Forest UCM RBM
Rigid Body Motion
Rigid Body
- 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.
let
- [math]\displaystyle{ \vec R }[/math] point to the center of mass of the object
- [math]\displaystyle{ \vec {r}_k }[/math] points to a mass element [math]\displaystyle{ m_k }[/math]
- [math]\displaystyle{ \vec{r}_k^{\prime} }[/math] points from the center of mass to the mass element [math]\displaystyle{ m_k }[/math]
The rotational position of the body is specified by an angle [math]\displaystyle{ \theta }[/math] that is drawn between the line OA on the body and the x-axis.
INSERT PICTURE HERE
Let
- [math]\displaystyle{ m_k = }[/math] the kth mass of a particle located at the distance [math]\displaystyle{ \vec {R}_k }[/math], from the origin, and moving with the velocity [math]\displaystyle{ \vec{v}_k }[/math] and angular velocity [math]\displaystyle{ \omega }[/math]. \psi represents the angle the point mass [math]\displaystyle{ m_k }[/math] makes with the line OA used as a reference point for the rotation angle [math]\displaystyle{ \theta }[/math] of the rigid body.
The particle moves about the rotation axis in a circle of radius
- [math]\displaystyle{ r_k =\sqrt{x_k^2 +y_k} }[/math]