95.9.2 parallel axis theorem - san jose state university parallel axis theorem.pdf · recall the...
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Parallel Axis TheoremStevenVukazich
SanJoseStateUniversity
x
Recall the Definition of the Moment of Inertia of an Area About an Axis
𝑑𝐴 y'
𝐼$ = &𝑦(𝑑𝐴�
�
= & 𝑦* + 𝑑 (𝑑𝐴�
�
x'
y d
Consider an axis x’ that is parallel to the x axis and passes through the centroid of the area. The distance between the two parallel axes is d
y = y' + d
𝐶
x
Expand and Examine Terms
𝑑𝐴 y'
𝐼$ = & 𝑦* + 𝑑 (𝑑𝐴 =&𝑦′(�
�
𝑑𝐴�
�
+ 2𝑑&𝑦*𝑑𝐴 + 𝑑(&𝑑𝐴�
�
�
�
x'
y d𝐶
Moment of Inertia of the area about the x' axis
First moment of the area about the x' axis = 0
Area
x
Parallel Axis Theorem
𝑑𝐴 y'𝐼$̅* = &𝑦′(
�
�
𝑑𝐴x'
y d𝐶
Centroidal Moment of Inertia
𝐼$ = 𝐼$̅* + 𝑑(𝐴
General Form
𝐼 = 𝐼 ̅ + 𝑑(𝐴
Parallel Axis Theorem
If we know the moment of inertia of a body about an axis passing through its centroid, we can calculate the body’s moment of inertia about any parallel axis
x
y
Example Problem
𝑏
ℎ
Find the Moment of Inertia of the of the shaded area about the x and yaxes shown. Use the Parallel Axis Theorem.
Note that this we have already found Ix , Iy and the location of the centroid for this shape using integration.
Moment of Inertia About Centroidal Axes
x𝐶
ℎ
13ℎ
23𝑏
y
b
x'
y'
Use Tabulated Solution for 𝐼 ̅
𝐼$̅* =136𝑏ℎ5
𝐼6̅* =136ℎ𝑏5
Moment of Inertia About Centroidal Axes
x𝐶
ℎ
13ℎ = 𝑑$
23𝑏 = 𝑑6
y
b
x'
y'
𝐼$̅* =136𝑏ℎ5
𝐼6̅* =136ℎ𝑏5
𝐼 = 𝐼 ̅ + 𝑑(𝐴
𝐴 =12𝑏ℎ
𝐼$ = 𝐼$̅* + 𝑑$(𝐴 =
136𝑏ℎ5 +
13ℎ
( 12𝑏ℎ
Moment of Inertia About the xAxis
𝐼$ = 𝐼$̅* + 𝑑$(𝐴 =
136𝑏ℎ5 +
13ℎ
( 12𝑏ℎ
𝐼$ =136𝑏ℎ5 +
118𝑏ℎ5
𝐼$ =336𝑏ℎ5 =
112𝑏ℎ5
Agrees with both the tabulated solution and our result from integration
x
y
𝑏
ℎ
Moment of Inertia About the yAxis
𝐼6 = 𝐼6̅* + 𝑑6(𝐴 =
136ℎ𝑏5 +
23𝑏
( 12𝑏ℎ
𝐼6 =136ℎ𝑏5 +
418ℎ𝑏5
𝐼6 =936ℎ𝑏5 =
14ℎ𝑏5
Agrees with our resultfrom integration
x
y
𝑏
ℎ