Dynamics of Simple Harmonic Motion in a Mass-Spring System

Explore the mass-spring system from equations to animations

This study focuses on the motion of a mass-spring system undergoing simple harmonic motion (SHM) in the absence of damping. Such a system exhibits oscillatory motion, where the restoring force of the spring is proportional to the position

x(t)

, as described by Hooke's law:

F(t)=-kx(t)

, where

is the spring constant. The governing equation of motion is derived from Newton's second law:

=-kx(t)

(1)

where

is the mass of the block. Dividing through by

, this simplifies to:

x(t)

(2)

where

ω=

k/m

is the angular frequency of the system. The solution to this equation depends on the initial conditions, including the initial position

x(0)=

and the initial velocity

x'(0)=

. Let us begin by creating a graphical representation of a mass-spring system.

Design static elements (wall, ground, boards) and create an oscillating spring with an attached mass:

In[1]:=

wall=Style[Cuboid[{-0.3,-1.5,-1},{0,1.5,1.5}],Cyan];ground=Style[Cuboid[{-0.3,-1.5,-1},{17.5,1.5,-0.7}],Cyan];board0=Style[Cuboid[{10,-1.5,-1},{10,1.5,1.5}],Red,Opacity[0.1]];boardR=Style[Cuboid[{15,-1.5,-1},{15,1.5,1.5}],Red,Opacity[0.1]];boardL=Style[Cuboid[{5,-1.5,-1},{5,1.5,1.5}],Red,Opacity[0.1]];fixedEnd=0;vibratingEnd=10;phase=0;amplitude=5;omega=5;spring[t_]:=ParametricPlot3D

;mass[t_]:=StyleCuboid

,Glow[Red];

Define and label the x-axis, mark the origin, and add appropriate tick marks:

In[2]:=

xAxis=Style[Arrow[{{0,0,-0.7},{19,0,-0.7}}],Black];xAxisText=Text[Style["

",FontSize14,Black],{19.3,0,-0.6}];zeroText=Text[Style["

",FontSize14,Black],{10,-0.7,-0.7}];freeTicks=Table[Line[{{x,0.3,-0.7},{x,-0.3,-0.7}}],{x,0,17,1}];

Represent and label the stretched position line, the position vector, and the force vector:

In[3]:=

stretchLine=Style[Line[{{15,0,-0.7},{15,0,3}}],Thick,Dashed];stretchLabel="Stretched Position";stretchPosition=Style[Arrow[{{10,0,2.5},{15,0,2.5}}],Black];stretchPositionText=Text[Style["

",FontSize14,Black],{12.5,0,3}];stretchForce=Style[Arrow[{{15,0,3.5},{10,0,3.5}}],Red];stretchForceText=Text[Style["

",FontSize14,Black],{12.5,0,4}];

Specify and label the equilibrium position line, and include text labels for the position and force:

In[4]:=

equilibriumLine=Style[Line[{{10,0,-0.7},{10,0,3}}],Thick,Dashed];equilibriumLabel="Equilibrium Position";equilibriumPositionText=Text[Style["

x=0

",FontSize14,Black],{10,0,3.7}];equilibriumForceText=Text[Style["

F = 0

",FontSize14,Black],{10,0,4.9}];

Establish and label the compressed position line, the position vector, and the force vector:

In[5]:=

compressLine=Style[Line[{{5,0,-0.7},{5,0,3}}],Thick,Dashed];compressLabel="Compressed Position";compressPosition=Style[Arrow[{{10,0,2.5},{5,0,2.5}}],Black];compressPositionText=Text[Style["

",FontSize14,Black],{7.5,0,3}];compressForce=Style[Arrow[{{5,0,3.5},{10,0,3.5}}],Red];compressForceText=Text[Style["

",FontSize14,Black],{7.5,0,4}];

Create visualizations for different positions of the spring-mass system and combine the individual visualizations into a single Row layout:

In[6]:=

stretchPlot=Show

;equilibriumPlot=Show

;compressPlot=Show

;Row[{stretchPlot,equilibriumPlot,compressPlot}]

Out[6]=

Express the SHM equation of motion (Eq. 2), specify initial conditions, and solve for

x(t)

using DSolveValue:

In[7]:=

eqn=x''[t]-

x[t];ic1=x[0]x0;ic2=x'[0]v0;sol=DSolveValue[{eqn,ic1,ic2},x[t],t]

Out[7]=

x0ωCos[tω]+v0Sin[tω]

Rewrite the solution in terms of amplitude

and phase angle ϕ using substitutions

=Acos(ϕ)

and

=-ωAsin(ϕ)

, and Simplify to obtain the standard SHM form:

In[8]:=

x[t_]=Simplify[sol/.{x0ACos[ϕ],v0-ωASin[ϕ]}];x[t]

Out[8]=

ACos[ϕ+tω]

Use D to get the velocity

v(t)

and acceleration

a(t)

as the first and second derivatives of position with respect to time, respectively:

In[9]:=

v[t_]=D[x[t],{t,1}];v[t]

Out[9]=

-AωSin[ϕ+tω]

In[10]:=

a[t_]=D[x[t],{t,2}];a[t]

Out[10]=

-A

Cos[ϕ+tω]

Express kinetic energy

KE(t)

in terms of velocity

v(t)

and potential energy

PE(t)

in terms of position

x(t)

, obtaining them as time-dependent functions:

In[11]:=

kE[t_]:=

v[t]

;kE[t]

Out[11]=

Sin[ϕ+tω]

In[12]:=

pE[t_]:=

x[t]

;pE[t]

Out[12]=

Cos[ϕ+tω]

Calculate the total energy (

tE=kE(t)+pE(t)

), considering that

ω=

k/m

In[13]:=

tE=(kE[t]+pE[t])/.ω

k/m

Out[13]=

Cos

t+ϕ

Sin

t+ϕ

Use TrigReduce to simplify the total energy expression by reducing trigonometric functions and verify that the energy remains constant:

In[14]:=

TrigReduce[tE]

Out[14]=

Now, let us proceed with a specific mass-spring system. Consider a body with a mass of

m=0.2

kg, attached to a spring with a spring constant

k=5

N/m. The body oscillates on a frictionless surface. It is stretched to

A=0.05

m to the right of the equilibrium position and then released from rest as shown in the figure below.

Add labeled ticks along the

-axis and annotate the initial position and initial velocity of the mass-spring system:

In[15]:=

labeledTicks=Table

;x0Text=Text[Style["

=x(0)=A=0.05

m",FontSize14,Black],{10,0,5.5}];v0Text=Text[Style["

=v(0)=0

",FontSize14,Black],{10,0,4.5}];

Create a 3D visualization showing the stretched spring, mass, labeled axis, and initial conditions for the given system:

In[16]:=

Show



Out[16]=

Extract and assign the value of ϕ using Chop to eliminate any numerical noise:

Formulate the position, velocity, and acceleration as time-dependent functions:

Set frame ticks, axes labels, and labels for position, velocity, and acceleration:

Formulate the kinetic and potential energies as time-dependent functions:

Generate a plot of kinetic energy and potential energy versus time, including labeled axes, legends, and grid lines for clarity:

Use Manipulate to dynamically display the spring-mass system, energies, position, velocity, and acceleration plots, along with a time slider to track real-time changes over one period:

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Citation:
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics. 9th Edition. (Boston, 2014).
- Sándor Kabai (2007), "Helical Spring" Wolfram Demonstrations Project. demonstrations.wolfram.com/HelicalSpring/

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Contributed by: Akram Masoud

Dynamics of Simple Harmonic Motion in a Mass-Spring System

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