Damped Harmonic Motion Energy Loss Calculator

This template is designed to calculate the energy loss in a damped harmonic motion system, which is where a spring or pendulum experiences a decrease in amplitude over time due to a damping force.
CalculationThe formula for energy loss in a damped harmonic motion system involves the initial energy, damping coefficient, mass and time, highlighting how these factors influence the energy dissipation in the system.
Inputs﻿﻿E0
:100.00J​
﻿
﻿﻿c
:5.00N/ms​
﻿
﻿﻿m
:10.00kg​
﻿
﻿﻿t
:5.00s​
﻿
Output﻿﻿E
:1.39e-9J​
﻿
﻿
E=E0×e−(cm)×tE=E_0\times e^{-(\frac{c}{m})\times t} E=E0​×e−(mc​)×t
Where:
﻿﻿E0​
 is the initial energy of the system, which sets the starting point for the energy decay process, measured in joules ﻿(J)
. It is usually defined by the system's initial conditions, such as the initial displacement and velocity.
﻿﻿c
 is the damping coefficient, which represents how quickly the system loses its energy. It is a measure of the damping force per unit velocity, in newtons per meter per second ﻿(N⋅m−1⋅s−1)
. A higher damping coefficient indicates more rapid energy loss.
﻿﻿m
 is the mass of the oscillating system, measured in kilograms ﻿(kg)
. It determines the system's inertia, affecting how quickly it responds to the damping forces.
﻿﻿t
 is the time over which damping occurs, measured in seconds ﻿(s)
﻿
﻿﻿E
 is the energy loss in the system, measured in joules ﻿(J)
﻿
ExplanationHarmonic motion is a fundamental concept in mechanics and physics, serving as a model for various motions such as spring oscillations and pendulum swings. A damped harmonic motion system is where a spring or pendulum experiences a decrease in amplitude over time due to a damping force, such as internal friction or air resistance. 
The energy loss in the system is given as:
﻿
E=E0×e−(cm)×tE=E_0\times e^{-(\frac{c}{m})\times t} E=E0​×e−(mc​)×t
The energy of a damped harmonic oscillator decreases exponentially over time, meaning the rate of energy loss is time-dependent. The longer the system oscillates, the more energy it loses. This is quantified using the exponential decay factor ﻿e−(mc​)×t
 in the formula.
﻿
Damped harmonic motion showing a decrease in amplitude over time
This template is particularly useful in scenarios such as:
Understanding the dynamics of damped oscillatory systems in physics and engineering, such as in mechanical engineering for machinery with damping components or in the automotive industry for developing suspension systems 
Analyzing the energy loss due to damping in mechanical systems
Related ResourcesFrequency of a Simple Harmonic Motion Calculator
Time Period of a Simple Harmonic Motion Calculator
Simple Harmonic Motion Calculator
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References:Halliday, 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.