Simple Harmonic Motion Calculator

This template calculates the displacement, velocity and acceleration of an object undergoing simple harmonic motion (SHM), a type of periodic motion where the restoring force is directly proportional to the displacement. Simple harmonic motion is commonly observed in systems like pendulums and springs. This calculator is essential for understanding the fundamental principles of oscillatory motion in physics and engineering.
CalculationInputs﻿﻿A
:10.00m​
﻿
﻿﻿ω
:4.00rad/s​
﻿
﻿﻿t
:1.00s​
﻿
Output﻿﻿y
:-7.57m​
﻿
﻿﻿v
:-26.15m/s​
﻿
﻿﻿a
:121.09m/s2​
﻿
Where:
﻿﻿A
 is amplitude, the maximum extent of the oscillation measured from the equilibrium position in meters ﻿(m)
﻿
﻿﻿ω
 is angular frequency, the rate at which the object oscillates measured in radians per second ﻿(rad/s)
. The formula is: ﻿ω=mk​​=2πf
 where ﻿f
 is the frequency of the SHM. Check out this template to find the frequency!
﻿﻿t
 is time, the specific time at which you want to calculate the object's motion measured in seconds ﻿(s)
﻿
﻿﻿y
 is displacement, the position of the object at a time ﻿t
 measured in meters ﻿(m)
. The formula is: ﻿y=A×sin(ωt)
﻿
﻿﻿v
 is velocity, the speed and direction of the object's motion at the time ﻿t
 measured in meters per second ﻿(m/s)
. The formula is: ﻿v=A×ω×cos(ωt)
﻿
﻿﻿a
 is acceleration, the rate of change of velocity at time ﻿t
 measured in meters per second squared ﻿(m/s2)
. The formula is: ﻿a=−A×ω2×sin(ωt)
﻿
ExplanationSimple Harmonic Motion (SHM) is a fundamental concept in mechanics and physics, serving as a model for various motions such as spring oscillations and pendulum swings.
﻿
Simple Harmonic Motion of a spring, indicating the amplitude (A), time period (T), frequency (f) and angular frequency (ω)
This tool is particularly useful in scenarios such as:
Studying the dynamics of springs, pendulums, and other systems exhibiting simple harmonic motion.
Designing mechanical and structural components where specific vibrational characteristics are required.
Educational settings where the principles of harmonic motion are being taught or demonstrated.
Related ResourcesDamped Harmonic Motion Energy Loss Calculator
Frequency of a Simple Harmonic Motion Calculator
Time Period of a Simple Harmonic Motion Calculator
Simple Harmonic Motion Calculator
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ReferencesHalliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics (10th ed.). Wiley.
Tipler, P. A., & Mosca, G. (2008). Physics for Scientists and Engineers (6th ed.). W.H. Freeman and Company.
Young, H. D., & Freedman, R. A. (2012). University Physics with Modern Physics (13th ed.). Pearson.
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