Conservation of angular momentum

Rotational Motion

The motion of a rigid object can be separated into

• The motion of the center mass = average location of mass. Classical physics still applies
• Rotations about the center of mass = We can treat these two motions independently. To describe the rotation of an object about an axis is completely analogous to ordinary motion. When mapping the angle from a given origin, $\theta =0$, The velocity of an $\theta$ vs. $t$ graph gives the angular speed. Which for all allowed angles $\theta$, is very similar to velocity. $\frac{d\theta }{dt}=\omega \frac{d\omega }{dt}=\alpha$ Where $\theta$ is the angle of the object (similar to displacement), $\omega$ is the angular velocity, and $\alpha$ is the angular acceleration. In an environment of rotational symmetry about the same axis, angular momentum is conserved. $L=I\omega$ Where, $L$ is the angular momentum about the axis $\omega$ is the angular velocity. (Similar to $p=mv$) , and $I$ is the moment of inertia. The moment of inertia tells us how hard it is to get the object to spin (similar to mass). For more compact, lighter objects with their center of mass closer to the axis of rotation, are easier to spin. The moment of inertial is proportional to mass and proportional to size${}^2$ . $I=M(\bar{R}{)}^2$ Where $M$ is the mass of the object, and $\bar{R}$ is the average distance from the axis squared. Another useful rearrangement is:

\\ I = \delta m \sum ^n_iR^2_i \end{gather}$$ Where $\frac M N$ is the mass of each piece. THIS ASSUMES EACH PIECE HAS THE SAME MASS. ![[Figure10.20.png]] Brownian motion for the particles of gas and dust in a space leads to the formation of galaxies and planets. The sum of all of their angular momentum causes them to spiral closer ![[protoplanetary_collapse.jpg]] ![[Animation-Pulsar-Spinning-Neutron-Star.webp]] ### Summary - Angular momentum is given by $L=I\omega$ (if no torque) - Changed by torque $\frac {dL}{dt} = \tau_{net} = F_{\perp}r$ - $\tau_{net} = I\alpha$ which is the analogue to $F=ma$.