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Perpendicular axis theorem

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The perpendicular axis theorem (or plane figure theorem) states that for a planar lamina the moment of inertia about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia about two mutually perpendicular axes in the plane of the lamina, which intersect at the point where the perpendicular axis passes through. This theorem applies only to planar bodies and is valid when the body lies entirely in a single plane.

Define perpendicular axes $x$ , $y$ , and $z$ (which meet at origin $O$ ) so that the body lies in the $xy$ plane, and the $z$ axis is perpendicular to the plane of the body. Let I_x, I_y and I_z be moments of inertia about axis x, y, z respectively. Then the perpendicular axis theorem states that^[1]

I_{z}=I_{x}+I_{y}

This rule can be applied with the parallel axis theorem and the stretch rule to find polar moments of inertia for a variety of shapes.

If a planar object has rotational symmetry such that $I_{x}$ and $I_{y}$ are equal,^[2] then the perpendicular axes theorem provides the useful relationship:

I_{z}=2I_{x}=2I_{y}

Derivation

[edit]

Working in Cartesian coordinates, the moment of inertia of the planar body about the $z$ axis is given by:^[3]

I_{z}=\int (x^{2}+y^{2})\,dm=\int x^{2}\,dm+\int y^{2}\,dm=I_{y}+I_{x}

On the plane, $z=0$ , so these two terms are the moments of inertia about the $x$ and $y$ axes respectively, giving the perpendicular axis theorem. The converse of this theorem is also derived similarly.

Note that $\int x^{2}\,dm=I_{y}\neq I_{x}$ because in $\int r^{2}\,dm$ , $r$ measures the distance from the axis of rotation, so for a y-axis rotation, deviation distance from the axis of rotation of a point is equal to its x coordinate.

References

[edit]

^ Tipler, Paul A. (1976). "Ch. 12: Rotation of a Rigid Body about a Fixed Axis". Physics. Worth Publishers Inc. ISBN 0-87901-041-X.
^ Obregon, Joaquin (2012). Mechanical Simmetry. Author House. ISBN 978-1-4772-3372-6.
^ K. F. Riley, M. P. Hobson & S. J. Bence (2006). "Ch. 6: Multiple Integrals". Mathematical Methods for Physics and Engineering. Cambridge University Press. ISBN 978-0-521-67971-8.

Perpendicular axis theorem

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from Grokipedia

The perpendicular axis theorem, also known as the plane figure theorem, is a fundamental relation in classical mechanics that applies to planar (two-dimensional) objects, stating that the moment of inertia

I_z

about an axis perpendicular to the plane of the object is equal to the sum of the moments of inertia

I_x

and

I_y

about two mutually perpendicular axes lying in that plane and intersecting at the same point.^[1]^[2] Mathematically, this is expressed as

I_z = I_x + I_y

, where the moments are calculated relative to the common origin, and the theorem holds for any lamina or thin rigid body with mass distribution confined to a single plane, assuming uniform density or arbitrary distribution as long as the axes are properly chosen.^[3]^[4] This theorem is particularly useful in rotational dynamics for simplifying calculations of moments of inertia, especially when the perpendicular-axis moment is known or easier to compute, allowing derivation of in-plane moments by subtraction or symmetry.^[2] For example, it is commonly applied to symmetric shapes like thin disks, rings, or rectangular plates; for a uniform thin disk of mass

M

and radius

R

, the moment about the central perpendicular axis is

\frac{1}{2}MR^2

, so each in-plane axis through the center yields

\frac{1}{4}MR^2

due to symmetry.^[3]^[4] The theorem derives from the basic definition of moment of inertia as

\int r^2 \, dm

, where the perpendicular distance

r^2 = x^2 + y^2

in the plane splits naturally into components along the x- and y-axes, making it a direct consequence of vector geometry in rigid body mechanics.^[2] It complements the parallel axis theorem for shifting axes and is essential in engineering applications such as analyzing the rotation of flywheels, blades, or other planar components in machinery.^[1]

Background Concepts

Definition of Moment of Inertia

The moment of inertia, often denoted as

I

, quantifies a rigid body's resistance to angular acceleration about a specific axis of rotation. For a continuous rigid body, it is defined mathematically as the integral

I = \int r^2 \, dm

, where

r

is the perpendicular distance from the axis to the infinitesimal mass element

dm

, and the integration is taken over the entire mass distribution of the body.^[5] This formulation arises from considering the body as composed of point masses, where the contribution of each is

m_i r_i^2

, summed or integrated accordingly.^[6] Physically, the moment of inertia serves as the rotational analog to mass in linear motion, determining the torque

\tau

required to produce a given angular acceleration

\alpha

via Newton's second law for rotation:

\tau = I \alpha

.^[7] It depends not only on the total mass but also on how that mass is distributed relative to the axis; masses farther from the axis contribute more significantly to

I

, increasing the body's reluctance to change its rotational speed. The SI unit of moment of inertia is kilogram square meter (kg·m²).^[8] For rotation about a fixed axis, the moment of inertia is a scalar quantity. However, in general three-dimensional rigid body dynamics, the full description requires the moment of inertia tensor, a 3×3 symmetric matrix that relates the angular velocity vector

\vec{\omega}

to the angular momentum vector $\vec{L}$

through $\vec{L} = \mathbf{I} \vec{\omega}$

=Iω

.^[9] This tensor accounts for the coupling between rotations about different axes, with its diagonal elements representing moments about the principal axes and off-diagonal elements (products of inertia) capturing cross-terms. The concept emerged in the 18th century through the work of Leonhard Euler, who formalized it in his 1765 treatise Theoria Motus Corporum Solidorum as "momentum inertiae," building on earlier ideas from János András Segner on gyroscopic motion and rigid body principles.^[10] Further refinements in the 19th century, including vector formulations by Louis Poinsot and William Rankine, solidified its role in classical mechanics.^[10] For planar bodies, where the mass distribution is confined to a single plane, the moment of inertia simplifies, often focusing on the axis perpendicular to that plane.^[8]

Moments for Planar Bodies

In the context of rigid body dynamics, planar bodies, also known as laminae, are idealized as two-dimensional objects with mass distributed uniformly or arbitrarily within the xy-plane and negligible thickness along the z-direction. This assumption simplifies the analysis by confining all mass elements to z = 0, allowing the moment of inertia to be computed via double integrals over the area of the lamina.^[11] The density function ρ(x, y) describes the mass per unit area, which may be constant for uniform laminae or vary for non-uniform distributions, enabling the total mass m to be expressed as m = ∬_D ρ(x, y) dA, where D is the region occupied by the body.^[12] The coordinate system is typically established with the origin at an arbitrary point or, preferably, the center of mass for rotational dynamics calculations, and the z-axis oriented perpendicular to the xy-plane. For axes lying in the plane of the lamina, the moment of inertia quantifies resistance to rotation about those directions: about the x-axis, it is I_x = ∬_D y² ρ(x, y) dA, where y represents the perpendicular distance from the axis to each mass element dm = ρ(x, y) dA; similarly, about the y-axis, I_y = ∬_D x² ρ(x, y) dA, with x as the distance.^[13] In contrast, the moment of inertia about the z-axis, perpendicular to the plane, accounts for distances in both in-plane directions and is given by I_z = ∬_D (x² + y²) ρ(x, y) dA.^[11] To illustrate, consider a single point mass m positioned at coordinates (x, y) within the plane, which serves as the fundamental building block for extended bodies. The moment of inertia of this point mass about the z-axis passing through the origin is simply I_z = m (x² + y²), where √(x² + y²) is the radial distance from the axis; about the x-axis, it would be I_x = m y², and about the y-axis, I_y = m x².^[11] This example highlights how the perpendicular axis configuration leverages the planar geometry, with distances measured solely in the xy-plane.^[14]

Statement of the Theorem

Mathematical Formulation

The perpendicular axis theorem states that for a lamina or planar mass distribution lying in the

xy

-plane, the moment of inertia

I_z

about an axis perpendicular to the plane (the

z

-axis) passing through a given point is equal to the sum of the moments of inertia

I_x

and

I_y

about two mutually perpendicular axes (the

x

- and

y

-axes) in the plane that intersect at the same point.^[1]^[15] In terms of the inertia tensor for such a two-dimensional mass distribution, the theorem corresponds to the relation among the diagonal elements:

I_{zz} = I_{xx} + I_{yy}

, where the off-diagonal elements involving the

z

-direction vanish due to the planarity.^[16]^[17] The axes must be mutually perpendicular and concurrent at the same point, which is often taken as the center of mass for convenience, though the theorem holds for any common intersection point in the plane.^[1]^[4] This relation applies exclusively to planar (two-dimensional) mass distributions; for three-dimensional bodies with thickness or extent along the perpendicular direction, the equality becomes an inequality

I_{zz} \leq I_{xx} + I_{yy}

, with equality only in the limiting case of zero thickness.^[16]^[18]

Geometric Interpretation

The perpendicular axis theorem provides an intuitive geometric understanding of how the moment of inertia about an axis perpendicular to a plane relates to those about in-plane axes. For a planar mass distribution in the xy-plane, consider a small mass element dm located at coordinates (x, y). The distance from this element to the z-axis (perpendicular to the plane) is the radial distance r = √(x² + y²), so the squared distance r² = x² + y² represents the perpendicular distance contributing to rotational resistance about the z-axis. This decomposition shows that the contributions to the moment of inertia from deviations along the x and y directions add directly, as the total "spread" of mass relative to the z-axis is the quadrature sum of the individual spreads along the perpendicular in-plane axes.^[15] This additive nature illustrates why the theorem holds for planar bodies: each mass element's contribution to the perpendicular moment I_z is the sum of its separate contributions to I_x and I_y, reflecting independent "resistances" to rotation from the x- and y-directions. For instance, a mass element far from the x-axis but close to the y-axis will primarily contribute to I_x (due to its y-deviation) and thus to I_z, but minimally to I_y; conversely, elements offset primarily in x will dominate I_y and I_z. This independent summation underscores the theorem's utility in visualizing how in-plane mass distributions build the overall rotational inertia about the out-of-plane axis.^[1] The geometric rationale draws a direct analogy to the Pythagorean theorem, where the hypotenuse squared (r²) equals the sum of the squares of the legs (x² + y²) in a right triangle formed by the axes and the position vector to the mass element. Just as perpendicular distances combine in quadrature to yield the total distance, the moments of inertia about perpendicular in-plane axes combine additively to give the perpendicular moment, providing a visual bridge between Euclidean geometry and rotational dynamics.^[15] To visualize this, imagine a coordinate system with the xy-plane containing the lamina and the z-axis piercing the origin perpendicularly. For example, mass elements clustered along the x-axis (where y=0) contribute significantly to I_y due to their distances from the y-axis (x-coordinates), while contributions to I_x are zero since y=0, so I_z = I_y. Off-axis elements, such as those in the first quadrant, contribute to both I_x and I_y via their respective distances, with the z-axis seeing the full Pythagorean combination. A conceptual diagram would show arrows from the origin along x, y, and z, with position vectors to sample masses highlighting how x- and y-offsets vectorially sum to r, emphasizing differential contributions to each moment.^[19]

Proof and Derivation

Derivation from First Principles

The perpendicular axis theorem relates the moments of inertia of a planar lamina about three mutually perpendicular axes, with two axes lying in the plane of the lamina and the third perpendicular to it. To derive this from first principles, begin with the general definition of the moment of inertia for a continuous mass distribution about an arbitrary axis, which is the integral of the squared perpendicular distance from the axis to each mass element

dm

. For a lamina confined to the

xy

-plane (where the

z

-coordinate is zero for all mass elements), consider the moments about the coordinate axes originating at some point in the plane. The moment of inertia

I_z

about the

z

-axis, perpendicular to the plane, is the integral over the entire mass distribution:

I_z = \int (x^2 + y^2) \, dm,

where

x

and

y

are the coordinates of the mass element

dm

relative to the origin, and the integral extends over the area of the lamina.^[2]^[16] Now, the moment of inertia

I_x

about the

x

-axis (lying in the plane) is defined as the integral of the squared distance to this axis. For points in the

xy

-plane, the perpendicular distance to the

x

-axis is

|y|

(with

z = 0

), so

I_x = \int (y^2 + z^2) \, dm = \int y^2 \, dm,

since

z = 0

. Similarly, the moment of inertia

I_y

about the

y

-axis is

I_y = \int (x^2 + z^2) \, dm = \int x^2 \, dm.

These expressions follow directly from the general three-dimensional definition of moment of inertia, specialized to the planar case where the thickness in the

z

-direction is negligible.^[2]^[16] Adding

I_x

and

I_y

yields

I_x + I_y = \int y^2 \, dm + \int x^2 \, dm = \int (x^2 + y^2) \, dm = I_z.

This equality holds without additional assumptions beyond the planarity of the lamina and the use of Cartesian coordinates. For a lamina with constant areal mass density

\sigma

, each mass element is

dm = \sigma \, dA

, where

dA = dx \, dy

is the differential area element in the

xy

-plane, so the integrals become double integrals over the region's boundaries:

I_z = \sigma \iint_R (x^2 + y^2) \, dx \, dy, \quad I_x = \sigma \iint_R y^2 \, dx \, dy, \quad I_y = \sigma \iint_R x^2 \, dx \, dy,

with

R

denoting the area of the lamina, confirming the relation

I_z = I_x + I_y

through direct integration. This step-by-step setup demonstrates the theorem's origin in the additive nature of squared distances in the plane.^[16]

Using Coordinate Axes

The perpendicular axis theorem can be derived using the inertia tensor in a Cartesian coordinate system for a planar body confined to the

xy

-plane, where the

z

-coordinate is zero for all mass elements. The inertia tensor

\mathbf{I}

for such a body, with respect to axes passing through a common point (typically the center of mass), has diagonal components given by

I_{xx} = \int (y^2 + z^2) \, dm = \int y^2 \, dm, \quad I_{yy} = \int (x^2 + z^2) \, dm = \int x^2 \, dm, \quad I_{zz} = \int (x^2 + y^2) \, dm,

since

z = 0

. The off-diagonal products of inertia, such as

I_{xy} = \int xy \, dm

, may be nonzero unless the axes are chosen as principal axes, where these terms vanish by diagonalization of the tensor.^[20]^[21] The theorem emerges directly from these components:

I_{zz} = \int x^2 \, dm + \int y^2 \, dm = I_{yy} + I_{xx}

. This relation holds for any pair of orthogonal axes

x

and

y

in the plane, as the sum

I_{xx} + I_{yy}

equals the invariant trace of the in-plane sub-tensor, which is independent of rotation about the

z

-axis and always matches

I_{zz}

. In principal axes, where off-diagonal terms are zero,

I_{xx}

and

I_{yy}

represent the principal moments, but the theorem's validity does not require this condition.^[22]^[16] For a general three-dimensional body, the relation does not hold as an equality because nonzero

z

-coordinates contribute to

I_{xx}

and

I_{yy}

via the

z^2

terms, making

I_{xx} + I_{yy} > I_{zz}

. The full trace of the inertia tensor,

I_{xx} + I_{yy} + I_{zz} = 2 \int (x^2 + y^2 + z^2) \, dm

, remains invariant under orthogonal transformations of the axes, but the planar simplification relies on the absence of

z

-contributions.^[16]^[21]

Applications and Examples

Simple Geometric Shapes

The perpendicular axis theorem provides a practical method to verify moments of inertia for simple planar shapes by relating the polar moment

I_z

to the in-plane moments

I_x

and

I_y

. For these examples, the shapes are assumed to be uniform laminae lying in the

xy

-plane with the centroid at the origin, and all axes pass through the centroid. The calculations demonstrate the theorem's relation

I_z = I_x + I_y

, where moments are computed using standard formulas derived from integration over the mass distribution. Consider a rectangular lamina of mass

M

, length

a

along the

x

-axis, and width

b

along the

y

-axis. The moment of inertia about the

x

-axis is

I_x = \frac{1}{12} M b^2

, as this integrates the squared distances in the

y

-direction. Similarly, about the

y

-axis,

I_y = \frac{1}{12} M a^2

, integrating squared distances in the

x

-direction. Applying the theorem yields

I_z = I_x + I_y = \frac{1}{12} M (a^2 + b^2)

. To verify independently, the polar moment

I_z = \int (x^2 + y^2) \, dm

over the rectangle directly computes to the same value, confirming the relation.^[23] For a thin rod of mass

M

and length

L

aligned along the

x

-axis, the negligible thickness implies

I_x \approx 0

, since distances in the

y

- and

z

-directions are zero. The moment about the

y

-axis is

I_y = \frac{1}{12} M L^2

, integrating squared

x

-distances from

-\frac{L}{2}

\frac{L}{2}

. By symmetry for the thin rod in the plane,

I_z = \frac{1}{12} M L^2

. The theorem gives

I_z = I_x + I_y \approx \frac{1}{12} M L^2

, matching the direct computation of

I_z = \int x^2 \, dm

, which equals

I_y

due to the rod's linearity. A uniform circular disk of mass

M

and radius

R

serves as another illustration. The moments about the diameter axes are

I_x = I_y = \frac{1}{4} M R^2

, obtained by integrating squared perpendicular distances over the disk area. The theorem predicts

I_z = I_x + I_y = \frac{1}{2} M R^2

. Independent calculation of the polar moment

I_z = \int r^2 \, dm

, using polar coordinates, yields

\frac{1}{2} M R^2

, verifying the equality and highlighting the theorem's utility for symmetric shapes.^[24] For an equilateral triangular lamina of mass

M

and side length

L

, the height is

H = \frac{\sqrt{3}}{2} L

H=23

L. Due to symmetry, $I_x = I_y = \frac{1}{24} M L^2$ , where $I_x$ is about the axis parallel to the base through the centroid (equivalent to $\frac{1}{18} M H^2$ ). The theorem gives $I_z = I_x + I_y = \frac{1}{12} M L^2$ . Direct integration for $I_z = \int (x^2 + y^2) \, dm$ over the triangular area confirms this result, as the contributions from $x^2$ and $y^2$ sum symmetrically.^[25]

Practical Uses in Physics and Engineering

The perpendicular axis theorem plays a key role in rotational dynamics by simplifying the computation of moments of inertia for planar objects, thereby facilitating calculations of torque and angular momentum in systems involving flat components such as turbine blades or solar panels. For instance, when analyzing the rotational stability of a thin disk or lamina under applied torques, the theorem allows engineers to derive the perpendicular moment directly from in-plane values, reducing computational complexity in dynamic simulations.^[1] This approach is particularly valuable in scenarios where direct measurement about the perpendicular axis is impractical, enabling efficient prediction of angular acceleration via Newton's second law for rotation.^[26] In engineering applications, the theorem finds an analog in the second moment of area for structural analysis, where it relates the polar moment to in-plane moments, aiding in the design of beams and shafts to resist torsional deflection. For beam deflection under twisting loads, this relation helps quantify shear stress distribution and ensures structural integrity in components like propeller shafts or bridge girders.^[27] Similarly, in flywheel design for energy storage systems, such as those in gyroscopes, the theorem is used to compute the moment of inertia for symmetric planar flywheels, optimizing rotational inertia to maintain stable angular momentum during operation.^[28] In satellite attitude control, it supports modeling the dynamics of planar solar arrays or debris panels, where accurate inertia tensors are essential for predicting tumbling motion and stabilizing orientation in orbit.^[29] For composite bodies, the theorem is applied to individual laminas to determine their in-plane moments before combining with the parallel axis theorem to account for offsets, which is crucial in assembling complex structures like multi-layer panels or rotor assemblies. This method streamlines the inertia calculation for offset components in machinery.^[1] In modern contexts, such as robotics, it aids in computing arm link inertias for cylindrical or planar segments, informing dynamic control algorithms to minimize energy loss during motion.^[30] Software tools like MATLAB leverage the theorem in 2D simulations for verifying moment calculations in robotic or mechanical designs, allowing rapid iteration on prototypes. Despite its utility, the theorem assumes perfect planarity, limiting its direct application to strictly 2D objects; for components with slight thickness, such as thin plates, approximations are used or full 3D inertia tensors are computed to avoid errors in dynamic predictions.^[1] These practical constraints highlight the need for validation against experimental data in real-world implementations. The theorem often builds on moments derived for simple geometric shapes to extend analysis to these engineered systems.

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Perpendicular axis theorem

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Perpendicular axis theorem

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Definition of Moment of Inertia

Moments for Planar Bodies

Statement of the Theorem

Mathematical Formulation

Geometric Interpretation

Proof and Derivation

Derivation from First Principles

Using Coordinate Axes

Applications and Examples

Simple Geometric Shapes

Practical Uses in Physics and Engineering

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