Velocity of a Simple Harmonic Motion Calculator

This tool calculates the instantaneous velocity of an object undergoing simple harmonic motion (SHM), a type of periodic motion where the restoring force is directly proportional to the displacement. 
CalculationInputs﻿﻿A
:1.00m​
﻿
﻿﻿f
:1.00Hz​
﻿
﻿﻿Φ
:0.00rad​
﻿
﻿﻿t
:1.00s​
﻿
Output﻿﻿ω
:6.28rad/s​
﻿
﻿﻿ω=2πf
﻿
﻿﻿v
:0.00m/s​
﻿
﻿﻿v=−A×ω×sin(ωt+ϕ)
 
Output graph
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﻿
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.
The instantaneous velocity, ﻿v
 in simple harmonic motion at time ﻿t
 is given by:
﻿
v(t)=−A×ω×sin⁡(ωt+ϕ)v(t)=-A\times\omega\times \sin(\omega t+\phi)v(t)=−A×ω×sin(ωt+ϕ)
This is the derivative of the instantaneous displacement, ﻿x
 in simple harmonic motion which is given by:
﻿
x(t)=A×cos⁡(ωt+ϕ)x(t)=A\times \cos(\omega t+\phi)x(t)=A×cos(ωt+ϕ)
Where:
Instantaneous velocity, ﻿v
 of simple harmonic motion at time ﻿t
 ﻿(m/s)
﻿
Amplitude, ﻿A
 is the maximum range of the oscillatory motion ﻿(m)
﻿
Frequency, ﻿f
 is the the number of oscillations per second ﻿(Hz)
. A frequency of 1 Hz means that one complete cycle of the wave occurs every second.
Angular velocity, ﻿ω
 is the rate of change of the motion's phase ﻿(rad/s)
. A higher angular velocity will lead to steeper and more rapid oscillations
Time, ﻿t
 is the particular instant at which the velocity is calculated, allowing velocity analysis at any point in the motion's cycle ﻿(s)
﻿
Phase, ﻿ϕ
 is the initial position in the oscillatory cycle at time zero ﻿(rad)
. A phase of zero means the wave starts at the origin, different phase values would shift the wave left or right along the time axis.
The velocity is a sinusoidal waveform meaning the object's velocity changes dynamically as it oscillates. That is. the velocity varies depending on the object's position in its oscillatory path, reaching its maximum velocity at the equilibrium point and diminishing to zero at the maximum displacement points. This characteristic velocity pattern directly results from the harmonic restoring force acting on the object.
﻿
Velocity vs time graph of Simple Harmonic Motion of a spring
This calculator has applications such as:
In mechanical engineering where the oscillation speed is critical for designing springs, shock absorbers, and other components.
In wave phenomena in physics and acoustics where the velocity of particles in the medium is affected by SHM principles.
In seismology to understand ground motion velocity during earthquakes, which often follows SHM patterns.
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.