- #1
guv
- 123
- 22
Let's start with an arbitrary solid body rotating around a fixed axis of rotation with angular velocity ##\vec \omega## in the ## \hat z## direction. For simplicity, let's say the origin O is on the axis of rotation. Take a look at the picture I sketched in the next post. Tried my best to be clear about the notations I used in my derivation.
By definition, $$\vec L = \int_{\Omega} \vec r \times (dm \; \vec v) = \int_{\Omega} \vec r \times (\vec \omega \times \vec r) dm$$
Where ##\vec r## is is position position vector of dm from origin O and ##\Omega## is the domain of integration (the entire solid body).
If the angle between axis of rotation and position vector is ##\theta## (imagine ##\hat z## points upward from O, ##\vec r## points to the upper right from O), then
$$\vec L = \int r^2 \omega sin(\theta) dm \hat L$$
where ##\hat L## is a unit vector from dm perpendicular from ##\vec r## (imagine this unit vector from dm going upper left from dm)
We can decompose this ##\hat L## into its z direction component and a radial component.
$$\vec L = \int r^2 \omega sin^2\theta \hat z dm + \int r^2 \omega sin\theta cos\theta (-\hat p) dm$$
Here is ##\hat p## is a unit vector pointing outward from dm perpendicular to axis of rotation.
If ##\omega## is a constant, then
$$\int r^2 \omega sin^2\theta \hat z dm = \omega \int r^2 sin^2\theta dm \hat z = I_{zz} \omega \hat z$$
This is the z component of the angular moment since the ##\vec \omega = (0, 0, \omega)##.
My problem is I couldn't really simply ##\int r^2 \omega sin\theta cos\theta (-\hat p) dm## into 0 when center of mass is on the axis of rotation. Can someone help? Thanks,
guv
By definition, $$\vec L = \int_{\Omega} \vec r \times (dm \; \vec v) = \int_{\Omega} \vec r \times (\vec \omega \times \vec r) dm$$
Where ##\vec r## is is position position vector of dm from origin O and ##\Omega## is the domain of integration (the entire solid body).
If the angle between axis of rotation and position vector is ##\theta## (imagine ##\hat z## points upward from O, ##\vec r## points to the upper right from O), then
$$\vec L = \int r^2 \omega sin(\theta) dm \hat L$$
where ##\hat L## is a unit vector from dm perpendicular from ##\vec r## (imagine this unit vector from dm going upper left from dm)
We can decompose this ##\hat L## into its z direction component and a radial component.
$$\vec L = \int r^2 \omega sin^2\theta \hat z dm + \int r^2 \omega sin\theta cos\theta (-\hat p) dm$$
Here is ##\hat p## is a unit vector pointing outward from dm perpendicular to axis of rotation.
If ##\omega## is a constant, then
$$\int r^2 \omega sin^2\theta \hat z dm = \omega \int r^2 sin^2\theta dm \hat z = I_{zz} \omega \hat z$$
This is the z component of the angular moment since the ##\vec \omega = (0, 0, \omega)##.
My problem is I couldn't really simply ##\int r^2 \omega sin\theta cos\theta (-\hat p) dm## into 0 when center of mass is on the axis of rotation. Can someone help? Thanks,
guv
Last edited: