Download Dynamics of Rotational Motion

Chapter 10
Dynamics of
Rotational Motion
PowerPoint® Lectures for
University Physics, Twelfth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by James Pazun
Modified by P. Lam 5_31_2012
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
Goals for Chapter 10
•  To see how torques cause rotational dynamics (just
as linear forces cause linear accelerations)
•  To calculate work done by a torque
•  To study angular momentum and its conservation
•  To relate rotational dynamics and angular
momentum
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Torque
•  A force (F) applied at
a distance ( r) and at
an angle (θ) will
generate a torque (τ ).
! ! !
" #r$F
!
!
r
!
Magnitude of torque =| " |= rFsin# .
! by the
Direction of torque is given
!
"right - hand rule"- see next slide.
Which force on the figure
produces the largest torque
about point O and which one
produces the smallest torque?
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"
Moment force and Lever arm
Compute cross product
using î,ĵ,k̂:
In this example:
!
r=rsin! î+rsin! ĵ
!
F = Fĵ
! !
r ! F = rsin! î+rcos! ĵ ! Fĵ
(
)
= rsin! î ! Fĵ+rcos! ĵ! Fĵ
=rFsin! k̂ + 0
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Direction of torque vector
•  Mathematically, the
direction of torque
vector is given by the
right-hand rule (RHR)
by convention.
•  Physical effect of torque
is tend to rotate the
object counterclockwise
or clockwise.
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Calculate an applied torque
•  Consider Example 10.1.
•  Refer to Figure 10.5.
!
Which angle should you use in | " |= rFsin# ?
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!
Τ = Iα is just like F = ma
•  A 9.0N force is applied to the wheel for 2s and then released.
•  What is the angular acceleration as a function of time?
•  What is the angular velocity as a function of time? (Given the
wheel was initially at rest)
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!
Another look at the unwinding cable
Given the mass (m) and
the wheel (M), find the
acceleration of m.
(Assume no airresistance or friction at
the axle of the wheel,
assume the no-slipping)
!
!
Concept 1: net F = ma
!
!
Concept 2 : net " = I#
!
!
Concept 3 : | a |= R | # |
(no slipping condition)
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The yo-yo - rolling without slipping
•  Calculate the yoyo s acceleration
and then the final v
after it has fallen a
distance h.
!
!
Concept 1: net F = Ma
!
!
Concept 2 : net " = I#
!
!
Concept 3 : | a |= R | # |
(no slipping condition)
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Energy method (translation + rotation)
1 2
1
K = mvCM + ICM " 2
2
2
!
Rolling without slipping condition " vCM = R#
2 '
$ vCM
1 2
1
" K = mvCM + ICM & 2 )
2
2
%R (
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Application of Conservation of Energy
•  Find speed of
yo-yo after
falling a distance
h.
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The race of objects with different moments of inertia
•  Use energy method to determine which object will
reach the bottom of the incline first (i.e. which object
reach the bottom with the largest speed?)
The object with the smallest moment of inertia will spend
less energy rotating and hence has more energy for
translation. The sphere has the smallest moment of inertia
=> will have the largest speed.
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Why conservation of mechanical energy works when there is friction?
•  What is the role of friction?
Answer: The role of static friction is to do negative work on
the center of motion while doing positive work on the rotation
motion. Net result, the work done by static friction is zero.
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Angular momentum
!
Angular momentum (L) for a moving point mass :
! ! ! !
!
L " r # p = r # (mv).
!
dL ! ! !
=r#F =%
$
dt
!
Angular momentum for a rotating rigid object
about a symmetry axis :
!
!
L = I"
!
!
dL d(I" ) !
=
=#
dt
dt
! !
If I = constant $ I% = # as before.
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!
Compare linear and rotational dynamics
Linear
!
!
F = ma (if m = constant)
!
!
p = mv
! dp!
F=
dt
For a system of objecys :
!
If net Fexternal = 0
!
then p total is conserved.
Rotational dynamics of a rigid object
!
!
" = I# (if I = constant)
!
!
L = I$ (valid for rotation about a
symmetry axis)
!
! dL
"=
dt
!
If net " external = 0,
!
then, L total is conserved.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
Conservation of Angular momentum
•  The frictionless platform ensures the external torque along
the z-direction is zero=>Lz is conserved
I1"1 = I2" 2 ; I # " $
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!
This is how a car s clutch works-conservation of
angular momentum
!
!
!
IA" A + IB" B = (IA + IB )"
!
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Conservation of angular momentum in daily devices:
Gyroscope - A fast spinning top whose angular
momentum vector points at a fixed direction in space =>
a navigation device.
Fast rotating bicycle wheels keep the bicycle stable.
What is the purpose of the small propeller at the back of
a helicopter?
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Gyroscopic precession
•  If external torque =/
=0, there could be
precession motion.
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