# 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?

We know

at = r

The torque due to a tangential force is clearly

= r Ft

= r ( m at )

= r m ( r )

= [ m r 2 ]

= I

where, again,

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,

= I

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,

= I