Moment of inertia is the mass-weighted squared distance from a rotation axis and determines rotational kinetic energy.
A uniform solid sphere of mass and radius has the indicated moment of inertia about any diameter. Integrating the squared perpendicular distance against uniform mass density gives . It is different from the value for a thin spherical shell and determines the rotational contribution to kinetic energy during rolling without slipping.
The mass second-moment tensor about a centre of mass is . It measures the shape of a mass distribution and appears in tidal torque theory. It differs from the mechanical inertia tensor, which is . For a uniform ellipsoid of mass and semiaxes , its principal values are .
The principal moments of inertia are the eigenvalues of the inertia tensor, measured along its principal axes.
For torque-free rigid-body motion, the kinetic-energy ellipsoid fixed in the body rolls without slipping on the invariable plane perpendicular to the conserved angular momentum. Its point of contact is the angular-velocity vector.
The moment about a parallel axis equals the centre-of-mass moment plus mass times the squared axis separation.
For a planar lamina in the -plane, the moments about three mutually perpendicular axes through one point satisfyThis follows pointwise from in the defining mass integrals.
For a uniformly filled, negligibly light disc of fixed radius , remaining mass , angular velocity , and tangential exhaust speed relative to its rim, angular-momentum balance givesAt zero relative exhaust speed, is constant.
For a uniform solid cone of height , base radius , and mass , whose symmetry axis points from vertex to base, the centre of mass lies at from the vertex. The principal moments at the vertex are
Consider the uniform solid of revolution about the -axis whose boundary isFor total mass , the center of mass lies a distance from the pointed origin, and the principal moments about that origin are
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Moment of inertia, often denoted by \( I \), is a physical quantity that measures how much an object resists rotational motion about a specific axis. It plays a similar role in rotational dynamics as mass does in linear dynamics. The moment of inertia depends on the mass distribution of an object relative to the axis of rotation.