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 a tangential force on it Ft,
This means the mass m will have a tangential acceleration, at
Ft = m at
For a moment, think of this as a mass m embedded in a massless disk -- like a thin sheet of plexiglass or a thin piece of aluminum. How can we relate this tangential accleration of the small mass m to the angular accelration of the entire "system" or disk?
at = r
The torque due to a tangential force is clearly
= r Ft
= r ( m at )
= r m ( r )
= [ m r 2 ]
I = m r2
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,
looks very much like the linear form
F = 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 is
I = [ mi ri2]
and, for an extended or continuous object, this becomes
r is still the perpendicular distance 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,
Examples Torque Work, Power, and Energy Return to ToC, Ch10, Rotation about a Fixed Axis (c) Doug Davis, 2001; all rights reserved