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.
Total Angular Momentum of a Rigid Body
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]
The total angular momentum about the point O is given as
- [math]\displaystyle{ \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]\displaystyle{ \vec {r}_k = \vec R + \vec{r}_k^{\;\; \prime} }[/math]
- [math]\displaystyle{ \vec L = \sum \vec {r}_k \times m_k \vec {\dot r}_k }[/math]
- [math]\displaystyle{ = \sum (\vec R + \vec{r}_k^{\;\; \prime}) \times m_k (\vec \dot R + \vec{\dot r}_k^{\;\; \prime}) }[/math]
- [math]\displaystyle{ = \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]\displaystyle{ \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]\displaystyle{ \vec P = }[/math] momentum of the center of Mass
- [math]\displaystyle{ \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]\displaystyle{ \sum m_k \vec{\dot r}_k^{\;\; \prime} = \sum m_k \left ( \vec {r}_k - \vec R\right ) = \sum m_k \vec {r}_k - \sum m_k \vec R = \vec {v}_{cm} - \vec{v}_{cm} = 0 }[/math]
- The location of the center of mass is at [math]\displaystyle{ \vec{ r}_k^{\;\; \prime} = 0 }[/math] the derivative is also zero
- [math]\displaystyle{ \sum m_k \vec{\dot r}_k^{\;\; \prime} = \sum m_k \left ( \vec {r}_k - \vec R\right ) = \sum m_k \vec {r}_k - \sum m_k \vec R = \vec {v}_{cm} - \vec{v}_{cm} = 0 }[/math]
- [math]\displaystyle{ \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]\displaystyle{ \vec L = \vec R \times \vec P + \sum \vec{r}_k^{\;\; \prime} \times m_k \vec{\dot r}_k^{\;\; \prime} }[/math]
- [math]\displaystyle{ = L_{\mbox{CM}} + L_{\mbox{about CM}} }[/math]
The total angular momentum is the sum of the angular momentum of the center of mass of a rigid body [math]\displaystyle{ L_{\mbox{CM}} }[/math] and the angular momentum of the rigid body about the center of mass [math]\displaystyle{ L_{\mbox{about CM}} }[/math]
Planet example
What is the total angular momentum of the earth orbiting the sun?
There are two components
[math]\displaystyle{ L_{\mbox{CM}} }[/math] = angular momentum of the earth orbiting about the sun
[math]\displaystyle{ L_{\mbox{about CM}} }[/math] = angular momentum of the earth orbiting about the earth's center of mass (Spin)