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10.12: Perpendicular-Axis Theorem

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Perpendicular-Axis Theorem
 
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10.12: Perpendicular-Axis Theorem

The perpendicular-axis theorem states that the moment of inertia of a planar object about an axis perpendicular to its plane is equal to the sum of the moments of inertia about two mutually perpendicular concurrent axes lying in the plane of the body.

Consider a circular disc of mass M and radius R lying along an x-y plane. The origin lies at the center of the disc, and the z-axis is perpendicular to the disc's plane. All three axes coincide at the disc's center. The moment of inertia of this disc about an axis passing through its center of mass and perpendicular to the disc is given by the following:

Equation1

According to the perpendicular axis theorem, the moment of inertia along the z-axis equals the sum of the moments of inertia along the x-axis and y-axis.

Equation2

The circular symmetry of the disc ensures that the moments of inertia about the planar axes are equal. So, the moment of inertia along the z-axis is twice the moment of inertia along the x-axis.

As a result, the moment of inertia of the disc along the x-axis is obtained as follows:

Equation3


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Moment Of Inertia About Z-axis = I_z = (1/2)MR^2 Moment Of Inertia About X-axis = I_x = (1/4)MR^2 Moment Of Inertia About Y-axis = I_y = (1/4)MR^2 According To The Perpendicular-axis Theorem: I_z = I_x + I_y (1/2)MR^2 = (1/4)MR^2 + (1/4)MR^2 (1/2)MR^2 = (1/2)MR^2

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