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21.4: Translation and Rotation of a Rigid Body Undergoing Fixed Axis Rotation

Last updated

Jul 20, 2022
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- 21.3: Translational and Rotational Equations of Motion
- 21.5: Work-Energy Theorem

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For the special case of rigid body of mass m, we showed that with respect to a reference frame in which the center of mass of the rigid body is moving with velocity $\overrightarrow{\mathbf{V}}_{c m}$ all elements of the rigid body are rotating about the center of mass with the same angular velocity $\overrightarrow{\boldsymbol{\omega}}_{\mathrm{cm}}$ For the rigid body of mass m and momentum $\overrightarrow{\mathbf{p}}=m \overrightarrow{\mathbf{V}}_{c m}$ the translational equation of motion is still given by Equation (21.2.1), which we repeat in the form

$\overrightarrow{\mathbf{F}}^{\mathrm{ext}}=m \overrightarrow{\mathbf{A}}_{\mathrm{cm}} \nonumber$

For fixed axis rotation, choose the z -axis as the axis of rotation that passes through the center of mass of the rigid body. We have already seen in our discussion of angular momentum of a rigid body that the angular momentum does not necessary point in the same direction as the angular velocity. However we can take the z -component of Equation (21.3.28)

$\tau_{c m, z}^{\mathrm{ext}}=\frac{d L_{c m, z}^{\mathrm{spin}}}{d t} \nonumber$

For a rigid body rotating about the center of mass with $\vec{\omega}_{\mathrm{cm}}=\omega_{\mathrm{cm}, z} \hat{\mathbf{k}}$ the z -component of angular momentum about the center of mass is

$L_{c m, z}^{\mathrm{spin}}=I_{\mathrm{cm}} \omega_{\mathrm{cm}, z} \nonumber$

The z -component of the rotational equation of motion about the center of mass is

$\tau_{c m, z}^{\mathrm{ext}}=I_{\mathrm{cm}} \frac{d \omega_{\mathrm{cm}, z}}{d t}=I_{\mathrm{cm}} \alpha_{\mathrm{cm}, z} \nonumber$

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