Revision as of 02:02, 19 November 2014

Rigid Body Motion

Rigidy Body: A Rigid Body is a system involving a large number of point masses, called particles, whose distances between pairs of point particles remains constant even when the body is in motion or being acted upon by external force.

Forces of Constraint: The internal forces that maintain the constant distances between the different pairs of point masses.

Consider a rigid body that rotates about a fixed z-axis with the origin at point O.

INSERT PICTURE HERE

let

point to the center of mass of the object

points to a mass element

points from the center of mass to the mass element

the angular momentum of mass element [math]m_k[/math] about the point O is given as

The total angular momentum about the point O is given as

This can be cast in term of the angular momentum about the center of mass and the angular momentum of the motion

@@ Line 30: / Line 30: @@
 :<math> \vec L = \sum \ell_k = \sum \vec {r}_k \times m \vec {\dot r}_k</math>
+This can be cast in term of the angular momentum about the center of mass and the angular momentum of the motion
-Let
+:<math>\vec {r}_k = \vec R + \vec{r}_k^{\prime}</math>
-:<math>m_k =</math> the kth mass of a particle located at the distance <math>\vec {R}_k</math>, from the origin, and moving with the velocity <math>\vec{v}_k</math> and angular velocity <math>\omega</math>. \psi represents the angle the point mass <math>m_k</math> makes with the line OA used as a reference point for the rotation angle <math>\theta</math> of the rigid body.
-The particle moves about the rotation axis in a circle of radius
-:<math>r_k =\sqrt{x_k^2 +y_k}</math>
 [[Forest_Ugrad_ClassicalMechanics#Rigid_Body_Motion]]