Moment of inertia

The moment of inertia, otherwise known as the mass moment of inertia, angular/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation by a given amount.

For the quantity also known as the "area moment of inertia", see Second moment of area.

Moment of inertia

kg⋅m²

lbf·ft·s²

$I={\frac {L}{\omega }}$

M L²

It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis.

For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3-by-3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.

In mechanical engineering, simply "inertia" is often used to refer to "inertial mass" or "moment of inertia".^[1]

Introduction[edit]

When a body is free to rotate around an axis, torque must be applied to change its angular momentum. The amount of torque needed to cause any given angular acceleration (the rate of change in angular velocity) is proportional to the moment of inertia of the body. Moments of inertia may be expressed in units of kilogram metre squared (kg·m²) in SI units and pound-foot-second squared (lbf·ft·s²) in imperial or US units.

The moment of inertia plays the role in rotational kinetics that mass (inertia) plays in linear kinetics—both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, and will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by $mr^{2}$ , where $r$ is the distance of the point from the axis, and $m$ is the mass. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that depends on the dimensions, shape and total mass of the object.

In 1673, Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a compound pendulum.^[2] The term moment of inertia ("momentum inertiae" in Latin) was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765,^[2]^[3] and it is incorporated into Euler's second law.

The natural frequency of oscillation of a compound pendulum is obtained from the ratio of the torque imposed by gravity on the mass of the pendulum to the resistance to acceleration defined by the moment of inertia. Comparison of this natural frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body.^[4]^[5]

The moment of inertia also appears in momentum, kinetic energy, and in Newton's laws of motion for a rigid body as a physical parameter that combines its shape and mass. There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations yield a 3 × 3 matrix of moments of inertia, called the inertia matrix or inertia tensor.^[6]^[7]

The moment of inertia of a rotating flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal, horizontal and vertical axes determine how steering forces on the control surfaces of its wings, elevators and rudder(s) affect the plane's motions in roll, pitch and yaw.

Measuring moment of inertia[edit]

The moment of inertia of a complex system such as a vehicle or airplane around its vertical axis can be measured by suspending the system from three points to form a trifilar pendulum. A trifilar pendulum is a platform supported by three wires designed to oscillate in torsion around its vertical centroidal axis.^[19] The period of oscillation of the trifilar pendulum yields the moment of inertia of the system.^[20]

The moment of inertia of a thin rod with constant cross-section $s$ and density $\rho$ and with length $\ell$ about a perpendicular axis through its center of mass is determined by integration.^: 1301 Align the $x$ -axis with the rod and locate the origin its center of mass at the center of the rod, then $I_{C,{\text{rod}}}=\iiint _{Q}\rho \,x^{2}\,dV=\int _{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}\rho \,x^{2}s\,dx=\left.\rho s{\frac {x^{3}}{3}}\right|_{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}={\frac {\rho s}{3}}\left({\frac {\ell ^{3}}{8}}+{\frac {\ell ^{3}}{8}}\right)={\frac {m\ell ^{2}}{12}},$ where $m=\rho s\ell$ is the mass of the rod.

[23]

The moment of inertia of a thin disc of constant thickness $s$ , radius $R$ , and density $\rho$ about an axis through its center and perpendicular to its face (parallel to its axis of ) is determined by integration.^[23]^: 1301 Align the $z$ -axis with the axis of the disc and define a volume element as $dV=sr\,dr\,d\theta$ , then $I_{C,{\text{disc}}}=\iiint _{Q}\rho \,r^{2}\,dV=\int _{0}^{2\pi }\int _{0}^{R}\rho r^{2}sr\,dr\,d\theta =2\pi \rho s{\frac {R^{4}}{4}}={\frac {1}{2}}mR^{2},$ where $m=\pi R^{2}\rho s$ is its mass.

rotational symmetry

The moment of inertia of the compound pendulum is now obtained by adding the moment of inertia of the rod and the disc around the pivot point $P$ as, $I_{P}=I_{C,{\text{rod}}}+M_{\text{rod}}\left({\frac {L}{2}}\right)^{2}+I_{C,{\text{disc}}}+M_{\text{disc}}(L+R)^{2},$ where $L$ is the length of the pendulum. Notice that the parallel axis theorem is used to shift the moment of inertia from the center of mass to the pivot point of the pendulum.

$i$ , $j$ is equal to 1, 2 or 3 for $x$ , $y$ , and $z$ , respectively,

$\mathbf {r} _{k}=\left(x_{1}^{(k)},x_{2}^{(k)},x_{3}^{(k)}\right)$ is the vector to the point mass $m_{k}$ from the point about which the tensor is calculated and

$\delta _{ij}$ is the .

Moment of inertia