37.4 Summary of Angular Momentum and Kinetic Energy « Previous | Next »

Fixed Axis Rotation vs. Translation and Rotation

Here we contrast the expressions of the angular momentum and kinetic energy for a rigid object rotating about a fixed axis vs. a rigid object translating and rotating. For the case of fixed axis rotation, the object is pivoted about point S, left figure. Both rods are identical, length $$d$$, mass $$m$$, moment of inertia about an axis passing through the center of mass $$I_{cm}$$, and about an axis passing through point $$S$$ is $$I_S$$. The plane of rotation is contained in he plane of the screen. The position vector of the center of mass measured with respect to $$S$$ is $$\vec{r}_{cm}$$. Angular Momentum $$\vec{L}_S = I_S\vec{\omega}$$ Kinetic Energy $$\displaystyle K = \frac{1}{2}\;I_S\omega^2$$ Angular Momentum $$\vec{L}_S = \vec{r}_s \times m\vec{v}_{cm} + I_{cm}\vec{\omega}$$ Kinetic Energy $$\displaystyle K = \frac{1}{2}mv_{cm}^2+\frac{1}{2}\;I_{cm}\omega^2$$

Note:

In previous questions you have shown that if an object rotates about a fixed axis passing through point $$S$$ and perpendicular to the plane of rotation, the angular momentum about point $$S$$ is also given by: $$\vec{L}_S = m\vec{r}_{cm} \times \vec{v}_{cm} + I_{cm} \vec{\omega}$$

This is true because the center of mass is moving in a circle of radius $$d/2$$ with center at point $$S$$ and with the same $$\vec{\omega}$$ as the object. As a result, the velocity of the center of mass has a magnitude $$\omega d/2$$ therefore $$\vec{L}_S$$ becomes $$\vec{L}_S = m\frac{d^2}{4} \vec{\omega} + I_{cm} \vec{\omega}$$. Using the parallel axis theorem $$\vec{L}_S = I_S\vec{\omega}$$.

You also showed that the kinetic energy for the object rotating about a fixed axis passing through point $$S$$ is expressed as $$\displaystyle K = \frac{1}{2}mv_{cm}^2+\frac{1}{2}\;I_{cm}\omega^2$$.

For the same reason as before, this is true because the speed of the center of mass is $$\omega d/2$$ and the parallel axis theorem.

To avoid confusion, we advise you to always write $$\vec{L}_S = I_S\vec{\omega}$$ and $$K = \frac{1}{2}I_S\omega^2$$ for the case of fixed axis rotation.

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