PreClass Notes: Chapter 11 - U of T Physics · 2015-08-21 1 PreClass Notes: Chapter 11 •From Essential University Physics 3rd Edition •by Richard Wolfson, Middlebury College •©2016

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PreClass Notes: Chapter 11

• From Essential University Physics 3rd Edition

• by Richard Wolfson, Middlebury College

• ©2016 by Pearson Education, Inc.

• Narration and extra little notes by Jason Harlow,

University of Toronto

• This video is meant for University of Toronto

students taking PHY131.

Outline

“So far we’ve ascribed direction to

rotational motion using the terms

‘clockwise’ and ‘counterclockwise.’

But that’s not enough: To describe

rotational motion fully we need to

specify the direction of the rotation

axis.”– R.Wolfson

• 11.1 Angular velocity

and Angular

acceleration vectors

• 11.2 Torque and the

Vector Cross Product

• 11.3 Angular

Momentum

• 11.4 Conservation of

Angular Momentum

• 11.5 Gyroscopes and

Precession

Rotating earth animation from https://brianin3d.wordpress.com/2011/03/17/animated-gif-of-rotating-earth-via-povray/

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The Angular Velocity Vector

The magnitude of the angular

velocity vector is ω.

S.I. Units are [rad/s]

The angular velocity vector 𝜔points along the axis of rotation in

the direction given by the right-

hand rule for rotation as

illustrated.

A bicycle is traveling toward the right.

What is the direction of the angular velocity of the

wheels?

A.left

B.right

C.into the screen

D.out of the screen

E.up

Got it?

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Direction of the Angular Acceleration

• Angular acceleration points in the direction of the

change in the angular velocity :

– The change can be in the same direction as the

angular velocity, increasing the angular speed.

– The change can be opposite the angular velocity,

decreasing the angular speed.

– Or it can be in an arbitrary direction, changing the

direction and speed as well.

0lim

t

d

t dt

Math: The Cross Product of Two Vectors

The scalar product is one way to multiply two vectors, giving a

scalar. A different way to multiple two vectors, giving a vector, is

called the cross product.

If vectors 𝐴 and 𝐵 have angle between them, their cross product is the vector:

𝐴 × 𝐵 = 𝐴𝐵 sin𝛼

with the direction given by the right-hand rule for cross products:

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Note that .

Instead, .

The cross product is perpendicular to the plane of and . The right-hand rule for cross products comes in several forms. Try them all to see which works best for you.

The Right-Hand Rule

The Torque Vector

We earlier defined torque τ = rFsinϕ.

r and F are the magnitudes of vectors, so this is a really a cross product:

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The Torque Vector

A tire wrench

exerts a torque

on the lug nuts.

• The figure shows a pair of force and radius vectors

and four torque vectors. Which of the numbered

torque vectors goes with the force and radius vectors?

A. Torque vector t1

B. Torque vector t2

C. Torque vector t3

D. Torque vector t4

Got it?

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Angular Momentum of a Particle

We define the particle’s angular momentum vector relative to the origin to be:

A particle of mass m is moving. The particle’s momentum vector makes an angle with the position vector.

Torque causes a particle’s angular momentum to change. This is the rotational equivalent of

and is a general statement of Newton’s second law for rotation.

If you take the time derivative of and use the definition of the torque vector, you find:

Angular Momentum of a Particle

Why this definition?

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Angular Momentum of a Rigid Body

on a fixed axle, or about an axis of

symmetry

For a rigid body, we can add the angular momenta of all the particles forming the object. If the object rotates

then it can be shown that

And it’s still the case that .

Conservation of Angular Momentum

An isolated system that experiences no net torque has

and thus the angular momentum vector is a constant.

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Got it?

If a figure skater pulls in her arms

while rotating, what happens to her

angular speed ω?

A. ω decreases

B. ω increases

C. ω stays the same

Precession

Consider a horizontal

gyroscope, with the disk

spinning in a vertical plane,

that is supported at only one

end of its axle, as shown.

You would expect it to simply

fall over—but it doesn’t.

Instead, the axle remains horizontal, parallel to the ground,

while the entire gyroscope slowly rotates in a horizontal plane.

This steady change in the orientation of the rotation axis is

called precession, and we say that the gyroscope precesses

about its point of support.

The precession frequency Ω is much less that the disk’s

rotation frequency ω.

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Gravity on a

Nonspinning Gyroscope

Shown is a nonspinning

gyroscope.

When it is released, the net

torque is entirely

gravitational torque.

Initially, the angular

momentum is zero.

Gravity acts to increase the

angular momentum gradually

in the direction of the torque,

which is the +x-direction.

This causes the gyroscope

to rotate around x and fall.

Gravity on a

Spinning Gyroscope

Slide 12-104

Shown is a gyroscope initially

spinning around the z-axis.

Initially, gravity acts to

increase the angular

momentum slightly in the

direction of the torque, which

is the +x-direction.

This causes the gyroscopes

angular momentum to shift

slightly in the horizontal plane.

The gravitational torque vector

is always perpendicular to the

axle, so dL is always

perpendicular to L.

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Precession Frequency

Problem 11.61 from Wolfson: Consider a rapidly spinning

gyroscope of mass m whose axis is precessing uniformly in a

horizontal circle of radius r. The spin angular momentum of

the gyroscope is L. Find the angular speed of precession

about the vertical axis.