Parallel axis theorem

The parallel axis theorem is a key principle in classical mechanics that allows the calculation of the moment of inertia of a rigid body about any axis parallel to an axis passing through its center of mass.^[1] It states that the moment of inertia I about the arbitrary axis equals the moment of inertia I_{cm} about the center-of-mass axis plus the product of the body's total mass M and the square of the perpendicular distance d between the two axes: I = I_{cm} + M d^2.^[2] This theorem simplifies computations for rotational dynamics by leveraging known central moments of inertia.^[3] In rigid body mechanics, the parallel axis theorem is essential for analyzing systems where rotation occurs about non-central axes, such as in pulleys, pendulums, or compound objects like dumbbells composed of multiple parts.^[4] For instance, it enables the determination of the moment of inertia for a uniform rod rotating about one end by adding M (L/2)^2 to the central value of (1/12) M L^2, yielding (1/3) M L^2.^[2] The theorem's derivation follows from the definition of moment of inertia as \sum m_i r_i^2, where shifting the axis introduces a cross term that simplifies to the M d^2 addition.^[1] It applies strictly to parallel axes and assumes the body is rigid, making it a cornerstone for problems involving torque, angular acceleration, and energy in rotational motion.^[3] Beyond mass moments of inertia in physics, an analogous form of the theorem extends to the second moment of area in structural engineering, relating area moments about parallel axes through the centroid, though the physics context predominates in its classical formulation.^[5] This versatility underscores its broad utility in both theoretical and applied sciences, from deriving rotational kinetic energy to designing mechanical systems.^[6]

For Mass Moment of Inertia

Statement and Conditions

The parallel axis theorem for the mass moment of inertia states that the moment of inertia I of a rigid body about any axis is equal to the moment of inertia I_\text{cm} about a parallel axis passing through the center of mass plus the product of the total mass M and the square of the perpendicular distance d between the axes:

I = I_\text{cm} + M d^2.

Here, I_\text{cm} is the moment of inertia about the axis through the center of mass parallel to the given axis, defined as I_\text{cm} = \int r_\perp^2 \, dm, where r_\perp is the perpendicular distance from the mass element dm to the axis. This holds for any rigid body, whether planar or three-dimensional, as long as the axes are parallel and the body is rigid. The distance d is the shortest (perpendicular) distance between the two parallel axes.^[2] The theorem assumes the center of mass is known and I_\text{cm} is calculated (or looked up) for the parallel axis through it. It applies to composite bodies by treating the total mass and overall center of mass, or summing contributions from parts after finding the system's center of mass. In rotational dynamics, this facilitates computing torques and angular motion for off-center rotations, such as in physical pendulums or flywheels. For planar bodies rotating about an axis perpendicular to the plane, the moment corresponds to the polar form, but the theorem itself is not limited to this case.^[2]

Derivation

The parallel axis theorem for mass moments of inertia can be derived directly from the definition, applicable to any rigid body. Consider an axis through point O in direction \mathbf{u} (unit vector), parallel to an axis through the center of mass G. Let \mathbf{D} be the vector from O to G. For a mass element dm at position \mathbf{r} relative to G, its position relative to O is \mathbf{r} - \mathbf{D}. The moment of inertia about the axis through O is

I = \int |( \mathbf{r} - \mathbf{D} ) \times \mathbf{u} |^2 \, dm.

Expanding,

|( \mathbf{r} - \mathbf{D} ) \times \mathbf{u} |^2 = | \mathbf{r} \times \mathbf{u} |^2 + | \mathbf{D} \times \mathbf{u} |^2 - 2 ( \mathbf{r} \times \mathbf{u} ) \cdot ( \mathbf{D} \times \mathbf{u} ).

The integral becomes

I = \int | \mathbf{r} \times \mathbf{u} |^2 \, dm + | \mathbf{D} \times \mathbf{u} |^2 \int dm - 2 ( \mathbf{D} \times \mathbf{u} ) \cdot \int ( \mathbf{r} \times \mathbf{u} ) \, dm.

The first term is I_\text{cm}. The second is M d^2, where d = | \mathbf{D} \times \mathbf{u} | is the perpendicular distance between axes. The cross term vanishes because \int \mathbf{r} \, dm = 0 at the center of mass, so \int \mathbf{r} \times \mathbf{u} \, dm = \left( \int \mathbf{r} \, dm \right) \times \mathbf{u} = 0. Thus,

I = I_\text{cm} + M d^2.

This holds without assuming planarity. For planar lamina in the xy-plane rotating about a z-axis (perpendicular to the plane), a coordinate-based derivation simplifies similarly, with the cross term \int x \, dm = 0 at the center of mass.^[2]

Examples

A classic application of the parallel axis theorem is the calculation of the moment of inertia for a thin uniform rod of length L and mass M rotating about an axis perpendicular to its length. The moment of inertia about an axis through the center of mass is I_\text{cm} = \frac{1}{12} M L^2.^[7] To find the moment of inertia about a parallel axis through one end of the rod, where the perpendicular distance from the center of mass is d = L/2, the theorem gives I_\text{end} = I_\text{cm} + M d^2 = \frac{1}{12} M L^2 + M (L/2)^2 = \frac{1}{3} M L^2.^[8] For a uniform rectangular plate of mass M, width a, and height b, the moment of inertia about an axis through the center of mass parallel to the width a (perpendicular to the height) is I_\text{cm} = \frac{1}{12} M b^2. Applying the parallel axis theorem to shift the axis to one edge parallel to the width, at a distance d = b/2 from the center of mass, yields I_\text{edge} = \frac{1}{12} M b^2 + M (b/2)^2 = \frac{1}{3} M b^2. In composite systems, the theorem facilitates combining moments for connected bodies. Consider a dumbbell consisting of two point masses m each, separated by a massless rod of length $2r, with the center of mass at the midpoint. The moment of inertia about the center of mass axis perpendicular to the rod is I_\text{cm} = 2 m r^2.^[4] For a parallel axis displaced by distance d from the center of mass along the rod, the theorem applies directly to the total system: I = I_\text{cm} + (2m) d^2 = 2 m r^2 + 2 m d^2. The parallel axis theorem holds for bodies with non-uniform density, provided the center of mass is correctly identified. Common pitfalls in applying the theorem include failing to ensure the axes are parallel or incorrectly measuring the perpendicular distance d between them, which invalidates the relation since it assumes rigid body rotation about parallel lines.^[8] For complex three-dimensional shapes, the theorem extends naturally within the inertia tensor framework to handle off-axis components.

For Second Moment of Area

Statement and Conditions

The second moment of area, also known as the area moment of inertia, quantifies the distribution of area relative to an axis in the plane of the cross-section, essential for analyzing resistance to bending in structural elements like beams. For a plane area in the xy-plane, the second moments about the x- and y-axes through a reference point are defined as

I_x = \int y^2 \, dA, \quad I_y = \int x^2 \, dA,

where x and y are the perpendicular distances from the differential area element dA to the respective axes, and the product moment is I_{xy} = \int x y \, dA. These apply to cross-sections of prismatic members, such as beams or columns, where bending occurs about axes in the plane perpendicular to the loading direction.^[9] The parallel axis theorem for the second moment of area states that the moment I about any axis equals the moment I_c about a parallel centroidal axis plus the product of the total area A and the square of the perpendicular distance d between the axes:

I = I_c + A d^2.

The centroidal axis must pass through the geometric centroid of the area, defined where the first moments \int x \, dA = 0 and \int y \, dA = 0. This relation facilitates calculations for arbitrary parallel axes in engineering analyses of non-centroidal bending.^[10] The theorem requires the axes to be parallel and lie in the plane of the area (or parallel to it for thin sections), assuming a rigid, planar distribution where thickness variations are negligible. It applies to homogeneous or composite shapes by treating components separately relative to a common axis. In beam theory, these moments determine flexural rigidity, with the theorem enabling computation for offset axes, such as in eccentric loading or built-up sections. Note that the polar second moment J = I_x + I_y follows a similar form but is covered separately for torsional applications.^[5]

Derivation

The parallel axis theorem for the second moment of area can be derived directly from the integral definition, assuming a planar area with the centroid at the origin and parallel axes in the plane.^[9] Consider I_x = \int y^2 \, dA about the centroidal x-axis. For a parallel x'-axis displaced by distance d in the y-direction, the perpendicular distance from a point at (x, y) to the new axis is y - d. Thus,

I_x' = \int (y - d)^2 \, dA = \int (y^2 - 2 d y + d^2) \, dA = \int y^2 \, dA - 2 d \int y \, dA + d^2 \int dA.

The first term is I_{x,c}, the last is d^2 A, and the cross term vanishes because \int y \, dA = 0 at the centroid, yielding I_x' = I_{x,c} + A d^2. For a general displacement with components (d_x, d_y), the theorem applies separately to I_x (using d_y) and I_y (using d_x), or to the product moment. This holds for planar areas under the assumptions of rigidity and well-defined centroidal properties.^[10] An alternative approach uses the definition of the centroid. Shifting the reference shifts the coordinates, and the expansion of the squared distance leads to the same result, with the linear terms zero by the centroid condition. For composite shapes, the theorem applies additively to each part's contribution.^[5]

Applications

The parallel axis theorem plays a crucial role in structural engineering for calculating the second moment of area (I) of composite cross-sections in beams, such as built-up members composed of multiple geometric shapes like plates, angles, or channels welded together. For such sections, the total second moment of area about the neutral axis is determined by summing the individual moments of inertia of each component about their own centroids (I_local) and adding the term A d² for each, where A is the cross-sectional area of the component and d is the perpendicular distance from its centroid to the overall neutral axis. This approach simplifies the analysis of complex shapes by avoiding direct integration over irregular geometries.^[11] A representative example is the I-beam, a common rolled or fabricated steel section used in building frames and bridges, where the flanges and web are treated as rectangular components. To find the second moment of area about the horizontal neutral axis passing through the beam's centroid, the I_local for each flange and the web is calculated separately, then adjusted using the parallel axis theorem to account for the offset distances from the section's centroid; this yields the total I required for deflection and stress predictions. In practice, this method allows engineers to verify or customize I-beam properties for specific loading conditions, such as shifting the reference axis from the centroid to an arbitrary parallel line for localized stress computations at supports or connections.^[12] The theorem is integral to the flexural stress formula in beam design, σ = (M y) / I, where σ is the normal bending stress at a distance y from the neutral axis, M is the bending moment, and I is the second moment of area about the bending axis. By enabling accurate computation of I for non-centroidal axes parallel to the principal one, the parallel axis theorem ensures reliable stress distribution predictions, which are essential for ensuring beam safety under transverse loads in applications like girders and columns.^[13] However, the theorem's application in beam analysis assumes elastic behavior and homogeneous, isotropic materials, limiting its validity to linear elastic regimes where deformations remain small and proportional to loads; it does not account for plastic deformation or nonlinear material responses, which require advanced methods like plastic section modulus calculations. Historically, the parallel axis theorem underpinned calculations in Euler-Bernoulli beam theory during 19th-century structural engineering, facilitating the design of iron and early steel bridges by allowing efficient determination of section properties for bending resistance in truss and girder systems.^[14]

For Polar Moment of Inertia

Statement and Conditions

The polar moment of inertia, often denoted as J, measures the distribution of area or mass relative to a reference axis perpendicular to the plane of interest, playing a key role in analyzing rotational resistance, particularly in torsion. For a plane area, it is defined as

J = \int r^2 \, dA,

where r is the radial distance from the axis to the differential area element dA. For a mass distribution in a plane, the analogous quantity is

J = \int r^2 \, dm,

with r the perpendicular distance from the axis to the differential mass element dm. These definitions apply to planar configurations, such as cross-sections of shafts or thin bodies undergoing rotation about an axis normal to their plane.^[15]^[16] The parallel axis theorem extends to the polar moment of inertia, stating that the polar moment J about any axis perpendicular to the plane equals the polar moment J_c about a parallel centroidal axis plus the product of the total area A (or mass M) and the square of the perpendicular distance d between the axes:

J = J_c + A d^2

for area moments, or equivalently J = J_c + M d^2 for mass moments. The centroidal axis must pass through the geometric centroid of the area or the center of mass, ensuring the theorem accounts for the shift in the reference point while preserving the parallel orientation of the axes. This relation holds for both area and mass distributions, facilitating calculations for non-centroidal axes in engineering analyses.^[5]^[9] The theorem requires that both axes be parallel and perpendicular to the plane containing the area or mass distribution, typically for thin sections where out-of-plane variations are negligible or for idealized planar dynamics. It assumes the body is rigid and the distribution is uniform in the sense that the centroidal properties are well-defined, making it applicable to composite shapes by applying the theorem to each component relative to a common axis. In planar torsional dynamics, such as the twisting of circular or non-circular shafts, the polar moment quantifies resistance to shear stresses, with the theorem enabling efficient computation of J for axes offset from the centroid, as in eccentric loading or built-up sections. The polar moment also relates directly to the second moments of area about orthogonal in-plane axes through the same point, where J = I_x + I_y, linking it to bending properties without altering the parallel axis application.^[5]^[17]

Derivation

The parallel axis theorem for the mass moment of inertia about an axis perpendicular to the plane of a lamina (often called the polar moment of inertia in this context) can be derived in multiple ways, assuming a planar mass distribution with the center of mass at the origin and the axes parallel and perpendicular to the plane.^[2] One approach leverages the relationship between the polar moment and the second moments about the principal axes. For a lamina in the xy-plane, the polar moment of inertia about the z-axis through the center of mass is I_{z,\text{cm}} = I_{x,\text{cm}} + I_{y,\text{cm}}, where I_{x,\text{cm}} = \int y^2 \, dm and I_{y,\text{cm}} = \int x^2 \, dm. Applying the parallel axis theorem to each second moment separately for a displacement of the origin by (d_x, d_y) yields I_x = I_{x,\text{cm}} + M d_y^2 and I_y = I_{y,\text{cm}} + M d_x^2, where M is the total mass. Thus, the polar moment about the displaced axis is

\begin{aligned} I_z &= I_x + I_y \\ &= (I_{x,\text{cm}} + M d_y^2) + (I_{y,\text{cm}} + M d_x^2) \\ &= I_{z,\text{cm}} + M (d_x^2 + d_y^2) \\ &= I_{z,\text{cm}} + M d^2, \end{aligned}

with d = \sqrt{d_x^2 + d_y^2} the perpendicular distance between the axes.^[2]^[9] A direct derivation uses integration in Cartesian coordinates. Consider the polar moment about the displaced z'-axis, parallel to the z-axis through the center of mass and located at (d, 0) for simplicity. The distance from a mass element dm at (x, y) to the new axis is \sqrt{(x - d)^2 + y^2}, so

I_z = \int [(x - d)^2 + y^2] \, dm = \int (x^2 - 2dx + d^2 + y^2) \, dm = \int (x^2 + y^2) \, dm - 2d \int x \, dm + d^2 \int dm.

The first term is I_{z,\text{cm}}, the last is d^2 M, and the cross term vanishes because \int x \, dm = 0 at the center of mass, yielding I_z = I_{z,\text{cm}} + M d^2. For a general displacement direction, the result generalizes to the same form.^[2] Alternatively, in polar coordinates centered at the center of mass, the distance to the displaced axis is r' = \sqrt{r^2 + d^2 - 2 r d \cos \theta}, where r and \theta locate dm relative to the center of mass. Squaring gives r'^2 = r^2 + d^2 - 2 r d \cos \theta, so

I_z = \int r'^2 \, dm = \int r^2 \, dm + d^2 M - 2 d \int r \cos \theta \, dm.

The integral \int r \cos \theta \, dm represents the first moment about the displacement direction and equals zero at the centroid, again resulting in I_z = I_{z,\text{cm}} + M d^2. This holds under the assumptions of planar symmetry (lamina in a plane) and axes perpendicular to that plane.^[2]

Inertia Tensor and Matrix Forms

Tensor Generalization

The inertia tensor, a second-rank tensor that characterizes the mass distribution of a rigid body with respect to rotational motion, is defined in Cartesian coordinates as

I_{ij} = \int \left( \delta_{ij} r^2 - x_i x_j \right) \, dm,

where \delta_{ij} is the Kronecker delta, r^2 = x_k x_k (summation over repeated indices implied), x_i are the coordinates of a mass element dm, and the integral is over the body's volume.^[18] This tensor encapsulates both the moments of inertia (diagonal elements) and products of inertia (off-diagonal elements).^[19] The parallel axis theorem generalizes to the inertia tensor by relating the tensor about an arbitrary point to that about the center of mass (CM). Specifically, for axes parallel to those through the CM and displaced by a vector \vec{a} (with components a_i) from the CM to the new origin, the tensor components satisfy

I_{ij} = I_{ij}^{\rm cm} + M \left( \delta_{ij} a^2 - a_i a_j \right),

where M is the total mass and I_{ij}^{\rm cm} is the inertia tensor evaluated at the CM.^[18]^[19] This relation holds for the full 3D tensor, applying to both principal and non-principal axes. The theorem requires that the axes be parallel, meaning the displacement \vec{a} is the same for all tensor components, and that the new axes pass through the chosen point while maintaining the same orientation as those through the CM.^[18] It applies to rigid bodies in arbitrary orientations, provided the coordinate system is body-fixed or inertial.^[19] To derive this, consider a coordinate shift \vec{r} = \vec{r}' + \vec{a}, where \vec{r}' are coordinates relative to the CM. Substituting into the tensor definition yields

I_{ij} = \int \left[ \delta_{ij} (r'^2 + 2 \vec{r}' \cdot \vec{a} + a^2) - (x_i' + a_i)(x_j' + a_j) \right] dm.

Expanding and integrating term by term, the cross terms involving \vec{r}' vanish because \int \vec{r}' \, dm = 0 at the CM, leaving the CM tensor plus the parallel shift contribution M (\delta_{ij} a^2 - a_i a_j).^[18]^[19] Each element I_{ij} follows analogously to the scalar case, but with the tensor structure preserving rotational invariance. The scalar parallel axis theorem is a special case corresponding to the diagonal components, while the tensor form fully accounts for products of inertia (off-diagonal elements), enabling accurate dynamics for bodies with skewed mass distributions or in non-principal coordinate systems.^[20] This makes it essential for 3D rigid body simulations, such as in aerospace engineering or robotics, where full rotational freedom is involved.^[18]

Matrix Formulation and Identities

The parallel axis theorem for the inertia tensor is expressed in matrix form as \mathbf{I} = \mathbf{I}_{cm} + M (a^2 \mathbf{1} - \vec{a} \vec{a}^T), where \mathbf{I} is the inertia tensor about the new origin, \mathbf{I}_{cm} is the inertia tensor about the center of mass, M is the total mass, \mathbf{1} is the 3×3 identity matrix, a^2 = |\vec{a}|^2, and \vec{a} is the position vector from the center of mass to the new origin.^[18]^[21] This compact representation facilitates the computation of the full 3×3 inertia matrix, which contains six independent elements due to its symmetry, for use in either body-fixed coordinates (rotating with the rigid body) or space-fixed coordinates (inertial frame).^[22]^[23] The inertia tensor \mathbf{I} is symmetric (\mathbf{I} = \mathbf{I}^T) and positive semi-definite, ensuring that the quadratic form \vec{\omega}^T \mathbf{I} \vec{\omega} \geq 0 for angular velocity \vec{\omega}, which corresponds to the rotational kinetic energy being non-negative.^[22] In the context of rigid body dynamics, the angular momentum is related to the angular velocity by \vec{L} = \mathbf{I} \vec{\omega}, where the parallel axis shift adjusts \mathbf{I} to compute \vec{L} about displaced points while preserving the linear momentum contributions separately.^[22]^[23] A key identity arising from the matrix form is the shift in the trace: \operatorname{tr}(\mathbf{I}) = \operatorname{tr}(\mathbf{I}_{cm}) + 2 M a^2, obtained by taking the trace of the added term M (a^2 \mathbf{1} - \vec{a} \vec{a}^T), which yields M (3 a^2 - a^2) = 2 M a^2 and quantifies the increase in the sum of the principal moments due to the displacement.^[18]^[22] For planar bodies confined to the xy-plane, this formulation specializes to the perpendicular axis theorem as I_{zz} = I_{xx} + I_{yy}, where the z-axis is normal to the plane, reflecting the trace relation in two dimensions.^[22] As an illustrative example, consider a uniform thin rectangular lamina of mass M, length $2\ell along x, and width $2w along y, lying in the xy-plane with center of mass at the origin. The inertia tensor at the center of mass is diagonal:

\mathbf{I}_{cm} = \begin{pmatrix} \frac{1}{3} M w^2 & 0 & 0 \\ 0 & \frac{1}{3} M \ell^2 & 0 \\ 0 & 0 & \frac{1}{3} M (\ell^2 + w^2) \end{pmatrix},

satisfying the perpendicular axis theorem for the I_{zz} component. To shift to an edge at x = \ell, y = 0, z = 0, take \vec{a} = (\ell, 0, 0) with a^2 = \ell^2. The added matrix is

M (\ell^2 \mathbf{1} - \vec{a} \vec{a}^T) = M \ell^2 \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix},

yielding the new inertia tensor

\mathbf{I} = \begin{pmatrix} \frac{1}{3} M w^2 & 0 & 0 \\ 0 & \frac{1}{3} M \ell^2 + M \ell^2 & 0 \\ 0 & 0 & \frac{1}{3} M (\ell^2 + w^2) + M \ell^2 \end{pmatrix} = \begin{pmatrix} \frac{1}{3} M w^2 & 0 & 0 \\ 0 & \frac{4}{3} M \ell^2 & 0 \\ 0 & 0 & \frac{4}{3} M \ell^2 + \frac{1}{3} M w^2 \end{pmatrix}.

This demonstrates how the off-axis shift introduces no cross terms but modifies the moments about the y- and z-axes, consistent with the theorem's application to non-principal axes.^[21]^[18]

Parallel axis theorem

For Mass Moment of Inertia

Statement and Conditions

Derivation

Examples

For Second Moment of Area

Statement and Conditions

Derivation

Applications

For Polar Moment of Inertia

Statement and Conditions

Derivation

Inertia Tensor and Matrix Forms

Tensor Generalization

Matrix Formulation and Identities

References