Simple Harmonic Motion Calculator
Calculate simple harmonic motion calculator solve spring-mass SHM oscillator equations for period T, frequency f, angular frequency omega, max acceleration a_max, and total energy E. Accurate physics formulas and unit conversions for engineers, students & technicians.
Calculate Spring-Mass SHM Oscillator
Enter your physical parameters below to compute verified SHM metrics.
Calculation Results
Calculated using verified physical methodology: Angular Frequency: \omega = \sqrt{\k / m}, Period: T = 2\pi \sqrt{\m / k}
Max Acceleration: a_{max} = \omega^2 A, Total Energy: E = \1 / 2kA^2
Quick Summary
The Simple Harmonic Motion Calculator evaluates spring-mass SHM oscillator metrics ($\omega = \sqrt{\k / m}$, $T = 2\pi \sqrt{\m / k}$, $a_{max} = \omega^2 A$, $E = \1 / 2kA^2$), solving period, frequency, max acceleration, and total oscillator energy.
Formula Explanation
Angular Frequency: \omega = \sqrt{\k / m}, Period: T = 2\pi \sqrt{\m / k}
Max Acceleration: a_{max} = \omega^2 A, Total Energy: E = \1 / 2kA^2
How It Works
The Simple Harmonic Motion Calculator computes natural angular frequency ($\omega = \sqrt{k/m}$) from mass ($m$) and spring constant ($k$). It determines oscillation period ($T = 2\pi / \omega$), frequency ($f = \omega / 2\pi$), maximum acceleration ($a_{max} = \omega^2 A$), and total energy ($E = 0.5 \cdot k \cdot A^2$).
Step-by-Step Worked Example
Practical Problem: Calculate SHM parameters for a 2.0 kg mass attached to a spring with $k = 200.0\text{ N/m}$ displaced by amplitude $A = 0.1\text{ meters}$.
- Step 1: Identify Input Parameters: Mass $m = 2.0\text{ kg}$, Spring Constant $k = 200.0\text{ N/m}$, Amplitude $A = 0.1\text{ m}$.
- Step 2: Calculate Natural Angular Frequency ($\omega$): $\omega = \sqrt{\200.0 / 2.0} = \sqrt{100.0} = 10.00\text{ rad/s}$.
- Step 3: Calculate Oscillation Period ($T$): $T = \2\pi / 10.00 = 0.6283\text{ seconds}$.
- Step 4: Calculate Frequency ($f$) & Maximum Acceleration ($a_{max}$): Frequency $f = \1 / 0.6283 = 1.592\text{ Hz}$. Max Acceleration $a_{max} = \omega^2 A = (10.0)^2 \times 0.1 = 10.00\text{ m/s}^2$.
- Step 5: Calculate Total Oscillator Mechanical Energy ($E$): $E = 0.5 \times 200.0 \times (0.1)^2 = 100.0 \times 0.01 = 1.000\text{ Joules}$.
Real-World Calculation Examples
Scenario 1: Industrial Spring Mass System
Parameters: $m = 2.0\text{ kg}$, $k = 200\text{ N/m}$, $A = 0.1\text{ m}$
Result: $T = 0.63\text{ s}$ ($f = 1.59\text{ Hz}$, $\omega = 10.0\text{ rad/s}$, $1.00\text{ J}$ energy). Industrial SHM.
Scenario 2: Automotive Shock Absorber Spring
Parameters: $m = 400.0\text{ kg}$ (quarter car), $k = 40,000\text{ N/m}$, $A = 0.05\text{ m}$
Result: $T = 0.63\text{ s}$ ($f = 1.59\text{ Hz}$, $\omega = 10.0\text{ rad/s}$, $50.00\text{ J}$ energy). Vehicle suspension SHM.
Scenario 3: Micro-Electro-Mechanical MEMS Sensor
Parameters: $m = 1.0\times 10^{-6}\text{ kg}$ (1 mg), $k = 1,000\text{ N/m}$, $A = 1.0\times 10^{-4}\text{ m}$
Result: $T = 0.000198\text{ s}$ ($f = 5,032.9\text{ Hz}$, $\omega = 31,623\text{ rad/s}$). High-frequency MEMS resonance.
Scenario 4: Heavy Machinery Isolation Base
Parameters: $m = 1,000\text{ kg}$, $k = 10,000\text{ N/m}$, $A = 0.02\text{ m}$
Result: $T = 1.99\text{ s}$ ($f = 0.503\text{ Hz}$, $\omega = 3.16\text{ rad/s}$, $2.00\text{ J}$ energy). Low-frequency vibration isolation.
Key Benefits of Using This Calculator
Complete SHM Parameter Suite
Solves period ($T$), frequency ($f$), angular frequency ($\omega$), $a_{max}$, and total energy ($E$) in one step.
Hooke's Law & Energy Conservation
Verifies restoring force $F = -k x$ and potential-kinetic energy exchange ($E = \1 / 2kA^2$).
Vibration Engineering Support
Ideal for designing automotive suspension springs, tuned mass dampers, and acoustic resonators.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is simple harmonic motion (SHM)?
Simple harmonic motion is repetitive back-and-forth oscillation where restoring force is directly proportional to displacement and acts toward equilibrium (F = -k * x).
What is the formula for angular frequency in SHM?
omega = sqrt(k / m), where k is spring constant in N/m and m is mass in kg.
What is the formula for period in a spring-mass system?
T = 2 * pi * sqrt(m / k).
What is maximum velocity and maximum acceleration in SHM?
v_max = omega * A (at equilibrium x = 0); a_max = omega² * A = (k * A) / m (at maximum displacement x = +/- A).
What is the total energy of a simple harmonic oscillator?
E_total = 0.5 * k * A² = 0.5 * m * v_max².
Does amplitude affect the period of a spring-mass oscillator?
No, in ideal linear SHM, period T depends only on mass m and spring constant k; amplitude A does not affect frequency or period.
What is the differential equation for SHM?
d²x/dt² + (k/m)*x = 0, which has sinusoidal solutions x(t) = A * cos(omega*t + phi).
What is damping in SHM?
Damping forces (like friction or fluid drag) dissipate mechanical energy over time, causing amplitude A to decay exponentially.
What is resonance in forced harmonic motion?
Resonance occurs when an external driving force frequency matches the natural frequency omega_0 = sqrt(k/m), resulting in maximum oscillation amplitude.
What is the SI unit of spring constant k?
Newtons per meter (N/m), equivalent to kg/s².