Acceleration of a Simple Harmonic Motion Calculator

This calculator calculates the instantaneous acceleration of an object undergoing simple harmonic motion (SHM), a type of periodic motion where the restoring force is directly proportional to the displacement. 
CalculationInputs﻿﻿A
:0.50​
﻿
﻿﻿f
:2.00​
﻿
﻿﻿t
:2.00​
﻿
﻿﻿Φ
:0.50rad​
﻿
Output﻿﻿ω
:12.57rad/s​
﻿
﻿﻿ω=2πf
﻿
﻿﻿α
:-69.29m/s2​
﻿
﻿﻿α=−A×ω2×cos(ωt+ϕ)
﻿
💡Note, phase angle (﻿ϕ
) sign convention
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 acceleration, ﻿α
 in simple harmonic motion at time ﻿t
 is given by:
﻿
a(t)=−A×ω2×cos⁡(ωt+ϕ)a(t)=-A\times \omega^2\times \cos(\omega t+\phi)a(t)=−A×ω2×cos(ωt+ϕ)
This is the derivative of the velocity, ﻿v
 and a second derivative of the displacement, ﻿x
 in simple harmonic motion which are given by:
﻿
v(t)=−A×ω×sin⁡(ωt+ϕ)x(t)=A×cos⁡(ωt+ϕ)v(t)=-A\times\omega\times \sin(\omega t+\phi)\\x(t)=A\times \cos(\omega t+\phi)v(t)=−A×ω×sin(ωt+ϕ)x(t)=A×cos(ωt+ϕ)
Where:
Amplitude (﻿A
): The maximum range of oscillatory motion ﻿(m)
﻿
Frequency (﻿f
): 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
): The rate of change of the motion's phase ﻿(rad/s)
 given by: ﻿ω=2πf
. A higher angular velocity will lead to steeper and more rapid oscillations
Time (﻿t
): The point in time at which the acceleration is calculated ﻿(s)
 
Phase angle (﻿ϕ
): 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 acceleration is a cosine waveform meaning the object's acceleration changes dynamically as it oscillates. We can write ﻿α(t)=−ω2x(t)
 meaning the acceleration acts in opposite direction to the displacement and is proportional to it.
﻿
Acceleration vs time graph of Simple Harmonic Motion of a spring
This calculator has applications such as:
Physics and engineering education, especially in wave mechanics and oscillations
Mechanical engineering for the design of springs, pendulums, and oscillatory systems
Research in areas like seismology and acoustics
Related ResourcesAcceleration of a Simple Harmonic Motion Calculator (related to the displacement)
Damped Harmonic Motion Energy Loss Calculator
Frequency of a Simple Harmonic Motion Calculator
Simple Harmonic Motion Calculator
Time Period of a Simple Harmonic Motion Calculator
Velocity of a Simple Harmonic Motion Calculator
Check out our full library of CalcTree templates here!
ReferencesHalliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
Young, H. D., & Freedman, R. A. (2019). University Physics with Modern Physics. Pearson.
Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.
Calculation

Inputs

Output

💡Note, phase angle (﻿ϕ) sign convention

Output graph

Explanation

Related Resources

References

💡Note, phase angle (

$ϕ$
) sign convention
