preclass notes: chapter 10, sections 10.1- 10jharlow/teaching/phy131f15/cl16v...2015-08-07 1...
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2015-08-07
1
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?
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