Newton's Laws For Planar Movement
Newton's laws for a rigid system of N particles, Pi, i = 1,..., N, can be written in terms of a resultant force and torque at a reference point R, to yield
where ri denotes the trajectory of each particle.
The kinematics of a rigid body yields the formula for the acceleration of the particle Pi in terms of the position R and acceleration A of the reference particle as well as the angular velocity vector ω and angular acceleration vector α 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 k 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 ei from the reference point R to a point ri and the unit vectors ti = kxei, so
This yields the resultant torque on the system as
where eixei = 0, and eixti = k is the unit vector perpendicular to the plane for all of the particles Pi .
Use the center of mass C as the reference point and define the moment of inertia relative to the center of mass IC, then the equation for the resultant torque simplifies to
The moment of inertia IC about an axis perpendicular to the movement of the rigid system and through the center of mass is known as the polar moment of inertia.
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