Short Description of Topic

claimed by Fabian Nunez Fall 2025
==The Main Idea==

The Parallel Axis Theorem is used to interpret the moment of inertia (I) for any axis parallel to the axis through the center line used to calculate the moment of inertia by using the object's mass and distance between the axes.

===A Mathematical Model===

The formula for the Parallel Axis Theorem is

<math>{I}_{parallel} = {I}_{cm} + Md^2</math>

where:

<math>{I}_{cm}</math> is the moment of inertia about the center of mass. (Usually original / easier to solve)

<math>M</math> is the total mass of the object.

<math>d</math> is the distance between the Parallel Axis and the Center of Mass axis.

===Proof===

If we start by setting our axis along a normal Cartesian plane, centered around the center of mass, we can define the moment of inertia about a line parallel to the center of mass as an integral with respect the object's mass.

<math>{I}_{cm} = \int r^2 dm</math>

where r is the distance between the center of mass and the point mass. Because of the Pythagorean theorem, <math>r= \surd (x^2+y^2)</math>

We can plug this in to redefine our equation as
<math>{I}_{cm} = \int \surd (x^2+y^2) dm</math>

Then, if we know D is the distance between our axes, we can plug this in to calculate our new axis

<math>{I}_{new axis} = \int \surd ((x-D)^2+y^2) dm</math>

This expands into

<math>{I}_{new axis} = \int \surd ((x^2+D^2+y^2-2Dx) dm</math>

We can solve this integral by separating the Distance terms, where the term <math> \int -2Dx dm</math> evaluates to 0 (the x coordinate of the center of mass), and the term <math> \int D^2 dm </math> is simplifies to <math>D^2 \int dm</math>. Our final solution is

<math>{I}_{new axis} = {I}_{cm} + MD^2</math>

==Examples==

Remember the formula for the Parallel Axis Theorem is

<math>{I}_{parallel} = {I}_{cm} + Md^2</math>

===Simple===
'''Problem'''

A thin rod of mass m and length L has a moment of interia around the center axis as <math>{I}_{center} = (1/12)md^2</math> Find the moment of inertia about a line passing through one end of the rod parallel to the center axis.

'''Solution'''
Distance d = L/2

<math>{I}_{parallel} = (1/3)ML^2 </math>
===Middling===
'''Problem'''
A disk of diameter D= 6 m and mass m = 10 kg has its moment of inertia about its center defines as <math>{I}_{cm} = (1/2)mR^2</math>. Find the moment of interia of a line parallel to the axis on the edge of the disk

'''Solution'''
1st, find distance from center to edge (radius).
Radius = R = D/2 = 3

2nd, setup formula

<math>{I}_{parallel} = {I}_{cm} + Md^2</math> => <math>{I}_{parallel} = (1/2)M(R^2) + M(D/2)^2</math> => <math>{I}_{parallel} = (3/2)MR^2</math>

3rd, plug in variables.

<math>{I}_{parallel} = 135 kg * m^2</math>

===Difficult===

'''Problem'''
I have a thin-rod-shaped stick of length L = 1 m and mass 5 kg. The moment of inertia of this stick about the center of mass is given by <math>{I}_{center} = (1/12)mL^2</math> [the axis is perpendicular to the length of the stick]. I apply massless adhesive to one end of a stick, then I touch the center of the top face of a cube with side length =.5 m and mass = 6 kg. What is total moment of inertia of a line parallel to the axis perpendicular to the stick's length of the combined object about the new center of mass?
The moment of inertia of the cube is <math>I = (1/6)ml^2</math>

'''Solution'''
1st, identify variables.
<math>{I}_{stick} = (1/12){m}_{stick}L^2</math>

<math>{m}_{stick} = 5 kg</math>

<math>{I}_{block} = (1/6){m}_{block}(S)^2</math>

<math>{m}_{block} = 6 kg</math>

Side length = S = .5 m

2nd, solve unknowns
Distance of new center of mass = <math>{\sum_{i=1}^n m_i {r}_i\over\sum_{i=1}^n m_i }. </math> = <math> 5*(0.5)+6*(-.25)/11 = 1/11</math>
(Remember the block is below the stick)

3rd, apply parallel axis theorem
<math>{I}_{stick,parallel} = {I}_{stick} + {m}_{stick}(.5-1/11)^2</math>

<math>{I}_{block,parallel} = {I}_{block} + {m}_{block}(.25+1/11)^2</math>

4th, plug in values

Total moment of inertia = <math>(1/12)*5*1^2 + 5(.5-1/11)^2 + (1/6)6(.5)^2 + 6(.25+1/11)^2 = \frac{581}{264}</math>

==Connectedness==
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==History==

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Parallel axis theorem

2025-12-02T22:11:25Z

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Parallel axis theorem

2025-12-02T20:18:55Z

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Parallel axis theorem

2025-12-02T20:05:42Z

FabianN2: /* The Main Idea */

Parallel axis theorem

2025-12-02T19:13:42Z

FabianN2: /* The Main Idea */

Parallel axis theorem

2025-12-01T16:47:34Z

FabianN2: /* The Main Idea */

Short Description of Topic

claimed by Fabian Nunez Fall 2025
==The Main Idea==

State, in your own words, the main idea for this topic

===A Mathematical Model===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

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===External links===

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==References==

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[[Category:Which Category did you place this in?]]

Parallel axis theorem

2025-12-01T16:45:50Z

FabianN2: Created page with "Short Description of Topic ==The Main Idea== Created by: Fabian Nunez Fall 2025 State, in your own words, the main idea for this topic Electric Field of Capacitor ===A Mathematical Model=== What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings. ===A Computational Model=== How do we..."

Short Description of Topic

==The Main Idea==

Created by: Fabian Nunez Fall 2025

State, in your own words, the main idea for this topic
Electric Field of Capacitor

===A Mathematical Model===

What are the mathematical equations that allow us to model this topic. For example <math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}</math> where '''p''' is the momentum of the system and '''F''' is the net force from the surroundings.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]