Newton's Second Law for Rotations
The rotational version of Newton's law is
where
![$ \alpha\isdeftext \dot{\omega}$](http://www.dsprelated.com/josimages_new/pasp/img2948.png)
![$ \tau $](http://www.dsprelated.com/josimages_new/pasp/img112.png)
![$ f_t$](http://www.dsprelated.com/josimages_new/pasp/img113.png)
![$ R$](http://www.dsprelated.com/josimages_new/pasp/img9.png)
![$ I$](http://www.dsprelated.com/josimages_new/pasp/img238.png)
![$ \tau $](http://www.dsprelated.com/josimages_new/pasp/img112.png)
![$ L=I\omega$](http://www.dsprelated.com/josimages_new/pasp/img2854.png)
![$ f$](http://www.dsprelated.com/josimages_new/pasp/img195.png)
![$ p$](http://www.dsprelated.com/josimages_new/pasp/img290.png)
![\begin{eqnarray*}
\tau &=& \dot{L} \,\eqss \, I\dot{\omega}\,\isdefss \, I\alpha\\ [5pt]
f &=& \dot{p} \,\eqss \, m\dot{v}\,\isdefss \, ma
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img2949.png)
To show that Eq.(B.28) results from Newton's second law
,
consider again a mass
rotating at a distance
from an axis
of rotation, as in §B.4.3 above, and
let
denote a tangential force on the mass, and
the corresponding tangential acceleration. Then we have, by Newton's
second law,
![$\displaystyle f_t \eqsp ma_t
$](http://www.dsprelated.com/josimages_new/pasp/img2951.png)
![$ R$](http://www.dsprelated.com/josimages_new/pasp/img9.png)
![$\displaystyle f_tR \eqsp ma_tR \isdefs m\dot{v}_tR \isdefs m\dot{\omega}R^2 \eqsp
I\dot{\omega} \eqsp I\alpha.
$](http://www.dsprelated.com/josimages_new/pasp/img2952.png)
![$ \omega=v_tR$](http://www.dsprelated.com/josimages_new/pasp/img2953.png)
![$ I=mR^2$](http://www.dsprelated.com/josimages_new/pasp/img2719.png)
![$ \tau=f_tR$](http://www.dsprelated.com/josimages_new/pasp/img2954.png)
![$\displaystyle \tau\eqsp I\alpha
$](http://www.dsprelated.com/josimages_new/pasp/img2955.png)
![$ f_t=ma_t$](http://www.dsprelated.com/josimages_new/pasp/img2956.png)
![$ f_t$](http://www.dsprelated.com/josimages_new/pasp/img113.png)
![$ a_t$](http://www.dsprelated.com/josimages_new/pasp/img2950.png)
![$ m$](http://www.dsprelated.com/josimages_new/pasp/img6.png)
In summary, force equals the time-derivative of linear momentum, and torque equals the time-derivative of angular momentum. By Newton's laws, the time-derivative of linear momentum is mass times acceleration, and the time-derivative of angular momentum is the mass moment of inertia times angular acceleration:
![$\displaystyle \dot{p_t}=ma_t\;\;\;\Leftrightarrow\;\;\; \dot{L}=I\alpha
$](http://www.dsprelated.com/josimages_new/pasp/img2957.png)
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Equations of Motion for Rigid Bodies
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Torque