Rotational KE, Angular Rotational KE, Angular MomentumMomentum

Rotational EnergyRotational Energy It is moving so it is a type of Kinetic

Energy (go back and rename the first)

Translational KE Rotaional KE

Example: cylinder rollingExample: cylinder rolling Consider a cylinder with radius R and mass M, rolling

w/o slipping down a ramp. Determine the ratio of the translational to rotational KE.

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Friction causes object to roll, but if it rolls w/o slipping friction does NO work!

W = F d cos q d is zero for point in contact

No dissipated work, energy is conserved

Need to include both translation and rotation kinetic energy.

Example: cylinder rollingExample: cylinder rolling Consider a cylinder with radius R and mass M, rolling

w/o slipping down a ramp. Determine the ratio of the translational to rotational KE.

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2MR VR

use and

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Translational:

Rotational:

Ratio:

Example: cylinder rollingExample: cylinder rolling What is the velocity of the cylinder at the bottom of the

ramp?

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Angular MomentumAngular Momentum

MomentumMomentum Angular MomentumAngular Momentum

p = mV L = I

conserved if Fext = 0 conserved if ext =0

Vector Vector!

units: kg-m/s units: kg-m2/s

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Direction of Angular MomentumDirection of Angular Momentum

Right Hand Rule Wrap fingers of right hand around direction of

rotation, thumb gives direction of angular momentum.

Example: Two DisksExample: Two Disks A disk of mass M and radius R rotates around the z axis

with angular velocity i. A second identical disk, initially not rotating, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity f. Find f.

i

z

f

z

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Example: Merry Go RoundExample: Merry Go RoundFour students (mass 70kg) are riding on a merry-go-round (solid disk of mass = 90kg rotating with angular velocity =3 rad/s. Initially all four students are on the outer edge. Suddenly 3 of students pull themselves to within 0.25m of the center. What is the final angular velocity of the merry-go-round?

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Before

1.2m

After

.25

Example: Merry Go RoundExample: Merry Go Round

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Before

1.2m

After

.25

Example: Merry Go RoundExample: Merry Go RoundWhat is the centripetal acceleration felt by each of the students?

17

Before

1.2m

After

.25

DemoDemoYou are sitting on a freely rotating bar-stool with your arms stretched out and a weight in each hand. Your professor gives you a twist and you start rotating around a vertical axis though the center of the stool. You can assume that the bearing the stool turns on is frictionless, and that there is no net external torque present once you have started spinning.

You now pull your arms and hands (and weights) close to your body.

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DemoDemo What happens to the angular momentum as you pull in your arms?

1. it increases 2. it decreases 3. it stays the same

L1 L2

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What happens to your angular velocity as you pull in your arms?

1. it increases 2. it decreases 3. it stays the same

DemoDemoWhat happens to your kinetic energy as you pull in your arms?

1. it increases 2. it decreases 3. it stays the same

1 2

I2 I1

L L

K 12

2I 12

2 2

II 1

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IL (using L = I )

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Turning the bike wheelTurning the bike wheelA student sits on a barstool holding a bike wheel. The wheel is initially spinning CCW in the horizontal plane (as viewed from above) L= 25 kg m2/s She now turns the bike wheel over. What happens?

A. She starts to spin CCW.B. She starts to spin CW.C. Nothing

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Start w/ angular momentum L pointing up from wheel. When wheel is flipped, no more angular momentum from it pointing up, so need to spin person/stool to conserve L!

Gyroscopic Motion:Gyroscopic Motion: Suppose you have a spinning gyroscope

in the configuration shown below: If the left support is removed, what will

happen??

pivotsupport

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Gyroscopic Motion...Gyroscopic Motion...

Suppose you have a spinning gyroscope in the configuration shown below:

If the left support is removed, what will happen?The gyroscope does not fall down!

pivot

g

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Gyroscopic Motion...Gyroscopic Motion... ... instead it precessesprecesses around its pivot

axis !

pivot

Bicycle wheel

SummarySummary

= I

L = I Right Hand Rule gives directionIf = 0, L is conserved