Rotational Dynamics

When you apply a force to a rigid body (i.e. one that maintains its form with no internal disruption) at a distance from an axis, the torque you create will cause ____.

In the translational world, F=ma. In the rotational world, ___=___ ___ I (Moment of Inertia) is the rotational analog of mass. It is kind of like mass, but with one DIFFERENCE!

I =  m i r i 2 I = m 1 r m 2 r m 3 r m 4 r 4 2 m3m3 r3r3 m1m1 r1r1 m2m2 r2r2 m4m4 r4r4 I has units of: ______

To find I for objects, either _______ and ______ or use ______________. Hoop of Mass M

To find I for objects, either _______ and ______ or use ______________. I =  m i r i 2 = Mr 2 Hoop of Mass M

To find I for objects, either _______ and ______ or use ______________. I = Mr 2 For a hoop:

For a uniform disk or cylinder: I = ½ Mr 2

For a uniform rod rotated at the center: I = ( 1/12 )ML 2

For a uniform rectangular block: I = ( 1/12 )M(a 2 +b 2 ) a b

For a uniform sphere: I = ( 2/5 )Mr 2

TPS: Which will get to the bottom of an incline (without slipping) faster, a 10 kg hoop or a 10 kg cylinder? (Each has the same radius.)

As the hoop and cylinder roll down the incline, they both lose the same amount of GPE. Where does the GPE go in each case?

Obviously, they each gain KE.

Strangely,neither does any work against friction, because they are __________, not sliding. (However, there may be drag.)

However, as the objects accelerate down the incline without slipping, friction causes the objects to change their rates of rotation.

What was the source of this rotational energy? _____ Energy is required for this process! Just as KE = ½mv 2 translationally, KE rot = _____. ½ I  2

So the total equation we need to consider is: PE TOP = KE T bot + KE R bot = ½ mv 2 + ½ I  2 (Energy may have been sapped by drag, as well.)

So based on all of this, which object would win???

The cylinder would win,

The cylinder would win, because the hoop has a larger _____.

Notice that you could also have determined the work required to create the rotation via the rotational analog of W = Fd: _____________ W = 

Notice (especially if you are into cars) that the analog of P = Fv is… P = 

The analog of p = mv is L =  L = 

L is known as ANGULAR MOMENTUM. Like Linear momentum, angular momentum has always been found to be _______________. L =     

The Law of Conservation of Angular Momentum states that for any situation in which  = 0, L is a constant. (Or, the total angular momentum of a system remains constant.)

So without an external unbalanced torque, an object’s rotational momentum will remain constant… Watch the Travis Pastrana Double Back Flip Clip Double Back Flip Clip

Finally, the analog of J = m  v = F  t is  =  t___ = 