Simple Harmonic Motion Calculator

Simple Harmonic Motion Guide

What's the key thing to understand about Mass-Spring System?

For a mass m on a spring with spring constant k: T = 2π√(m/k). f = 1/T = (1/2π)√(k/m). ω = 2πf = √(k/m). The period is independent of amplitude — this is a key property of SHM. Example: m=0.5kg, k=20 N/m. T = 2π√(0.5/20) = 2π×0.158 = 0.994 s. f = 1/0.994 = 1.006 Hz. ω = 2π/0.994 = 6.32 rad/s. With amplitude A=0.1m: v_max = Aω = 0.1×6.32 = 0.632 m/s. a_max = Aω² = 0.1×39.9 = 3.99 m/s².

What do I need to know about Simple Pendulum?

For a pendulum of length L: T = 2π√(L/g). Valid for small angles (< 15°) — at larger angles, the period increases slightly. Pendulum length of 1.0m on Earth: T = 2π√(1.0/9.81) = 2π×0.319 = 2.006 s ≈ 2 seconds. The seconds pendulum (T=2s exactly) has length exactly 0.994m. Note: the period of a pendulum is independent of its mass and of the amplitude (for small angles). This was used historically in precision clocks.

What should I know about SHM Equations of Motion?

Displacement: x = A cos(ωt + φ). Velocity: v = −Aω sin(ωt + φ). Acceleration: a = −Aω² cos(ωt + φ) = −ω²x. The key relationship: a = −ω²x. Acceleration is proportional to displacement and directed toward equilibrium — the defining equation of SHM. Energy: total mechanical energy E = ½kA². At equilibrium (x=0): all KE = ½mv². At amplitude (x=A): all PE = ½kA². At any point: ½mv² + ½kx² = ½kA². Velocity at displacement x: v = ω√(A² − x²).

What's the key thing to understand about Damping and Resonance?

Real oscillators lose energy to damping (friction, air resistance, internal heating). Light damping: amplitude decreases slowly. Heavy damping: returns to equilibrium without oscillating. Critical damping: returns to equilibrium fastest without oscillating — used in door closers, car suspension. Resonance: when a driving frequency equals the natural frequency f₀ = (1/2π)√(k/m), amplitude increases dramatically. In lightly damped systems, this can cause structural failure. Examples: the Tacoma Narrows Bridge collapse in 1940 is the most famous real-world case of resonance-driven structural failure, though the actual mechanism was more complex than simple wind-driven resonance alone.

What should I know about Energy in Simple Harmonic Motion?

A system undergoing simple harmonic motion continuously exchanges energy between kinetic and potential forms while its total mechanical energy remains constant (assuming no damping). For a mass-spring system, the total energy is E = ½kA², where k is the spring constant and A is the amplitude — this total is reached entirely as potential energy (½kx²) at the extremes of motion, where displacement x equals the amplitude A and velocity is momentarily zero, and entirely as kinetic energy (½mv²) at the equilibrium position, where displacement is zero and speed is at its maximum. At any point in between, the energy splits between the two forms, always summing to the same constant ½kA². This relationship directly explains why maximum speed in SHM occurs exactly at the equilibrium point (x = 0) and why the oscillating object momentarily stops at each extreme of its swing (x = ±A) — the system has traded all of its kinetic energy for potential energy at that instant, exactly as this calculator's amplitude and angular frequency values predict when used to find velocity at any given displacement.

What's the key thing to understand about Phase in SHM?

The general equation describing simple harmonic motion is x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ (phase angle, or phase constant) determines where in its oscillation cycle the system starts at time t = 0. If an oscillator is released from its maximum displacement with no initial velocity, φ = 0, and the motion simplifies to the simple cosine form x(t) = A cos(ωt). If instead the oscillator starts at the equilibrium position moving with maximum speed, φ = −π/2 (90° out of phase with the cosine case), and the motion is better described by a sine function, x(t) = A sin(ωt) — mathematically the same family of motion, just shifted in time. The phase constant becomes especially important when comparing two oscillating systems, such as two pendulums of the same period released at different moments, or the voltage and current in an AC circuit, where the phase difference between two oscillating quantities (rather than either one's absolute phase) determines important physical behaviour, such as power delivered in an electrical circuit.

What should I know about Angular Frequency, Frequency, and Period in SHM?

Simple harmonic motion is described through three closely linked quantities: period T (seconds per oscillation), frequency f = 1/T (oscillations per second), and angular frequency ω = 2πf = 2π/T (radians per second) — the rate the underlying sinusoidal phase advances. For a mass-spring system, ω = √(k/m); for a simple pendulum at small angles, ω = √(g/L). Keeping these three straight matters because formulas for velocity and acceleration in SHM (v_max = ωA, a_max = ω²A) use angular frequency specifically, not the everyday frequency in hertz.

Worked Example: Full Calculation for a Mass-Spring Oscillator

A 0.5 kg mass attached to a spring with spring constant k = 200 N/m is pulled to an amplitude of 0.1 m and released. The angular frequency is ω = √(k/m) = √(200/0.5) = √400 = 20 rad/s. From that, the period is T = 2π/ω = 2π/20 ≈ 0.314 s, and the frequency is f = 1/T ≈ 3.18 Hz. The maximum speed, reached as the mass passes through equilibrium, is v_max = Aω = 0.1 × 20 = 2 m/s, and the maximum acceleration, reached at the extremes of the swing, is a_max = Aω² = 0.1 × 400 = 40 m/s² — four times gravity. The total mechanical energy of the system, which stays constant throughout the motion (ignoring friction), is E = ½kA² = ½ × 200 × 0.1² = ½ × 200 × 0.01 = 1 J. At any instant, that 1 J is split between kinetic and elastic potential energy depending on position, but the sum always equals 1 J — a useful check when verifying a calculator's output at any given point in the cycle.

Table: SHM Quantities Compared — Springs vs. Pendulums

Both mass-spring systems and simple pendulums undergo simple harmonic motion, but the formula for angular frequency depends on completely different physical quantities in each case.

Energy Exchange Over One Cycle

In an ideal (frictionless) SHM system, energy continuously trades between kinetic and potential forms while the total stays fixed. At the extremes of motion (x = ±A), velocity is zero, so all the energy is potential: E = ½kA². At the equilibrium point (x = 0), the spring or pendulum is undeformed, so all the energy is kinetic: E = ½mv_max². At any intermediate position x, the energy splits as E = ½kx² + ½mv², and solving for the speed at that position gives v = ω√(A² − x²) — a formula that lets you find the velocity anywhere along the swing, not just at the extremes or the center. For the 0.5 kg, k = 200 N/m, A = 0.1 m example above, at the halfway point x = 0.05 m, the speed would be v = 20 × √(0.1² − 0.05²) = 20 × √(0.01 − 0.0025) = 20 × √0.0075 ≈ 20 × 0.0866 ≈ 1.73 m/s — noticeably less than the maximum of 2 m/s, because part of the energy at that point is still stored as potential energy in the stretched or compressed spring.

Table: Period Formulas for Common Oscillating Systems

While mass-spring systems and simple pendulums are the two classic SHM examples, the same mathematical pattern — a restoring force proportional to displacement, producing a period independent of amplitude — shows up in several other physical systems.

A notable consequence of the pendulum formula is that a pendulum clock runs slightly fast at the equator and slightly slow near the poles, because measured gravitational acceleration g varies by about 0.5% across Earth's surface — enough to matter for precision timekeeping, though not enough to notice in everyday pendulum swings.