page_title: Moment of inertia
section_title: Newton's laws
hierarchical_section_title: Moment of inertia / Motion in a fixed plane / Rigid body / Newton's laws
caption_attribution_description: English: John Deere Model D Tractor - Submitted by and originally used on www.antiquefarming.com. Our mission is to preserve the past so the future can enjoy. This John Deere D is known as a "Spoker D" due to the spoked flywheel. Beginning in 1926 the flywheel was changed to a solid flywheel. The change was mainly due to injuries when starting the tractor to the farmers hands when they could become caught in the spokes of the flywheel.
context_page_description: The moment of inertia, otherwise known as the mass moment of inertia, angular mass or rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis; similar 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.
  It is an extensive 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. 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.
context_section_description: Newton's laws for a rigid system of  particles, , can be written in terms of a resultant force and torque at a reference point , to yield
  where  denotes the trajectory of each particle.
  The kinematics of a rigid body yields the formula for the acceleration of the particle  in terms of the position  and acceleration  of the reference particle as well as the angular velocity vector  and angular acceleration vector  of the rigid system of particles as,
  For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along  perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors  from the reference point  to a point  and the unit vectors , so
  This yields the resultant torque on the system as
  where , and  is the unit vector perpendicular to the plane for all of the particles .
  Use the center of mass  as the reference point and define the moment of inertia relative to the center of mass , then the equation for the resultant torque simplifies to
caption: A 1920s John Deere tractor with the spoked flywheel on the engine. The large moment of inertia of the flywheel smooths the operation of the tractor.
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