Physics · Ch 6 — System of Particles and Rotational Motion

Summary

Centre of mass of a system is the weighted average position: $\vec{R}_{\text{CM}} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$ . For a rigid body, it behaves as if the entire mass is concentrated there for translational motion.
Newton’s second law for a system: $\vec{F}_{\text{ext}} = M \vec{a}_{\text{CM}}$ , where $\vec{F}_{\text{ext}}$ is the net external force. Internal forces cancel in pairs and do not affect the CM motion.
Angular velocity $\vec{\omega}$ and angular acceleration $\vec{\alpha}$ : $\vec{\alpha} = \frac{d\vec{\omega}}{dt}$ . For pure rotation about a fixed axis, every particle has the same $\omega$ .
Torque about a point: $\vec{\tau} = \vec{r} \times \vec{F}$ . Magnitude $\tau = rF\sin\theta$ . Net external torque causes angular acceleration.
Angular momentum of a particle: $\vec{L} = \vec{r} \times \vec{p}$ . For a rigid body rotating about a fixed axis: $L = I\omega$ , where $I$ is the moment of inertia.
Relation between torque and angular momentum: $\vec{\tau}_{\text{ext}} = \frac{d\vec{L}}{dt}$ . If $\vec{\tau}_{\text{ext}} = 0$ , angular momentum is conserved.
Moment of inertia $I = \sum m_i r_i^2$ for discrete masses; $I = \int r^2 dm$ for continuous bodies. It depends on the axis of rotation. …