**Newton's Laws For Planar Movement**

Newton's laws for a rigid system of N particles, *P _{i}*,

*i*= 1,...,

*N*, can be written in terms of a resultant force and torque at a reference point

**R**, to yield

where **r**_{i} denotes the trajectory of each particle.

The kinematics of a rigid body yields the formula for the acceleration of the particle `P _{i}` 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 **e**_{i} from the reference point **R** to a point **r**_{i} and the unit vectors **t**_{i} = *k*x**e**_{i}, so

This yields the resultant torque on the system as

where **e**_{i}x**e**_{i} = 0, and **e**_{i}x**t**_{i} = *k* is the unit vector perpendicular to the plane for all of the particles `P _{i}` .

Use the center of mass **C** as the reference point and define the moment of inertia relative to the center of mass `I _{C}`, then the equation for the resultant torque simplifies to

The moment of inertia `I _{C}` 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*.

Read more about this topic: Inertia Tensor, Polar Moment of Inertia

### Famous quotes containing the words movement, newton and/or laws:

“... contemporary black women felt they were asked to choose between a black *movement* that primarily served the interests of black male patriarchs and a women’s *movement* which primarily served the interests of racist white women.”

—bell hooks (b. c. 1955)

“The next Augustan age will dawn on the other side of the Atlantic. There will, perhaps, be a Thucydides at Boston, a Xenophon at New York, and, in time, a Virgil at Mexico, and a *Newton* at Peru. At last, some curious traveller from Lima will visit England and give a description of the ruins of St Paul’s, like the editions of Balbec and Palmyra.”

—Horace Walpole (1717–1797)

“Our books of science, as they improve in accuracy, are in danger of losing the freshness and vigor and readiness to appreciate the real *laws* of Nature, which is a marked merit in the ofttimes false theories of the ancients.”

—Henry David Thoreau (1817–1862)