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Chapter 21. Angular Momentum. Definitions. P = period in seconds (sec) f = frequency in Hertz (Hz or rev/sec) w = angular speed ( rad /sec) P=1/f w = 2 p f. I = Rotational Inertia.

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Chapter 21 l.jpg

Chapter 21

Angular Momentum


Definitions l.jpg

Definitions

  • P = period in seconds (sec)

  • f = frequency in Hertz (Hz or rev/sec)

  • w = angular speed (rad/sec)

  • P=1/f

  • w = 2 p f


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I = Rotational Inertia

  • An object rotating about an axis tends to remain rotating unless interfered with by some external influence.

  • This influence is called torque.

  • Rotation adds stability to linear motion.

    • Examples:

      • spinning football

      • bicycle tires

      • Frisbee


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  • The greater the distance between the bulk of an object's mass and its axis of rotation, the greater the rotational inertia.

  • Examples:

    • Tightrope walker

    • Inertia Bars

    • Ring and Disk on an Incline

    • Metronome


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Conservation of Angular Momentum

  • angular momentum = rotational inertia ´ rotational velocity

  • L = I w

  • Newton's first law for rotating systems:

    • “A body will maintain its state of angular momentum unless acted upon by an unbalanced external torque.”


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Direction of

Motion

Centripetal

Type of

Force

Centrifugal

Type of

Force


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Centripetal

  • …is applied by some object.

  • Centripetal means "center seeking".

Centrifugal

  • …results from a natural tendency.

  • Centrifugal means "center fleeing".

  • This is a fictitious force. Why?


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Examples

Centripetal

Type of Force

CentrifugalType of Force

  • Bucket

  • Earth’s gravity

  • Road Friction

  • Hanger

  • Space Station Floor

  • Nature

  • Nature

  • Nature

  • Nature

  • Nature

  • water in bucket

  • moon’s orbit

  • car on circular path

  • coin on a hanger

  • jogging in a space station


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Circular Motion

  • The linear speed is directly proportional to both rotational speed and radial distance.

    v = w r

  • What are two ways that you can increase your linear speed on a rotating platform?

    • Answers:

      • Move away from the rotation axis.

      • Have the platform spin faster.


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Centripetal Acceleration

  • “Turning acceleration”

    ac = v2/r = w2 r

  • What are two ways that you can increase the “false gravity” in a spinning space station?

    • Answers:

      • Move away from the rotation axis.

      • Have the space station spin faster.


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Moments of Inertia Table 21.2

Hoop or Ring

Solid Cylinder


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Moments of Inertia Table 21.2

Mass on a Rod

m

R


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Moments of Inertia Table 21.2

Short Barbell

Long Barbell


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Moments of Inertia Table 21.2

Rectangular Parallelepiped

b

c

a

Which axis of a book has the highest moment of inertia?


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Moments of Inertia Table 21.2

Solid Sphere

Problem 21.6


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Problem 21.9

Ff

Free

Body

Diagram

FN

Fg


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Problem 21.9

Ff

Free

Body

Diagram

FN

Fg

[maximum]


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Problem 21.9


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Problem 21.10

FN

Free

Body

Diagram

H2O

Fg

Note: “Minimum angular speed” means that FN=0.


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What if you ran in a spinning space station?


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What if you ran in a spinning space station?Would you feel heavier or lighter or no different?


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What if you ran the other way?

You’d lose “weight”!


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Project Two Update

  • We will now have meetings outside of class time to work on Project Two.

  • We will meet in room 315A for these meetings. Be sure to sign in when you arrive.

  • We will attempt to build 2 engines per team, rather than 1 for each person.


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Project Two Update

  • Week 1

    • Piston and Pressure Vessel Construction

  • Week 2

    • Crankshaft and Flywheel Construction

  • Week 3

    • Final Adjustments and Testing


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Project Two Update

  • Take 3 minutes to select a meeting time for your team for this week.

  • Select an alternate time from the list below as well.

    Tuesday

    • 12:00

    • 4:00

    • 5:00

      Wednesday

    • 2:30

    • 3:30

    • 4:30

    • 5:30

      Thursday

    • 11:00

    • 12:00

    • 5:00

      Friday

    • 12:00

    • 1:00

    • 2:00

    • 3:00

    • 4:00


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  • Examples:

    • 1. ice skater spin

    • 2. cat dropped on back

    • 3. Diving

    • 4. Collapsing Stars (neutron stars)


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Torque

  • Torque is the product of the force and lever-arm distance, which tends to produce rotation.

  • Torque = force ´ lever arm

    • Examples:

      • wrenches

      • see-saws


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Center of Mass

  • The center of mass of an object is the average position of mass.

  • Objects tend to rotate about their center of mass.

  • Examples:

    • Meter stick

    • Map of Texas

    • Rotating Hammer

    • Rolling Double-Cone


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Stability

  • For stability center of gravity must be over area of support.

  • Examples:

    • Tower of Pisa

    • Touching toes with back to wall

    • Meter stick over the edge


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Circular Motion

  • Linear speed - the distance moved per unit time. Also called simply speed.

  • Rotational speed - the number of rotations or revolutions per unit time.

  • Rotational speed is often measured in revolutions per minute (RPM).


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Linear versus Rotational MotionEquation Sheet for Exam 4

  • Momentum and Impulse

  • Conservation of Momentum

  • Newton’s Second Law

  • Distance

  • Speed

  • Acceleration

  • vf = vi + at

  • xf = xi +vit + ½ at2


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