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.

INSERT PICTURE HERE

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

[math]\displaystyle{ \ell_k = \vec {r}_k \times \vec {p}_k = \vec {r}_k \times m \vec {\dot r}_k }[/math]

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]

Forest_Ugrad_ClassicalMechanics#Rigid_Body_Motion