### Newton's Second Law for Rotations

The rotational version of Newton's law is

 (B.28)

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,

Multiplying both sides by gives

where we used the definitions and . Furthermore, the left-hand side is the definition of torque . Thus, we have derived

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:

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Equations of Motion for Rigid Bodies
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Torque