Physics 218, Lecture XXI 1

Physics 218Lecture 21Dr. David Toback

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Exam Next Tuesday

Today’s Topics notnot covered on the exam, but are a review

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General Rotation

• Chapter 11• Today’s lecture will be about extending the

angular stuff we did last week.• In many ways it’s a summary

–Some math–Torque–Angular Momentum–Kepler’s 2nd Law

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Vector Cross Product

This is the last way of multiplying vectors we will see

A B vs.B A

A A

:Check

B" intoA Swing"

Rule-Hand-Right fromDirection

SinB AC

B A C

Pneumonic: Matrix

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Torque Redefined

A better way to define Torque

This gives us the magnitude and the direction

FR

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Example of Cross Product

The location of a body is length r from the origin and at an angle from the x-axis. A force F acts on the body purely in the y direction.

What is the Torque on the body?

z

x

y

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Angular Momentum

Again we use the Cross Product:

p r

L

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Angular Motion of a Particle

Determine the angular momentum, L, of a particle,with mass m and speed v, moving in uniform circular motion with radius r.

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Kepler’s 2nd Law

2nd Law: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out area in equal periods of time

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A bullet strikes a cylinderA bullet of speed V and

mass m strikes a solid cylinder of mass M and inertia ½MR2, at radius R and sticks. The cylinder is anchored at point 0 and is initially at rest.

What is of the system after the collision?

Is energy Conserved?

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Atwood’s Machine

A pulley with a fixed center (at point O), radius R and moment of inertia I, has a massless rope wrapped around it (no slipping). The rope has two masses, m1 and m2 attached to its ends. Assume m2>m1

• What is the acceleration of the system?

• Do some checks.

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Next Period: Exam• There is an exam here, next class period (Tuesday)• Same as usual:

– Problems from the Textbook, Homework and Lecture

– Formula sheet handed out

– All formulas you use must be derived from those on the formula sheet

• Topics are: Chapters 8-10– Conservation of Energy

– Momentum and Collisions

– Rotational Motion

• Today’s Topics notnot covered, but are a review

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Next Tuesday: Exam• There is an exam here, not next class period, but Tuesday• Same as usual:

– Problems from the Textbook, Homework and Lecture– Formula sheet handed out– All formulas you use must be derived from those on the

formula sheet

• Topics are: Chapters 8-10– Conservation of Energy– Momentum and Collisions– Rotational Motion

• Today’s Topics notnot covered, but are a review • Next week’s topics: Oscillations

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Why does the Bicycle Wheel Turn to the Right?

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Angular Momentum

Again we use the Cross Product:

Derivation of

= dL/dt

p r

L

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L for a system of many bodies

• Have to be careful with Angular Momentum– = dl/dt for a single particle

– = dl/dt) for a system of many particles

– All internal torques cancel because of Newton’s law (all internal forces are equal and opposite)

• Reference Frame matters. Only true for:– The origin is an inertial Reference Frame

– The center of mass

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L for a Rigid Body

Find the angular momentum, L, for this body given that it is rotating around the Z axis with angular velocity