Simple Harmonic Motion Calculator

General Physics

Calculate various parameters of Simple Harmonic Motion (SHM), including period, frequency, velocity, and acceleration for mass-spring systems and pendulums.

Practical Examples

Explore these real-world scenarios to understand how the calculator works.

Mass-Spring System Period

Mass-Spring System

Calculate the period and frequency of a 2 kg mass attached to a spring with a constant of 8 N/m.

Calculation Type: Period and Frequency

System Type: Mass-Spring System

Mass: 2 kg

Spring Constant: 8 N/m

Pendulum on Earth

Simple Pendulum

Find the period and frequency of a 0.5 m long pendulum on Earth.

Calculation Type: Period and Frequency

System Type: Simple Pendulum

Length: 0.5 m

Gravity: 9.81 m/s²

Spring's Motion at 1.2s

Spring Motion Parameters

A spring with a 0.5 kg mass and 50 N/m constant is displaced by 0.1 m. Find its position, velocity, and acceleration at t=1.2s, assuming a zero phase angle.

Calculation Type: Motion Parameters (Position, Velocity, Acceleration)

System Type: Mass-Spring System

Amplitude: 0.1 m

Mass: 0.5 kg

Spring Constant: 50 N/m

Time: 1.2 s

Phase Angle: 0 rad

Pendulum Swing

Pendulum Motion Parameters

A 1.0 m pendulum is released with an amplitude of 0.2 m. Find its position, velocity, and acceleration at t=0.5s, assuming zero phase angle and standard gravity.

Calculation Type: Motion Parameters (Position, Velocity, Acceleration)

System Type: Simple Pendulum

Amplitude: 0.2 m

Length: 1.0 m

Gravity: 9.81 m/s²

Time: 0.5 s

Phase Angle: 0 rad

Other Titles
Understanding the Simple Harmonic Motion Calculator: A Comprehensive Guide
Dive deep into the principles of SHM, from basic definitions to complex applications, and learn how to use this calculator effectively.

What is Simple Harmonic Motion (SHM)?

  • Defining the Core Concept
  • Key Characteristics of SHM
  • The Restoring Force
Simple Harmonic Motion (SHM) is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. In simpler terms, it's a back-and-forth movement through an equilibrium (or central) position, where the maximum displacement on one side is equal to the maximum displacement on the other. The interval of time for each complete vibration is constant.
Key Characteristics of SHM
1. Period (T): The time taken to complete one full oscillation. It is measured in seconds (s).
2. Frequency (f): The number of oscillations completed per unit of time. It is the reciprocal of the period (f = 1/T) and is measured in Hertz (Hz).
3. Amplitude (A): The maximum displacement from the equilibrium position. It is a measure of the intensity of the oscillation.
4. Angular Frequency (ω): A measure of rotational speed, expressed in radians per second (rad/s). It's related to frequency by ω = 2πf.
The Restoring Force
The defining feature of SHM is the restoring force, described by Hooke's Law: F = -kx, where 'F' is the restoring force, 'k' is a positive constant (like the spring constant), and 'x' is the displacement from equilibrium. The negative sign indicates that the force always acts to pull or push the system back towards its equilibrium position.

Conceptual Examples

  • A child on a swing (approximates SHM for small angles).
  • A mass bobbing up and down on a spring.
  • The vibrations of a tuning fork.

Step-by-Step Guide to Using the SHM Calculator

  • Selecting Calculation and System Types
  • Entering Input Parameters
  • Interpreting the Results
1. Select Your Goal
Begin by choosing what you want to calculate from the 'Calculation Type' dropdown. You can either solve for the fundamental properties ('Period and Frequency') or for the state of the system at a specific moment ('Motion Parameters').
2. Define Your System
Next, tell the calculator which physical system you are analyzing. Choose 'Mass-Spring System' if your problem involves a mass and a spring, or 'Simple Pendulum' for a mass swinging from a string. The required input fields will change based on this selection.
3. Provide the Known Values
Fill in the input fields with the data from your problem. Ensure you are using the correct units as specified (e.g., meters, kilograms, seconds). Forgetting a required field will trigger a validation error. For instance, to find the period of a spring system, you must provide both mass and the spring constant.
4. Calculate and Analyze
Click the 'Calculate' button. The results will appear below, showing the calculated values with their corresponding units. If you chose to calculate motion parameters, you will see the position, velocity, and acceleration at the specified time 't'.

Usage Scenarios

  • A student calculating the period of a pendulum for a lab report.
  • An engineer designing a shock absorber (a form of damped oscillator).
  • A physicist modeling the vibration of a molecule.

Mathematical Derivations and Formulas

  • Equations for Mass-Spring Systems
  • Equations for Simple Pendulums
  • General Equations of Motion
Mass-Spring System Formulas
For a mass (m) on a spring with constant (k):
- Angular Frequency: ω = √(k / m)
- Period: T = 2π / ω = 2π √(m / k)
- Frequency: f = 1 / T = (1 / 2π) √(k / m)
Simple Pendulum Formulas (Small Angle Approximation)
For a pendulum of length (L) under gravity (g):
- Angular Frequency: ω = √(g / L)
- Period: T = 2π / ω = 2π √(L / g)
- Frequency: f = 1 / T = (1 / 2π) √(g / L)
General Equations of Motion
Given amplitude (A), angular frequency (ω), and phase angle (φ):
- Position at time t: x(t) = A cos(ωt + φ)
- Velocity at time t: v(t) = -Aω * sin(ωt + φ)
- Acceleration at time t: a(t) = -Aω² cos(ωt + φ)
- Maximum Velocity: vmax = Aω
- Maximum Acceleration: amax = Aω²

Formula Application

  • If m=1kg and k=100 N/m, then T = 2π * √(1/100) ≈ 0.628 s.
  • If L=9.81m on Earth (g=9.81 m/s²), then T = 2π * √(9.81/9.81) = 2π ≈ 6.28 s.