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 = miri2

I = m1r12 + m2r2

2 + m3r32 + m4r4

2

m3r3

m1r1

m2

r2

m4r4

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 = miri2 = Mr2

Hoop of Mass M

To find I for objects, either _______ and ______ or use ______________.

I = Mr2

For a hoop:

For a uniform disk or cylinder:

I = ½ Mr2

For a uniform rod rotated at the center:

I = (1/12)ML2

For a uniform rectangular block:

I = (1/12)M(a2 +b2)a b

For a uniform sphere:

I = (2/5)Mr2

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 = ½mv2 translationally, KErot = _____. ½I2

So the total equation we need to consider is:

PETOP = KETbot + KERbot = ½ mv2 +

½I2

(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 isL = 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

Finally, the analog of J = mv = Ft is

= t ___ =