Short Answer:
The Perpendicular Axis Theorem states that the moment of inertia of a plane area about an axis perpendicular to its plane is equal to the sum of the moments of inertia of the area about two mutually perpendicular axes lying in its plane and intersecting at a point where the perpendicular axis passes.
Mathematically, it is written as:
where is the moment of inertia about the perpendicular axis, and and are the moments of inertia about the two perpendicular axes in the plane.
Detailed Explanation :
Perpendicular Axis Theorem
In Engineering Mechanics, the Perpendicular Axis Theorem is an important principle that applies to plane (two-dimensional) figures such as thin plates, laminae, or any flat surface. It is used to calculate the moment of inertia of a plane area about an axis that is perpendicular to its plane (usually called the z-axis) by knowing the moments of inertia about two perpendicular axes (x-axis and y-axis) that lie in the same plane.
This theorem is widely used in mechanical and structural engineering to analyze the rotational properties of thin bodies and is particularly useful for determining the polar moment of inertia, which measures a body’s resistance to twisting or torsional deformation.
Statement of the Theorem
The Perpendicular Axis Theorem can be stated as:
“The moment of inertia of a plane area about an axis perpendicular to its plane is equal to the sum of its moments of inertia about any two mutually perpendicular axes lying in the plane and intersecting at the point where the perpendicular axis meets the plane.”
Mathematically,
where,
- = Moment of inertia about the perpendicular axis (z-axis),
- = Moment of inertia about the x-axis,
- = Moment of inertia about the y-axis.
This relationship holds only for plane (flat) figures and not for three-dimensional bodies.
Derivation of the Perpendicular Axis Theorem
Let the plane area lie in the -plane. Consider a small elemental area at a point having coordinates .
The moment of inertia of this element about the x-axis is given by:
and about the y-axis:
Now, the moment of inertia of the same element about the z-axis (which is perpendicular to the plane) is given by:
where is the distance of the element from the z-axis.
From geometry,
Therefore,
Integrating over the whole area,
Hence,
This proves the Perpendicular Axis Theorem.
Physical Meaning
The theorem shows that the resistance of a plane area to rotation about an axis perpendicular to its plane (the z-axis) is equal to the combined resistances to rotation about two perpendicular axes lying in the plane (x and y axes).
It means the polar moment of inertia (I_Z) represents the total rotational resistance of the area about the perpendicular axis and is obtained by summing the two rectangular moments of inertia (I_X and I_Y).
Thus, the theorem helps in determining the distribution of an area relative to all three axes passing through the same point.
Important Notes
- The theorem is only valid for plane areas (laminae) and not for three-dimensional bodies.
- The three axes — X, Y, and Z — must intersect at the same point, usually the centroid of the area.
- The Z-axis must be perpendicular to the plane containing the X and Y axes.
- It is very useful for finding the polar moment of inertia, which is used in torsional stress analysis and rotational motion studies.
Applications of the Perpendicular Axis Theorem
- Calculation of Polar Moment of Inertia (J):
The theorem directly gives the polar moment of inertia,
which is used in torsional analysis of shafts, where resistance to twisting is needed.
- Composite Shapes:
For plane figures made up of multiple simple shapes (rectangles, circles, triangles, etc.), the theorem helps in calculating the total M.I. about the z-axis. - Structural Design:
Used to find moments of inertia for thin structural parts such as plates, beams, and cross-sections. - Mechanical Components:
Helps in analyzing thin rotating discs, flywheels, and rotors by determining how mass is distributed about different axes. - Balancing of Rotating Systems:
Useful in determining inertia properties for dynamic balancing of planar bodies.
Example
Example:
Find the moment of inertia of a circular plate of radius about its perpendicular axis passing through the center.
Solution:
For a circular plate,
Now, using the Perpendicular Axis Theorem,
Hence, the moment of inertia of the circular plate about the perpendicular axis through its center is:
This example shows how the theorem simplifies the calculation of the polar moment of inertia.
Relation to Polar Moment of Inertia
The polar moment of inertia () is defined as the moment of inertia of a plane area about an axis perpendicular to its plane and passing through a given point.
By the Perpendicular Axis Theorem,
This value is particularly important in the torsional analysis of shafts, where the ability to resist twisting depends directly on .
Importance of the Theorem
- Simplifies the computation of moment of inertia for complex plane shapes.
- Helps in understanding the relationship between bending and torsional resistance.
- Essential in designing machine components like discs, rotors, and flywheels.
- Used in mechanical analysis to ensure balanced rotation and structural stability.
Conclusion
In conclusion, the Perpendicular Axis Theorem is a fundamental principle in mechanics that relates the moment of inertia of a plane area about a perpendicular axis to the sum of its moments of inertia about two mutually perpendicular axes in the plane. It provides a simple and effective way to determine the polar moment of inertia and is widely applied in the design and analysis of mechanical and structural components. This theorem is vital in understanding the distribution of area and its resistance to rotational motion in planar systems.