11-05-25 Cross Products, Translational Angular Momentum

Angular Momentum

• Along with momentum and energy, angular momentum is a fundamental quantity that helps us to predict motion.

• We will see how angular momentum is associated with translational and rotational motion

Vector Cross Product

• Vector cross product: Way of multiplying two vectors to get a third vector that encodes the right-hand rule

$\overrightarrow A × \overrightarrow B =$ <A_yB_z - A_zB_y, A_zB_x -A_xB_z,A_xB_y-A_yB_x>

$|\overrightarrow A ×\overrightarrow B| = |\overrightarrow A| |\overrightarrow B| \sin (\theta)$

Warning: $\overrightarrow A × \overrightarrow B\ne \overrightarrow B × \overrightarrow A$

Vector Cross Product - Construction

• The vector cross product assumes the right-hand-rule:

$\overline x× \overline y = \overline z$

\overline y× \overline x  = -\overline z

$\overline y × \overline z = \overline x$

$\overline z × \overline y = - \overline x$

\overline z × \overline x = \overline y

\overline x × \overline z = -\overline y

\overline x × \overline x = 0

\overline y × \overline y = 0

\overline z × \overline z = 0

• It also assumes the distributive law holds:

\overrightarrow A × \overrightarrow B = (A_x \overline x + A_y \overline y + A_z \overline z) × (B_x \overline x + B_y \overline y + B_z \overline z)

=(A_x B_x \overline x × \overline x) + (A_x B_y \overline x × \overline y) + (A_xB_z \overline x × \overline z) + (A_y B_x \overline y × \overline x) + (A_yB_y \overline y × \overline y) + (A_y B_z \overline y × \overline z) + (A_z B_x \overline z × \overline x) + (A_z B_y \overline z × \overline y) + (A_z B_z \overline z × \overline z)

• Simplifying:

\overrightarrow A × \overrightarrow B = (A_xB_x \cdot 0) + (A_xB_y \cdot \overline z) + (A_xB_z \cdot -\overline y) + (A_yB_x \cdot -\overline z) + (A_yB_y\cdot 0) + (A_yB_z \cdot \overline x) + (A_zB_x \cdot \overline y) + (A_zB_y \cdot - \overline x) + (A_zB_z \cdot 0)

• Grouping like terms gives us:

\overrightarrow A × \overrightarrow B = (A_yB_z - A_zB_y)\overline x + (A_zB_x - A_x B_z) \overline y + (A_xB_y - A_yB_x)\overline z

Torque as a vector cross product

• The torque created by a force is a measure of the twist (around a reference point) that that force exerts on the system:

\overrightarrow \tau_a = \overrightarrow r_A × \overrightarrow F

Angular Momentum around generic points

• The Momentum Principle can be used to tell us what torque around a generic point A will do to a system:

Translational Angular Momentum

• Translational angular momentum characterizes the ‘rotation’ of a system around a reference point

• When $\theta = 0 \degree$, the motion is entirely toward/away from A

• When $\theta = 90^\degree$, the motion is entirely ‘around’ A