Acceleration of a Simple Harmonic Motion Calculator (related to the displacement)

This calculator calculates the instantaneous acceleration of an object undergoing simple harmonic motion (SHM), as a function of it's displacement from the equilibrium position.
CalculationInputs﻿﻿x
:1.00m​
﻿
﻿﻿f
:2.00Hz​
﻿
Output﻿﻿α
:-157.91m/s2​
﻿
﻿﻿α=−4 π2 f2 x
﻿
Where:
Displacement (﻿x)
: The displacement of the object undergoing SHM from it's equilibrium position ﻿(m)
﻿
Frequency (﻿f
): The number of oscillations per second ﻿(Hz)
 of the object undergoing SHM. A frequency of ﻿1 Hz
 means that one complete cycle of the wave occurs every second
Acceleration (﻿α
): The acceleration of the object undergoing SHM when the object is at distance ﻿x
 from it's equilibrium position ﻿(m2/s)
﻿
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 equations for the displacement ﻿x(t)
, velocity ﻿v(t)
 and acceleration ﻿α(t)
 of an object undergoing simple harmonic motion at time ﻿t
 are given by:
﻿
x(t)=A×cos⁡(ωt)v(t)=−A×ω×sin⁡(ωt)a(t)=−A×ω2×cos⁡(ωt)x(t)=A\times \cos(\omega t)\\v(t)=-A\times\omega\times \sin(\omega t)\\a(t)=-A\times \omega^2\times \cos(\omega t)x(t)=A×cos(ωt)v(t)=−A×ω×sin(ωt)a(t)=−A×ω2×cos(ωt)
Where:
Angular velocity (﻿ω
): The rate of change of the motion's phase ﻿(rad/s)
 is 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 displacement, velocity and acceleration is calculated ﻿(s)
 
Amplitude (﻿A
): The maximum range of oscillatory motion ﻿(m)
﻿
We can see that the velocity ﻿v(t)
 is the first derivative of the object's displacement and the acceleration ﻿α(t)
 is the second derivate of the object's displacement (or the first derivative of the object's velocity). We can re-write the acceleration ﻿α(t)
 in terms of the displacement ﻿x(t)
:
﻿
a(t)=−ω2×x(t)a(t)=-\omega^2\times x(t)a(t)=−ω2×x(t)
﻿
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
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
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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.