Newton's Second Law for Rotations

The rotational version of Newton's law is

$\displaystyle \tau \eqsp I\alpha, \protect$

(B.28)

where $\alpha\isdeftext \dot{\omega}$ denotes the angular acceleration. As in the previous section, $\tau$ is torque (tangential force

times a moment arm

), and

is the mass moment of inertia. Thus, the net applied torque $\tau$ equals the time derivative of angular momentum $L=I\omega$ , just as force

equals the time-derivative of linear momentum

$\begin{eqnarray*} \tau &=& \dot{L} \,\eqss \, I\dot{\omega}\,\isdefss \, I\alpha\\ [5pt] f &=& \dot{p} \,\eqss \, m\dot{v}\,\isdefss \, ma \end{eqnarray*}$

To show that Eq. $\,$ (B.28) results from Newton's second law , consider again a mass rotating at a distance from an axis of rotation, as in §B.4.3 above, and let denote a tangential force on the mass, and the corresponding tangential acceleration. Then we have, by Newton's second law,

$\displaystyle f_t \eqsp ma_t$

Multiplying both sides by

gives

$\displaystyle f_tR \eqsp ma_tR \isdefs m\dot{v}_tR \isdefs m\dot{\omega}R^2 \eqsp I\dot{\omega} \eqsp I\alpha.$

where we used the definitions $\omega=v_tR$ and

. Furthermore, the left-hand side is the definition of torque $\tau=f_tR$ . Thus, we have derived

$\displaystyle \tau\eqsp I\alpha$

from Newton's second law

applied to the tangential force

and acceleration

of the mass

In summary, force equals the time-derivative of linear momentum, and torque equals the time-derivative of angular momentum. By Newton's laws, the time-derivative of linear momentum is mass times acceleration, and the time-derivative of angular momentum is the mass moment of inertia times angular acceleration:

$\displaystyle \dot{p_t}=ma_t\;\;\;\Leftrightarrow\;\;\; \dot{L}=I\alpha$

Previous: Torque
Next: Equations of Motion for Rigid Bodies

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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See Also

Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Rigid-Body Dynamics
          Newton's Second Law for Rotations