### Newton's Second Law for Rotations

The rotational version of Newton's law is

where denotes the

*angular acceleration*. As in the previous section, is

*torque*(tangential force times a moment arm ), and is the

*mass moment of inertia*. Thus, the net applied torque equals the time derivative of angular momentum , just as force equals the time-derivative of linear momentum :

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,

*torque*. Thus, we have derived

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:

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Equations of Motion for Rigid Bodies

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Torque