Page Description Equation
328 linear speed center of mass for pure rolling motion \[ v_{\mathrm{CM}}=R\omega \]
329 linear acceleration center of mass for pure rolling motion \[ a_{\mathrm{CM}}=R\alpha \]
330 total kinetic energy of a rigid body rolling on a rough surface (no slipping) \[ K=\frac{1}{2}I_{\mathrm{CM}}\omega^2+\frac{1}{2}Mv_{\mathrm{CM}}^2 \]
332 torque. rotation is always in the counterclockwise direction when facing the tip of the torque vector. \[ \vect{\tau}=\vect{r}\times\vect{F} \]
333 cross product \[ \vect{A}\times\vect{B}=AB\sin{\theta}\,\vect{u},\\ \text{ where } 0\le\theta\le\pi \]
333 magnitude of cross product \[ \left|\vect{A}\times\vect{B}\right|=AB\sin{\theta} \]
333 cross product identity \[ \vect{A}\times\vect{B}=-\vect{B}\times\vect{A} \]
333 parallel, antiparallel vectors \[ \vect{A}\times\vect{B}=\vect{0},\\ \text{ if }\ \theta=180^\circ \text{ or }\ \theta=0^\circ \]
333 cross product identity \[ \vect{A}\times\vect{A}=\vect{0} \]
333 norm of cross product \[ \left|\vect{A}\times\vect{B}\right|=AB,\\ \text{ if }\ \theta=90^\circ \]
333 distribution for cross products \[ \vect{A}\times\left(\vect{B}+\vect{C}\right)=\vect{A}\times\vect{B}+\vect{A}\times\vect{C} \]
333 derivative product rule for cross products \[ \frac{d}{dt}\left(\vect{A}\times\vect{B}\right)=\vect{A}\times\frac{d\vect{B}}{dt}+\frac{d\vect{A}}{dt}\times\vect{B} \]
334 cross product identity \[ \vect{i}\times\vect{i}=\vect{j}\times\vect{j}=\vect{k}\times\vect{k}=\vect{0} \]
334 cross product identity \[ \vect{i}\times\vect{j}=-\vect{j}\times\vect{i}=\vect{k} \]
334 cross product identity \[ \vect{j}\times\vect{k}=-\vect{k}\times\vect{j}=\vect{i} \]
334 cross product identity \[ \vect{k}\times\vect{i}=-\vect{i}\times\vect{k}=\vect{j} \]
334 cross product identity \[ \vect{A}\times\left(-\vect{B}\right)=-\vect{A}\times\vect{B} \]
334 cross product \[ \begin{align*} \vect{A}\times\vect{B} &=\begin{vmatrix} \vect{i}&\vect{j}&\vect{k}\\ A_x&A_y&A_z\\ B_x&B_y&B_z \end{vmatrix}\\ &=\vect{i} \begin{vmatrix} A_y&A_z\\ B_y&B_z \end{vmatrix} -\vect{j} \begin{vmatrix} A_x&A_z\\ B_x&B_z\\ \end{vmatrix} +\vect{k} \begin{vmatrix} A_x&A_y\\ B_x&B_y\\ \end{vmatrix}\\ & \begin{split} =\left(A_yB_z-A_zB_y\right)\vect{i} &-\left(A_xB_z-A_zB_x\right)\vect{j}\\ &+\left(A_xB_y-A_yB_x\right)\vect{k} \end{split} \end{align*} \]
335 angular momentum \[ \vect{L}\equiv\vect{r}\times\vect{p} \]
335 torque \[ \sum\vect{\tau}=\frac{d\vect{L}}{dt} \]
336 total angular momentum of a system of particles is the vector sum of the angular momenta of the individual particles \[ \vect{L}=\sum_{i}\vect{L}_i \]
338 $z$ component of angular momentum of a rigid object rotating about a fixed $z$ axis. \[ L_z=I\omega \]
338 net external torque \[ \sum\tau_{\mathrm{ext}}=\frac{d\vect{L}}{dt}=I\alpha \]
341 law of conservation of angular momentum \[ I_i\omega_i=I_f\omega_f=\mathrm{constant} \]

Notes

right-hand rule for determining the direction of the cross product vector

The total angular momentum of a system can vary with time only if a net external torque is acting on the system.