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]\vec R[/math] point to the center of mass of the object
- [math]\vec {r}_k[/math] points to a mass element [math]m_k[/math]
- [math]\vec{r}_k^{\;\;\prime}[/math] points from the center of mass to the mass element [math]m_k[/math]
the angular momentum of mass element [math]m_k[/math] about the point O is given as
- [math]\ell_k = \vec {r}_k \times \vec {p}_k = \vec {r}_k \times m \vec {\dot r}_k[/math]
The total angular momentum about the point O is given as
- [math] \vec L = \sum \ell_k = \sum \vec {r}_k \times m_k \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
- [math]\vec {r}_k = \vec R + \vec{r}_k^{\;\; \prime}[/math]
- [math] \vec L = \sum \vec {r}_k \times m_k \vec {\dot r}_k[/math]
- [math] = \sum (\vec R + \vec{r}_k^{\;\; \prime}) \times m_k (\vec \dot R + \vec{\dot r}_k^{\;\; \prime})[/math]
- [math] = \sum \vec R \times m_k \vec \dot R + \sum \vec R \times m_k \vec{\dot r}_k^{\;\; \prime} + \sum \vec{r}_k^{\;\; \prime} \times m_k \vec \dot R +\sum \vec{r}_k^{\;\; \prime} \times m_k \vec{\dot r}_k^{\;\; \prime} [/math]
- [math]\sum \vec R \times m_k \vec \dot R = \vec R \times \sum m_k \vec \dot R = \vec R \times M \vec \dot R = \vec R \times \vec P[/math]
- [math]\vec P =[/math] momentum of the center of Mass
- [math]\sum \vec R \times m_k \vec{\dot r}_k^{\;\; \prime} = \vec R \times \sum m_k \vec{\dot r}_k^{\;\; \prime} 0[/math]
- [math]\sum m_k \vec{\dot r}_k^{\;\; \prime} = 0[/math] : The location of the center of mass is at [math]\vec{ r}_k^{\;\; \prime} = 0[/math] the derivative is also zero
- [math]\sum \vec{r}_k^{\;\; \prime} \times m_k \vec \dot R = \sum m_k \vec{r}_k^{\;\; \prime} \times \vec \dot R =0 [/math] : The location of the CM is at 0
- [math] \vec L = \sum \vec R \times m_k \vec \dot R + \sum \vec{r}_k^{\;\; \prime} \times m_k \vec{\dot r}_k^{\;\; \prime} [/math]
Forest_Ugrad_ClassicalMechanics#Rigid_Body_Motion