7. Angular Momentum

Measure of the amount of rotational motion an object has about a particular axis

 

Rotating Rigid Body

$$\vec{L} = \vec{r} \times \vec{\rho}$$ $$\vec{L} = \vec{r} \times (m \vec{v}) = m \times \vec{v}^2$$

For a rotating rigid body, the total momentum is given by

$$\vec{L} = I \omega$$

 

Rate of Change of Angular Momentum

The rate of change of angular momentum is equal to the net torque

$$\dfrac{d\vec{L}}{dt} = \dfrac{d\vec{r}}{dt} \vec{\rho} + \vec{r}\dfrac{d\vec{\rho}}{dt}$$

assuming $\vec{r}$ is constant and we know $\dfrac{d\vec{\rho}}{dt} = \vec{F}$ from linear rate of change of momentum

then

$$\dfrac{d\vec{L}}{dt} = \vec{r} \times \vec{F} = \vec{\tau}$$

 

Conservation of Angular Momentum

When the net external torque acting on a system is zero, the total angular momentum of the system is constant

if

$$\sum \vec{\tau} = 0$$

then

$$\dfrac{d\vec{L}}{dt} = 0$$

and

$$\vec{L} = \text{constant}$$