Physics · Classical Mechanics · Oscillations & Waves
Simple Harmonic Motion Calculator
Calculates the period, frequency, angular frequency, maximum velocity, and maximum acceleration of a simple harmonic oscillator given mass and spring constant.
Calculator
Formula
T is the period (s), m is the mass (kg), k is the spring constant (N/m), f is the frequency (Hz), ω is the angular frequency (rad/s), A is the amplitude (m), v_max is the maximum velocity (m/s), and a_max is the maximum acceleration (m/s²).
Source: Halliday, Resnick & Krane — Physics, 5th Edition, Chapter 15; NIST Classical Mechanics Reference.
How it works
Simple harmonic motion describes the oscillation of a system where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction — expressed by Hooke's Law as F = −kx. This linear restoring force produces sinusoidal motion in time, making SHM the idealized model for a wide range of real oscillating systems. The motion is characterized by a fixed amplitude (maximum displacement), a well-defined period, and conserved total mechanical energy.
The period of oscillation is given by T = 2π√(m/k), where m is the mass of the oscillating object in kilograms and k is the spring constant in Newtons per meter. The frequency f = 1/T gives the number of complete oscillations per second, while the angular frequency ω = √(k/m) describes how rapidly the oscillator cycles in radians per second. Crucially, the period and frequency depend only on mass and spring constant — not on amplitude — which is a defining feature of SHM. The maximum velocity occurs at the equilibrium position and equals v_max = Aω, while the maximum acceleration occurs at the turning points and equals a_max = Aω².
In practice, SHM analysis is applied across mechanical engineering (vibration isolation, automotive suspension), civil engineering (building resonance and earthquake resistance), electrical engineering (LC tank circuits), and physics (atomic force microscopy, optical cavities). Understanding the relationship between mass, stiffness, and oscillation frequency allows engineers to tune systems away from dangerous resonance frequencies or deliberately design resonant systems like quartz oscillators for precise timekeeping.
Worked example
Consider a spring-mass system with a mass of 0.5 kg, a spring constant of 20 N/m, and an amplitude of 0.1 m.
Step 1 — Angular Frequency: ω = √(k/m) = √(20 / 0.5) = √40 ≈ 6.3246 rad/s
Step 2 — Period: T = 2π / ω = 2π / 6.3246 ≈ 0.9935 s
Step 3 — Frequency: f = 1/T ≈ 1.0065 Hz
Step 4 — Maximum Velocity: v_max = Aω = 0.1 × 6.3246 ≈ 0.6325 m/s
Step 5 — Maximum Acceleration: a_max = Aω² = 0.1 × 40 = 4.0000 m/s²
This system oscillates just over once per second, reaching peak speed of about 0.63 m/s at the center and a maximum acceleration of 4 m/s² at the turning points — approximately 0.4g.
Limitations & notes
This calculator assumes an ideal, undamped simple harmonic oscillator. In real systems, energy is lost to friction and air resistance (damping), causing amplitude to decay over time and shifting the effective resonant frequency. For lightly damped systems the period approximation remains accurate, but heavily damped or critically damped systems require a modified analysis using the damped angular frequency ω_d = √(ω₀² − γ²), where γ is the damping coefficient. The model also assumes the spring obeys Hooke's Law perfectly (linear elasticity), which breaks down at large deformations or near the elastic limit of the material. For pendulum systems, the SHM approximation is only valid for small angles (generally less than 15°). Additionally, this calculator does not account for the mass of the spring itself; for springs with non-negligible mass, an effective mass correction of approximately m_eff = m + m_spring/3 should be applied.
Frequently asked questions
Does the period of simple harmonic motion depend on amplitude?
No — one of the defining properties of ideal SHM is that the period T = 2π√(m/k) is independent of amplitude. Whether the oscillator is displaced by 1 cm or 10 cm, it takes exactly the same time to complete one cycle, as long as the system remains in the linear (Hooke's Law) regime.
What is the difference between frequency and angular frequency?
Frequency (f) measures the number of complete oscillation cycles per second in hertz (Hz). Angular frequency (ω) measures the same oscillation in radians per second and is related by ω = 2πf. Angular frequency is more convenient in mathematical formulations of SHM because the displacement can be written as x(t) = A cos(ωt + φ), making calculus operations straightforward.
Where does maximum velocity occur in simple harmonic motion?
Maximum velocity occurs at the equilibrium position (x = 0), where all the potential energy stored in the spring has been converted to kinetic energy. At the turning points (x = ±A), velocity is zero and potential energy is at its maximum. This energy interchange is a hallmark of conservative oscillatory systems.
How does doubling the mass affect the period?
Since T = 2π√(m/k), doubling the mass multiplies the period by √2 ≈ 1.414. For example, if the original period was 1.0 s, doubling the mass gives a new period of approximately 1.414 s. Mass and period are related by a square root, not a linear relationship.
Can this calculator be used for a simple pendulum?
Not directly. A simple pendulum undergoing small-angle oscillations has a period T = 2π√(L/g), where L is the pendulum length and g is gravitational acceleration. This is analogous to the spring formula with L replacing m and g replacing k, but the inputs and physics differ. For pendulum calculations, use a dedicated pendulum period calculator.
Last updated: 2025-01-15 · Formula verified against primary sources.