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PreClass Notes: Chapter 10, Sections 10.1-

10.3

• 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

“You’re sitting on a rotating planet.

The wheels of your car rotate. …

Even molecules rotate. Rotational

motion is commonplace throughout

the physical universe.” – R.Wolfson

• 10.1 Angular Velocity and

Acceleration

• 10.2 Torque

• 10.3 Rotational Inertia and

the Analog of Newton’s

Second Law

rotation plane

rota

tion a

xis

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© 2012 Pearson Education, Inc. Slide 1-3

Angular Position

© 2012 Pearson Education, Inc. Slide 1-4

Angular Velocity

Instantaneous: d

dt

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© 2012 Pearson Education, Inc. Slide 1-5

Angular Velocity and Linear Velocity

Angular Acceleration

• Angular acceleration is the rate of change of

angular velocity.

Average:

t Instantaneous:

d

dt

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Acceleration

• Angular and tangential

acceleration

– The linear

acceleration of a point

on a rotating body is

proportional to its

distance from the

rotation axis:

– A point on a rotating

object also has radial

acceleration:

a

t r

a

r

v2

r 2r

Constant Angular Acceleration

• Problems with constant angular acceleration are exactly

analogous to similar problems involving linear motion in

one dimension.

– The same equations apply, with the substitutions:

x , v , a

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Torque

• The tendency of a force to

cause rotation is called

torque.

• Torque depends upon three factors:

– Magnitude of the force

– The direction in which it acts

– The point at which it is applied on the object

Image by John Zdralek, retrieved Jan.10 2013 from http://en.wikipedia.org/wiki/File:1980_c1980_Torque_wrench,_140ft-lbs_19.36m-

kg,_nominally_14-20in,_.5in_socket_drive,_Craftsman_44641_WF,_Sears_dtl.jpg ]

Torque

• The equation for Torque is

• where θ is the angle between the force vector

and the vector from the rotation axis to the force

application point.

rF sin

r sinθ is called the “lever arm”.

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Torque—Example 1 of 3

• Lever arm is less than length of handle because of direction of force.

Torque—Example 2 of 3

• Lever arm is equal to length of handle.

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Torque—Example 3 of 3

• Lever arm is longer than length of handle.

Got it?

• The forces in the figures all

have the same magnitude.

Which force produces zero

torque?

A. The force in figure (a)

B. The force in figure (b)

C. The force in figure (c)

D. All of the forces produce

torque.

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Rotational Inertia

• An object rotating about

an axis tends to remain

rotating about the same

axis at the same rotational

speed unless interfered

with by some external

influence.

• The property of an object to resist changes

in its rotational state of motion is called

rotational inertia (symbol I).

• The SI unit of rotational inertia is kg m2.

[Image downloaded Jan.10, 2013 from http://images.yourdictionary.com/grindstone ]

Rotational Inertia

Depends upon:

• mass of object.

• distribution of mass around

axis of rotation.

– The greater the distance

between an object’s mass

concentration and the axis, the

greater the rotational inertia.

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• Rotational inertia I is

the rotational analog of

mass.

• Rotational

acceleration, torque,

and rotational inertia

combine to give the

rotational analog

of Newton’s second

law:

=I

Finding Rotational Inertia

• For a single point mass m, rotational inertia is the

product of mass with the square of the distance R

from the rotation axis: I mR2.

• For a system of discrete masses,

the rotational inertia is the sum of

the rotational inertias of the

individual masses:

I m

ir

i

2

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Got it?

• Consider the dumbbell in the figure. How would its

rotational inertia change if the rotation axis were at

the center of the rod?

A. I would increase.

B. I would decrease.

C. I would remain the same.

Finding Rotational Inertia

• For continuous matter, the

rotational inertia is given by an

integral over the distribution of

matter:

I r2 dm

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Rotational Inertias of Simple Objects

Example 10.5 Rotational Inertia by Integration: A rod

Find the rotational inertia of a uniform, narrow rod of mass M and

length L about an axis through its center and perpendicular to the rod.

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Rotational Inertias of Simple Objects

Rotational Inertias of Simple Objects

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Rotational Inertias of Simple Objects

Rotational Inertias of Simple Objects

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Rotational Inertias of Simple Objects

© 2012 Pearson Education, Inc. Slide 1-28

Parallel-Axis Theorem

• The parallel-axis theorem states that

where d is the distance from the center-of-mass axis

to the parallel axis and M is the total mass of the

object.

• If we know the rotational inertia Icm about an axis

through the center of mass of a body, the parallel-axis

theorem allows us to calculate the rotational inertia I

through any parallel axis.

2

cmI I Md

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© 2012 Pearson Education, Inc. Slide 1-29

Parallel-Axis

Theorem

Example 10.8 Rotational Dynamics: De-

Spinning a Satellite

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Example 10.9 Rotational and Linear Dynamics:

Into the Well

A solid cylinder of mass M and radius R is mounted on a frictionless

horizontal axle over a well, as shown. A rope is wrapped around the

cylinder and supports a bucket of mass m. The bucket is released from

rest. What is the bucket’s acceleration as it falls down the well shaft?