Simple Harmonic Motion Calculator
Calculate displacement, velocity, acceleration and more for SHM systems
Results
Displacement (x)
0 m
x(t) = A cos(ωt + φ)
Velocity (v)
0 m/s
v(t) = -Aω sin(ωt + φ)
Acceleration (a)
0 m/s²
a(t) = -Aω² cos(ωt + φ)
Angular Frequency (ω)
0 rad/s
ω = 2πf
Period (T)
0 s
T = 2π/ω
Total Energy (E)
-
E = ½kA²
SHM Graphs
SHM Equations
Fundamental Equations of Simple Harmonic Motion
| Parameter | Equation | Description |
|---|---|---|
| Displacement | x(t) = A cos(ωt + φ) | Position at time t |
| Velocity | v(t) = -Aω sin(ωt + φ) | First derivative of displacement |
| Acceleration | a(t) = -Aω² cos(ωt + φ) | Second derivative of displacement |
| Angular Frequency | ω = 2πf = √(k/m) | Related to frequency and spring system |
| Period | T = 2π/ω = 1/f | Time for one complete oscillation |
| Total Energy | E = ½kA² | Constant total mechanical energy |
Example Calculations
Common SHM System Examples
Example 1: Mass-Spring System
Given:
- Amplitude A = 0.1 m
- Frequency f = 5 Hz
- Phase angle φ = 0 radians
- Time t = 2 s
- Spring constant k = 100 N/m
Example 2: Simple Pendulum
Given:
- Amplitude A = 0.5 m
- Angular frequency ω = 1.4 rad/s
- Phase angle φ = π/4 radians
- Time t = 1.5 s
Example 3: Vertical Spring
Given:
- Amplitude A = 0.2 m
- Frequency f = 2 Hz
- Phase angle φ = 90°
- Time t = 0.75 s
- Spring constant k = 50 N/m
Help & Information
About Simple Harmonic Motion
Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
Common SHM Systems
- Mass-Spring System: A mass attached to a spring that can stretch and compress
- Simple Pendulum: A weight suspended from a pivot that swings back and forth
- Torsional Pendulum: An object suspended by a wire that twists and untwists
How to Use This Calculator
- Enter the amplitude of oscillation (maximum displacement)
- Provide either angular frequency (ω) or regular frequency (f)
- Set the phase angle (starting position of the oscillation)
- Enter the time at which you want to calculate the motion parameters
- Optionally provide the spring constant to calculate total energy
- Click "Calculate" to see the results
Tip: Check the "Show Graphs" box to visualize how displacement, velocity, and acceleration change over time.