/T. Higher frequency means faster oscillations. Natural frequency is a fundamental property of the system.
Angular Frequency (ω): Rate of phase change, measured in radians per second (rad/s). Related to frequency by ω = 2πf = 2π/T. For mass-spring systems, ω = √(k/m). Appears naturally in the differential equation of motion.
Mass (m): Mass of the oscillating object, measured in kilograms (kg). Heavier masses produce longer periods for a given spring constant. Mass appears in the square root, so period scales as √m.
Spring Constant (k): Stiffness of the spring, measured in Newtons per meter (N/m). Stiffer springs (larger k) produce shorter periods. Spring constant measures the restoring force per unit displacement.
Amplitude (A): Maximum displacement from equilibrium, measured in meters (m). For ideal SHM, period is independent of amplitude—this isochronous property distinguishes SHM from other oscillations.
Simple harmonic motion occurs when a system experiences a restoring force proportional to displacement: F = -kx. This linear relationship ensures that the acceleration is always directed toward equilibrium and proportional to displacement, resulting in sinusoidal motion. The negative sign indicates the force opposes displacement, always pulling the system back toward equilibrium.
The period formula T = 2π√(m/k) shows that heavier masses oscillate more slowly (longer period), while stiffer springs produce faster oscillations (shorter period). The square root dependence means that to double the period, you must quadruple the mass or reduce the spring constant by a factor of four. The independence of amplitude (isochronism) is a remarkable property: whether oscillating with small or large amplitude, the period remains constant—this was crucial for pendulum clocks.
In simple harmonic motion, energy oscillates between kinetic and potential forms. At maximum displacement, all energy is potential (E = (1/2)kA²). At equilibrium, all energy is kinetic (E = (1/2)mv²max). The total mechanical energy remains constant (conserved) in ideal SHM, assuming no damping. This energy conservation allows calculation of velocity at any position: v = ±ω√(A² - x²).
Simple harmonic motion was understood through studies of pendulums and springs, with Galileo making crucial observations about pendulum isochronism. The mathematical formulation developed through the work of Newton, Hooke, and others. SHM became fundamental to understanding oscillations, waves, and later quantum mechanics, where wave functions exhibit harmonic behavior. The concept of resonance, where driven systems respond maximally at the natural frequency, was crucial for engineering applications.
Period: s (seconds) - time for one complete cycle
Frequency: Hz (Hertz) - cycles per second, f = 1/T
Angular Frequency: rad/s - ω = 2πf = 2π/T = √(k/m)
Mass: kg (kilograms)
Spring Constant: N/m (Newtons per meter)
Relationship: T = 2π√(m/k), ω = √(k/m), f = (1/(2π))√(k/m)
T = 2π √(m/k)
Where:
Use case: Calculate the period of oscillation for a mass-spring system. This is the fundamental formula for SHM period, valid when damping is negligible. The period is independent of amplitude (isochronous). Essential for analyzing oscillatory systems, designing spring-mass systems, and understanding periodic motion.
ω = √(k/m)
Where:
Use case: Calculate angular frequency from spring constant and mass. Angular frequency appears in the differential equation of motion (m(d²x/dt²) + kx = 0) and in the solution x(t) = A cos(ωt + φ). Related to period by ω = 2π/T. Essential for mathematical analysis and when connecting to other oscillatory systems.
f = (1/(2π)) √(k/m)
Where:
Use case: Calculate frequency (oscillations per second) from mass and spring constant. Frequency is the reciprocal of period: f = 1/T. Useful when frequency measurements are available or when analyzing resonance phenomena. Higher frequencies correspond to stiffer springs or lighter masses.
k = m(2π/T)² = 4π²m/T²
Where:
Use case: Calculate spring constant when period and mass are known. Rearranged from T = 2π√(m/k). Useful for determining spring stiffness experimentally by measuring oscillation period. Essential for spring characterization, quality control, and experimental physics.
Simple harmonic motion calculations are essential across numerous physics, engineering, and technology fields. Here's a comprehensive overview of practical applications:
| Industry | Applications | Importance |
|---|---|---|
| Mechanical Engineering | Vibration isolation, shock absorbers, suspension systems, spring-mass dampers, vibration control, machine tool dynamics | Critical for controlling vibrations and improving system performance |
| Structural Engineering | Building dynamics, seismic analysis, tuned mass dampers, vibration mitigation, structural resonance analysis | Essential for earthquake resistance and vibration control |
| Electronics & Sensors | Accelerometers, gyroscopes, MEMS devices, oscillators, timing circuits, resonant sensors | Vital for sensor design and timing applications |
| Automotive | Vehicle suspension, engine mounts, vibration damping, ride comfort, NVH (Noise, Vibration, Harshness) analysis | Important for vehicle performance and comfort |
| Physics Research | Quantum mechanics, wave mechanics, oscillation experiments, resonance studies, harmonic analysis | Fundamental for understanding periodic motion and waves |
| Physics Education | Oscillation demonstrations, laboratory experiments, wave mechanics, periodic motion studies, resonance demonstrations | Essential for teaching fundamental physics concepts |
For a mass–spring system, Newton’s law gives m x¨ + k x = 0. The solution is sinusoidal with angular frequency ω = √(k/m), period T = 2π/ω, and frequency f = ω/(2π). Initial conditions set amplitude and phase. Energy oscillates between kinetic (1/2 m v²) and potential (1/2 k x²) with total constant in the ideal case.
Adding damping c x˙ yields m x¨ + c x˙ + k x = 0. Light damping slightly increases period and reduces amplitude over time; critical and over-damping remove oscillations. Driven SHM under F₀ cos(Ωt) exhibits resonance near Ω ≈ ω, amplitude limited by damping.
Coupled masses and springs produce multiple natural frequencies and mode shapes. Diagonalizing the system matrix reveals independent modal coordinates that each follow SHM, a cornerstone of vibration analysis.
Series combination: k_eq = (1/Σ (1/k_i)); parallel combination: k_eq = Σ k_i. For multiple masses, use effective mass where relevant (e.g., a moving platform or pulley systems) before computing T.
Time many oscillations and divide to reduce timing error. Keep amplitudes small to maintain linear behavior. Estimate k from load–deflection measurements (k = ΔF/Δx) and verify computed T against observation.
SHM underlies seismometer dynamics, sensor calibration, clock mechanisms, and vibration isolation. Identifying ω enables tuned mass dampers and filters, improving comfort, precision, and structural safety.
Given:
Step 1: Use T = 2π √(m/k)
T = 2π √(1.0/100) = 2π √(0.01) = 2π × 0.1 ≈ 0.628 s
Final Answer
≈ 0.628 s
Given:
T = 2π √(0.25/25) = 2π √(0.01) ≈ 0.628 s
Final Answer
≈ 0.628 s
Given:
T = 2π √(2/50) = 2π √(0.04) = 2π × 0.2 ≈ 1.257 s
Final Answer
≈ 1.257 s
Given:
T = 2π √(0.5/200) = 2π √(0.0025) = 2π × 0.05 ≈ 0.314 s
Final Answer
≈ 0.314 s
Given:
T = 2π √(1/(4π²)) = 1 s → f = 1 Hz
Final Answer
T = 1 s, f = 1 Hz
Given:
T₂ = T₁/√2 (doubling k halves T by √2)
Final Relation
T₂ = T₁ / √2
Disclaimer: The calculators and tools are for educational purposes. Verify results independently before professional use.