Name: moment of inertia for a flat disc
Uploaded: 2025-05-22T09:06:08.000Z
Duration: 1 min 16 s
Channel: JoVE

The perpendicular axis theorem correlates the moments of inertia of planar objects around axes that are mutually perpendicular and concurrent.
The 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 axes lying in the plane of the body.
Consider a circular hoop of mass M and radius R lying along an x-y plane. The z-axis is perpendicular to the hoop&#39;s plane, and all three axes are concurrent at the hoop&#39;s center.
The moment of inertia about the z-axis equals the mass multiplied by the square of its radius.
The circular symmetry of the hoop ensures that the moments of inertia about the planar axes are equal.
Using the perpendicular axis theorem, the moment of inertia along the x-axis equals half the moment of inertia along the z-axis.
As a result, the moment of inertia of the hoop along a planar axis equals half the product of its mass and its radius squared.

perpendicular-axis theorem

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:
<img alt="Moment of inertia formula, Iz=MR&#178;/2, equation." src="/files/ftp_upload/14704/14704_Equation_1v2.svg" style="text-align: center; height: 42px;" data-altupdatedat="1754904545000" data-alt="Moment of inertia equation \( I_z = \frac{MR^2}{2} \); physics, mechanics formula." />
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.
<img alt="Angular momentum conservation diagram with formula Lz=Lx+Ly, educational physics concept." src="/files/ftp_upload/14704/14704_Equation_2v2.svg" style="text-align: center; height: 20px;" data-altupdatedat="1754904545000" data-alt="Angular momentum equation \( L_z = L_x + L_y \); physics concept diagram." />
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:
<img alt="Moment of inertia equation for a solid cylinder; shows Ix = MR&#178;/4; physics diagram." src="/files/ftp_upload/14704/14704_Equation_3v2.svg" style="text-align: center; height: 42px;" data-altupdatedat="1754904545000" data-alt="Moment of inertia equation \(I_x = \frac{MR^2}{4}\), static equilibrium analysis." />

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 ...

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KEY TERMS AND CONCEPTS

moment of inertia for a flat disc

The moment of inertia of a flat disc can be found using the perpendicular-axis theorem. For a planar object, the moment of inertia about an axis perpendicular to its plane equals the sum of the moments of inertia about two mutually perpendicular axes in the plane of the body.

For a circular disc with mass M and radius R, the disc lies in the x-y plane. The origin is at the center of the disc. The z-axis is perpendicular to the disc’s plane, and all three axes meet at the center.

For this disc, the moment of inertia about the axis through its center of mass and perpendicular to the disc follows the theorem. The moment of inertia along the z-axis equals the sum of the moments of inertia along the x-axis and y-axis. Because the disc is circular and symmetric, the moments of inertia about the two planar axes are equal. So, the moment of inertia along the z-axis is twice the moment of inertia along the x-axis. From this relation, the moment of inertia of the disc along the x-axis can be determined.

Moment of Inertia for a Flat Disc

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The moment of inertia of a flat disc can be found using the perpendicular-axis theorem. For a planar object, the moment of inertia about an axis perpendicular to its plane equals the sum of the moments of inertia about two mutually perpendicular axes in the plane of the body. For a circular disc with mass M and radius R, the disc lies in the x-y plane. The origin is at the center of the disc. The z-axis is perpendicular to the disc’s plane, and all three axes meet at the center. For this...

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key terms and concepts

Learn how the perpendicular-axis theorem links the x, y, and z moments of inertia for a circular disc centered on the origin in the xy-plane.

Moment of Inertia for a Flat Disc | JoVE K12

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Moment of Inertia for a Flat Disc

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