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PHY 113 C Fall 2013-- Lecture 12 110/08/2013
PHY 113 C General Physics I11 AM - 12:15 PM MWF Olin 101
Plan for Lecture 12:
Chapters 10 & 11 – Rotational motion, torque, and angular momentum
1. Torque
2. Angular momentum
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PHY 113 C Fall 2013-- Lecture 12 210/08/2013
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PHY 113 C Fall 2013-- Lecture 12 310/08/2013
Angular motion
angular “displacement” q(t)
angular “velocity” angular “acceleration”
dtd(t) ωα
dtd(t) θω
“natural” unit == 1 radian
Relation to linear variables: sq = r (qf-qi)
vq = r w
aq = r a
so180 radians 3.14159 radians
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PHY 113 C Fall 2013-- Lecture 12 410/08/2013
Special case of constant angular acceleration: a = a0:
w(t) = wi + a0 t
q(t) = qi + wi t + ½ a0 t2
( w(t))2 wi2 + 2 a0 (q(t) - qi )
angular “displacement” q(t)
angular “velocity” angular “acceleration”
dtd(t) ωα
dtd(t) θω
q(t)w(t)
Rotational motion
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PHY 113 C Fall 2013-- Lecture 12 510/08/2013
Webassign Assignment #10:
A centrifuge in a medical laboratory rotates at an angular speed of 3800 rev/min. When switched off, it rotates 48.0 times before coming to rest. Find the constant angular acceleration of the centrifuge.
Special case of constant angular acceleration: a = a0:
w(t) = wi + a0 t
q(t) = qi + wi t + ½ a0 t2
( w(t))2 wi2 + 2 a0 (q(t) - qi )
Recall:
( )( )( )
( ) revrad 2
s 60min 1
rev 482rev/min 3800
2
0ωFor 222
qqwa
i
i
t
(t)
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PHY 113 C Fall 2013-- Lecture 12 610/08/2013
Review of rotational energy associated with a rigid body
( )
iii
iii
iii
iiirot
rmI
Irm
rmvmK
2
222
22
here w21
21
21
21
:energy Rotational
ww
w
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PHY 113 C Fall 2013-- Lecture 12 710/08/2013
Note that for a given center of rotation, any solid object has 3 moments of inertia; some times two or more can be equal
ji
k
iclicker exercise:Which moment of inertia is the smallest? (A) i (B) j (C) k
d dm m
IB=2md2 IC=2md2IA=0
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PHY 113 C Fall 2013-- Lecture 12 810/08/2013
( )( ) ( )( ) ( )( ) ( )( )( ) 222222 24222223 mkgxmIi
iiyy
From Webassign:
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PHY 113 C Fall 2013-- Lecture 12 910/08/2013
Digression – use of rotational energy in energy storage http://www.beaconpower.com/products/about-flywheels.asp
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PHY 113 C Fall 2013-- Lecture 12 1010/08/2013
Beacon Power company (went bankrupt in 2011)
Continuing efforts – http://www.scientificamerican.com/article.cfm?id=energy-storage-role-in-electric-grid
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PHY 113 C Fall 2013-- Lecture 12 1110/08/2013
22
21
21
:object rolling ofenergy kinetic Total
CM
CMrottotal
MvI
KKK
w
CMvRdtdR
dtds
dtd
wq
qw
: thatNote
( )
22
222
21
21
21
CM
CM
CMrottotal
vMRI
MvRRI
KKK
w
CM CM
Physics of rolling --
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PHY 113 C Fall 2013-- Lecture 12 1210/08/2013
Kinetic energy associated with rotation:
i
iirot rmIIK 2221 w
Distance to axis of rotation
Rolling: rotcomtot KKK
222
1 1
:slipping no is thereIf
comtot
com
vMR
IMK
Rv
w
Rolling motion reconsidered:
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PHY 113 C Fall 2013-- Lecture 12 1310/08/2013
Three round balls, each having a mass M and radius R, start from rest at the top of the incline. After they are released, they roll without slipping down the incline. Which ball will reach the bottom first?
AB C
22 0.1 MRMRI A 22 5.0
21 MRMRIB
22 4.052 MRMRIC
( )2
22
/12
01210
MRIghv
vMR
IMMgh
UKUK
CM
CM
ffii
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PHY 113 C Fall 2013-- Lecture 12 1410/08/2013
How can you make objects rotate?
Define torque:
t = r x F
t = rF sin q
r
F
q
αarτFraF
Imm
qsinr
q
F sin q
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PHY 113 C Fall 2013-- Lecture 12 1510/08/2013
Another example of torque:
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PHY 113 C Fall 2013-- Lecture 12 1610/08/2013
clockwise)(counter :T from Torque
)(clockwise :T from Torque
222
2
111
1
TR
TR
t
t
( )clockwise)(counter 52
(1)(5)(0.5)(15)
: torqueTotal15N 0.5m,
5N 1m, :Example
1122
22
11
Nm .Nm
TRTR
TRTR
t
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PHY 113 C Fall 2013-- Lecture 12 1710/08/2013
Example form Webassign #11
X
t1
t3
t2
iclicker exerciseWhen the pivot point is O, which torque is zero?
A. t1?B. t2?C. t3?
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PHY 113 C Fall 2013-- Lecture 12 1810/08/2013
Newton’s second law applied to rotational motion
dtdM
dtdm CM
totali
ii
ii
vFvF
Newton’s second law applied to center-of-mass motion
mi Fi
ri
dtdm
dtdm i
iiiii
iivrFrvF
( )
axis) principalabout rotating(for αωτ
rωrτ
rωvFrτ
IdtdI
dtdm
total
iiii
ii
iii
i
iidmI 2
di
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PHY 113 C Fall 2013-- Lecture 12 1910/08/2013
An example:
A horizontal 800 N merry-go-round is a solid disc of radius 1.50 m and is started from rest by a constant horizontal force of 50 N applied tangentially to the cylinder. Find the kinetic energy of solid cylinder after 3 s.
K = ½ I w2 t I a w wi at = atIn this case I = ½ m R2 and t = FR
R F
( ) JsN
Nmg
tFgRItFt
IFRIIK
Rg
mgItI
FRtIFR
625.275)3(80050m/s8.9
/21
21
21
21
22
222
2
2222
2
w
awa
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PHY 113 C Fall 2013-- Lecture 12 2010/08/2013
Re-examination of “Atwood’s” machine
T1
T2
T2T1
IT1-m1g = m1a
T2-m2g = -m2a
t =T2R – T1R = I a = I a/R
R
212
12
212
12
τ
a
I/Rmmmm
RIg
I/Rmmmmg
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PHY 113 C Fall 2013-- Lecture 12 2110/08/2013
Another example: Two masses connect by a frictionless pulley having moment of inertia I and radius R, are initially separated by h=3m. What is the velocity v=v2= -v1 when the masses are at the same height? m1=2kg; m2=1kg; I=1kg m2 ; R=0.2m .
m1 m2v1 v2
hh/2
( ) ( )
( ) ( )( ) ( ) ( )( ) sm
RImmmmv
hgmhgm
vvmvmghm
UKUK
RI
ffii
/19.02.0/112
12
/
0
:energy ofon Conservati
2
221
21
21
221
1
2212
2212
121
1 2
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PHY 113 C Fall 2013-- Lecture 12 2210/08/2013
Tl
2
21 MRI
IrTrT lu
a
Example from Webassign #10
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PHY 113 C Fall 2013-- Lecture 12 2310/08/2013
Newton’s law for torque:
αωτ IdtdItotal
F
fs( )
FfMRI
IMRFf
IRfaRIaIRf
MafF
s
s
sCMCMs
CMs
312
21
2
2
cylinder, solid aFor
/11
/
a
Note that rolling motion is caused by the torque of friction:
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PHY 113 C Fall 2013-- Lecture 12 2410/08/2013
Bicycle or automobile wheel:
t
fs
( )
τ/Rf
MRI
I/MRτ/Rf
RIaIfMaf
s
s
CMs
CMs
21
For
1
/αR-τ
2
2
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PHY 113 C Fall 2013-- Lecture 12 2510/08/2013
iclicker exercise:What happens when the bicycle skids?
A. Too much torque is appliedB. Too little torque is appliedC. The coefficient of kinetic friction is too smallD. The coefficient of static friction is too smallE. More than one of these
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PHY 113 C Fall 2013-- Lecture 12 2610/08/2013
Vector cross product; right hand rule
qsinBACBAC
ˆ ˆ ˆ ˆ ˆ ˆ 0ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
i i j j k ki j j i kj k k j ik i i k j
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PHY 113 C Fall 2013-- Lecture 12 2710/08/2013
ˆ ˆ ˆ ˆ ˆ ˆ 0ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
i i j j k ki j j i kj k k j ik i i k j
For unit vectors:
i
k
j
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PHY 113 C Fall 2013-- Lecture 12 2810/08/2013
y z x yx zx y z
y z x yx zx y z
A A A AA AA A A
B B B BB BB B B
ˆ ˆ ˆˆ ˆ ˆ
i j kA B i j k
More details of vector cross products:
( ) ( ) ( )y z z y x z z x x y y xA B A B A B A B A B A Bˆ ˆ ˆ A B i j k
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PHY 113 C Fall 2013-- Lecture 12 2910/08/2013
iclicker exercise:What is the point of vector products
A. To terrify physics studentsB. To exercise your right handC. To define an axial vectorD. To keep track of the direction of rotation
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PHY 113 C Fall 2013-- Lecture 12 3010/08/2013
( ) ( )prvrvrarτFr
aF
dtd
dtmd
dtdmm
m
From Newton’s second law:
( ) ( )prvr dtd
dtmd
ConsiderIs this
A. Wrong?B. Approximately right?C. Exactly right?
iclicker exercise:
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PHY 113 C Fall 2013-- Lecture 12 3110/08/2013
From Newton’s second law – continued – conservation of angular momentum:
( )
(constant)
0 0 If
:Define
L
Lτ
prL
prτFr
dtd
dtd
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PHY 113 C Fall 2013-- Lecture 12 3210/08/2013
Torque and angular momentum
Define angular momentum:
For composite object: L = Iw
constant 0 then 0 If
LLτ
Lωτ
dtd
dtd
dtdI
total
total
prL
Newton’s law for torque:
In the absence of a net torque on a system,
angular momentum is conserved.
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PHY 113 C Fall 2013-- Lecture 12 3310/08/2013
A student sits on a rotatable stool holding a spinning bicycle wheel with angular momentum Li. What happens when the wheel is inverted?
iclicker exercise:
(a) The student will remain at rest.
(b) The student will rotate counterclockwise.
(c) The student will rotate clockwise.
counterclockwise
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PHY 113 C Fall 2013-- Lecture 12 3410/08/2013
More details:
wheelbf
wheelwheelbf
wheelibiwheelfbf
LL
LLL
LLLL
2
0
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PHY 113 C Fall 2013-- Lecture 12 3510/08/2013
Other examples of conservation of angular momentum
mm
d1 d1
mm
d2 d2
I1=2md12 I2=2md2
2
I1w1=I2w2 w2=w1 I1/I2
w1 w2
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PHY 113 C Fall 2013-- Lecture 12 3710/08/2013
Webassign problem:A disk with moment of inertia I1 is initially rotating at angular velocity wi. A second disk having angular momentum I2, initially is not rotating, but suddenly drops and sticks to the second disk. Assuming angular moment to be conserved, what would be the final angular velocity wf?
( )
21
1
211
III
III
if
fi
ww
ww
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PHY 113 C Fall 2013-- Lecture 12 3810/08/2013
Summary – conservation laws we have studied so far
Conserved quantity Necessary conditionLinear momentum p Fnet = 0Angular momentum L tnet = 0
Mechanical energy E No dissipative forces