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Euler Equation \begin{align*} &\mathbf{I}\,{\dot{\omega}}+\mathbf\omega\times \left(\mathbf{I}\,\mathbf\omega\right)=\mathbf\tau\tag 1 \end{align*} and the kinematic equation \begin{align*} &\mathbf …

Starting with $$\vec{v}=\vec{\omega}\times \vec{r}$$ with: $\vec{v}=\frac{d\vec{s}}{dt}$ and $\vec{\omega}=\frac{d\vec{\Phi}}{dt}$ you get: $$\frac{d\vec{s}}{dt}=\frac{d\vec{\Phi}}{dt}\times …

The Euler equations Euler equation inertial frame at the CM \begin{align*} &\frac{d}{dt}\left(\mathbf I\,\mathbf\omega\right)=\mathbf\tau\\ &\Rightarrow\\ &\mathbf I\,\mathbf{\dot{\omega}}+\mathbf{\do …

Assume the tire is rolling with out slipping on the road, you apply constant torque $~\tau~$ at the center of the tire. thus the center of the tire angular acceleration is: $$\alpha_T= \frac {\tau}{I …

lets obtain the center of mass velocities $~v_1~$ and $~v_2~$ after the pencil collied with the floor . in both cases the momenta and the energies are conserved. the heights after the collision are p …

Calculation the Car Wheels Load $F_G=m\,g$ car weight $F_V$ wheel front load $F_H$ wheel rear load $a_A$ car acceleration $a_D$ car deceleration take the sum of the torques about point V you …

To calculate the equation of motion we obtain the sum of the torques about point A, because we don't have to take care about the contact force. first I obtain the vector u from point B to A $$\v …

This figure can help you to solve your problem using parallel axis transformation . You can choose the x,y,z coordinate system at arbitrary point p, not necessarily center of mass, but all coordinate …

The equations of motion for rigid body rotation are: $I\,\dot{\vec{\omega}}+\vec{\omega}\times I\,\vec{\omega}=\vec{\tau}$ How i can calculate this equations using Lagrangian method ? If i use …

you have two objects with mass $~m$ and $M$ that collide. first I write the equations of motions during the collision $$m\,\frac{dv_1}{dt}=F_c\tag 1$$ $$M\,\frac{dv_2}{dt}=-F_c\tag 2$$ where $F_c$ is …

To answer your equations first I will write down the equation of motion, assuming that the tube is rotate with constant angular velocity $~\omega$ The EOM: $$m\,\ddot{s}+F_\mu-\omega^2\,m\,s-\sin(\om …

$\vec v=\vec \omega\times \vec r$ is always valid , not only in circular motion Circular motion the line element is $$s=\varphi\,r$$ $\Rightarrow$ $$v=\frac{ds}{dt}=\frac{d\varphi}{dt}\,r=\omega\,r$ …

look at this the projection of the vector $~\vec a~$ towards $~\vec b~$ is $$\vec a_p=a\,\cos(\phi)\,\hat b$$ with $$\vec a\cdot\vec b=a\,b\,\cos(\phi)\quad \Rightarrow\\ \cos(\phi)=\frac{\vec a\cdo …

We first write the EOM's for the center mass $C_m$ Translation $$m\,\vec{a}=\vec{F}\tag 1$$ and Rotation $$I_{cm}\,\vec{\alpha}_{cm}+\vec{\omega}_{cm} \times (I_{cm}\,\vec{\omega}_{cm})=\vec …

this animation is result of simulation the equation of motion, I used Euler- Lagrange with non holonomic constraint equation (rolling condition). Ring rolling on paraboloid surface Paraboloid param …