## Newton's Second Law

## for Rotation about a Fixed Axis

Just because it moves in a circle, an object of mass m must have a centeral force on it -- a radial force, a force toward the center, or a centripetal force.

In addition,consider an object of mass m that has atangentialforce on it F_{t},This means the mass

mwill have atangential acceleration, a_{t}F _{t}= m a_{t}For a moment, think of this as a mass

membedded in a massless disk -- like a thin sheet of plexiglass or a thin piece of aluminum. How can we relate thistangential acclerationof the small massmto theangular accelrationof the entire "system" or disk?We know

a _{t}= rThe

torquedue to a tangential force is clearly= r F _{t}= r ( m a

_{t})= r m ( r )

= [ m r

^{2}]= I

where, again,

I = m r ^{2}is the moment of intertia, the rotational equivalent of the mass, indeed, the "rotational mass".

Notice, of course, that this rotational version of Newton's Second Law,

= I looks

verymuch like the linear formF = m a with which we are already quite familiar.

This immediate example had only a single mass

m. If more masses are involved then, just as before, the moment of inertia isI = [ m _{i}r_{i}^{2}]and, for an extended or continuous object, this becomes

ris still theperpendiculardistance from the axis of rotation.Typically the density will be a constant so we will have

Once the moment of inertia is found or given, we are back to applying Newton's Second Law -- this time for

rotation,= I

Examples TorqueWork, Power, and EnergyReturn to ToC, Ch10, Rotation about a Fixed Axis(c) Doug Davis, 2001; all rights reserved