rotational dynamics. when you apply a force to a rigid body (i.e. one that maintains its form with...
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![Page 1: 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,](https://reader035.vdocuments.net/reader035/viewer/2022062322/5697bf891a28abf838c8a440/html5/thumbnails/1.jpg)
Rotational Dynamics
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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 ____.
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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!
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I = miri2
I = m1r12 + m2r2
2 + m3r32 + m4r4
2
m3r3
m1r1
m2
r2
m4r4
I has units of: ______
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To find I for objects, either _______ and ______ or use ______________.
Hoop of Mass M
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To find I for objects, either _______ and ______ or use ______________.
I = miri2 = Mr2
Hoop of Mass M
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To find I for objects, either _______ and ______ or use ______________.
I = Mr2
For a hoop:
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For a uniform disk or cylinder:
I = ½ Mr2
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For a uniform rod rotated at the center:
I = (1/12)ML2
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For a uniform rectangular block:
I = (1/12)M(a2 +b2)a b
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For a uniform sphere:
I = (2/5)Mr2
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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.)
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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?
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Obviously, they each gain KE.
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Strangely,neither does any work against
friction, because they are __________, not
sliding.(However, there may be drag.)
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However, as the objects accelerate down the incline
without slipping, friction causes the objects to change
their rates of rotation.
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What was the source of this
rotational energy?
_____
Energy is required for this process! Just as KE = ½mv2 translationally, KErot = _____. ½I2
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So the total equation we need to consider is:
PETOP = KETbot + KERbot = ½ mv2 +
½I2
(Energy may have been sapped by drag, as well.)
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So based on all of this, which object would win???
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The cylinder would
win,
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The cylinder would win, because the hoop has a larger
_____.
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Notice that you could also have determined the work
required to create the rotation via the rotational
analog of W = Fd:_____________ W =
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Notice (especially if you are into cars) that the analog of P = Fv is…
P =
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The analog of p = mv isL = L =
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L is known as ANGULAR MOMENTUM. Like Linear
momentum, angular momentum has always
been found to be _______________.
L =
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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.)
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So without an external unbalanced torque, an
object’s rotational momentum will remain
constant…Watch the Travis Pastrana
Double Back Flip Clip
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Finally, the analog of J = mv = Ft is
= t ___ =