Parallel axis theorem

The mass moment of inertia of a body around an axis can be determined from the mass moment of inertia around a parallel axis through the center of mass.

Suppose a body of mass m is rotated about an axis z passing through the body's center of mass. The body has a moment of inertia I_cm with respect to this axis. The parallel axis theorem states that if the body is made to rotate instead about a new axis z′, which is parallel to the first axis and displaced from it by a distance d, then the moment of inertia I with respect to the new axis is related to I_cm by

${\displaystyle I=I_{\mathrm {cm} }+md^{2}.}$

Explicitly, d is the perpendicular distance between the axes z and z′.

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

Parallel axes rule for area moment of inertia

The parallel axes rule also applies to the second moment of area (area moment of inertia) for a plane region D:

${\displaystyle I_{z}=I_{x}+Ar^{2},}$

where I_z is the area moment of inertia of D relative to the parallel axis, I_x is the area moment of inertia of D relative to its centroid, A is the area of the plane region D, and r is the distance from the new axis z to the centroid of the plane region D. The centroid of D coincides with the centre of gravity of a physical plate with the same shape that has uniform density.

Polar moment of inertia of a body around a point can be determined from its polar moment of inertia around the center of mass.

The mass properties of a rigid body that is constrained to move parallel to a plane are defined by its center of mass R = (x, y) in this plane, and its polar moment of inertia I_R around an axis through R that is perpendicular to the plane. The parallel axis theorem provides a convenient relationship between the moment of inertia I_S around an arbitrary point S and the moment of inertia I_R about the center of mass R.

Recall that the center of mass R has the property

${\displaystyle \int _{V}\rho (\mathbf {r} )(\mathbf {r} -\mathbf {R} )\,dV=0,}$

where r is integrated over the volume V of the body. The polar moment of inertia of a body undergoing planar movement can be computed relative to any reference point S,

${\displaystyle I_{S}=\int _{V}\rho (\mathbf {r} )(\mathbf {r} -\mathbf {S} )\cdot (\mathbf {r} -\mathbf {S} )\,dV,}$

where S is constant and r is integrated over the volume V.

In order to obtain the moment of inertia I_S in terms of the moment of inertia I_R, introduce the vector d from S to the center of mass R,

${\displaystyle {\begin{aligned}I_{S}&=\int _{V}\rho (\mathbf {r} )(\mathbf {r} -\mathbf {R} +\mathbf {d} )\cdot (\mathbf {r} -\mathbf {R} +\mathbf {d} )\,dV\\&=\int _{V}\rho (\mathbf {r} )(\mathbf {r} -\mathbf {R} )\cdot (\mathbf {r} -\mathbf {R} )dV+2\mathbf {d} \cdot \left(\int _{V}\rho (\mathbf {r} )(\mathbf {r} -\mathbf {R} )\,dV\right)+\left(\int _{V}\rho (\mathbf {r} )\,dV\right)\mathbf {d} \cdot \mathbf {d} .\end{aligned}}}$

The first term is the moment of inertia I_R, the second term is zero by definition of the center of mass, and the last term is the total mass of the body times the square magnitude of the vector d. Thus,

${\displaystyle I_{S}=I_{R}+Md^{2},\,}$

which is known as the parallel axis theorem.[2]

The inertia matrix of a rigid system of particles depends on the choice of the reference point.[3] There is a useful relationship between the inertia matrix relative to the center of mass R and the inertia matrix relative to another point S. This relationship is called the parallel axis theorem.

Consider the inertia matrix [I_S] obtained for a rigid system of particles measured relative to a reference point S, given by

${\displaystyle [I_{S}]=-\sum _{i=1}^{n}m_{i}[r_{i}-S][r_{i}-S],}$

where r_i defines the position of particle P_i, i = 1, ..., n. Recall that [r_i − S] is the skew-symmetric matrix that performs the cross product,

${\displaystyle [r_{i}-S]\mathbf {y} =(\mathbf {r} _{i}-\mathbf {S} )\times \mathbf {y} ,}$

for an arbitrary vector y.

Let R be the center of mass of the rigid system, then

${\displaystyle \mathbf {R} =(\mathbf {R} -\mathbf {S} )+\mathbf {S} =\mathbf {d} +\mathbf {S} ,}$

where d is the vector from the reference point S to the center of mass R. Use this equation to compute the inertia matrix,

${\displaystyle [I_{S}]=-\sum _{i=1}^{n}m_{i}[r_{i}-R+d][r_{i}-R+d].}$

Expand this equation to obtain

${\displaystyle [I_{S}]=\left(-\sum _{i=1}^{n}m_{i}[r_{i}-R][r_{i}-R]\right)+\left(-\sum _{i=1}^{n}m_{i}[r_{i}-R]\right)[d]+[d]\left(-\sum _{i=1}^{n}m_{i}[r_{i}-R]\right)+\left(-\sum _{i=1}^{n}m_{i}\right)[d][d].}$

The first term is the inertia matrix [I_R] relative to the center of mass. The second and third terms are zero by definition of the center of mass R,

${\displaystyle \sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )=0.}$

And the last term is the total mass of the system multiplied by the square of the skew-symmetric matrix [d] constructed from d.

The result is the parallel axis theorem,

${\displaystyle [I_{S}]=[I_{R}]-M[d]^{2},}$

where d is the vector from the reference point S to the center of mass R.[3]

• Christiaan Huygens
• Jakob Steiner
• Moment of inertia
• Perpendicular axis theorem
• Rigid body dynamics
• Stretch rule

Wikimedia Commons has media related to Steiner's parallel axis theorem.

• Parallel axis theorem
• Moment of inertia tensor
• Video about the inertia tensor