# resistance of a flywheel to change of angular momentum

I've got a flywheel with angular momentum $\vec L$. Now I'd like to change the orientation of $\vec L$ by an angle $φ$ (in degrees). How do I calculate the angular momentum $\vec M$ for that change?

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Hint: Say you have a position vector $\vec{r} =(x,y,z)$, and you rotate it an angle $\varphi$. What would be the change $|\Delta\vec{r}|$ in position, as a function of $r=|\vec{r}|$ and $\varphi$? – Qmechanic May 10 '11 at 23:29

Rate of change of angular momentum is equal to sum of applied moments.

$$\vec{M} = \dfrac{\rm{d}\vec{L}}{\rm{d}t}$$

To impose a diffrence in angular momentum, you have to apply an moment action

$$\int\vec{M}\,{\rm d}t={\rm Rot}(\hat{k},\varphi)\,\vec{L}-\vec{L}$$

and if the moment is constant, it can be taken outside the integral.

=== Euler Equations ===

Or you can use the equations of motion

$$\vec{M}=I\,\vec{\alpha}+\vec{\omega}\times I\,\vec{\omega}$$

expressed in an inertial frame, and use $\vec{L}=I\,\vec{\omega}$ at any instant.

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