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PHY 113 A Fall 2012 -- Lecture 25 110/31/2012
PHY 113 A General Physics I9-9:50 AM MWF Olin 101
Plan for Lecture 25:
Review: Chapters 10-13, 15
1. Advice on how to prepare for exam
2. Review of rotational motion, angular momentum, static equilibrium, simple harmonic motion, universal gravitational force law
PHY 113 A Fall 2012 -- Lecture 25 310/31/2012
Format of Friday’s exam
What to bring:1. Clear, calm head2. Equation sheet (turn in with exam)3. Scientific calculator4. Pencil or pen(Note: labtops, cellphones, and other electronic equipment must be off or in sleep mode.)
Timing:May begin as early as 8 AM; must end ≤ 9:50 AM
Probable exam format 4-5 problems similar to homework and class examples;
focus on Chapters 10-13 & 15 of your text. Full credit awarded on basis of analysis steps as well as
final answer
PHY 113 A Fall 2012 -- Lecture 25 410/31/2012
Examples of what to include on equation sheetGiven information
on examSuitable for equation sheet
Universal or common constants (such as g, G, ME, MS, RE…)
Basic equations from material from earlier Chapters: Newton’s laws, energy, momentum, center of mass
Particular constants (such as k,m,I… )
Simple derivative and integral relationships, including trigonometric functions
Unit conversion factors such as Hz and rad/s
Definition of moment of inertia, torque, angular momentum, rotational kinetic energy
Newton’s law for rotational motion; combination of rotational and center of mass motion
Equations describing simple harmonic motion and driven harmonic motion
Newton’s universal gravitation force law and corresponding gravitational potential energy
Gravitational stable circular orbits
PHY 113 A Fall 2012 -- Lecture 25 510/31/2012
Possible extra review session on Thursday:
iclicker question:Which of the following possible times would work with your schedules (vote for one)?
A. 2 PMB. 3 PMC. 4 PMD. Prefer to meet individually or in small groups
in my office (Olin 300).
PHY 113 A Fall 2012 -- Lecture 25 610/31/2012
Rotations: Angular variables
angular “displacement” q(t)
angular “velocity”
angular “acceleration” dtd
(t)ω
α dtd
(t)θ
ω
“natural” angular unit = radian
Relation to linear variables:
sq = r (qf-qi)
vq = r w
aq = r a
s
PHY 113 A Fall 2012 -- Lecture 25 710/31/2012
Object rotating with constant angular velocity (a = 0)
w
v=0v=Rw
Kinetic energy associated with rotation:
“moment of inertia”
iii
iii
iii
rmI
IrmvmK
2
22
1222
122
1
:where
;ωω
Rmi
vi=riw
PHY 113 A Fall 2012 -- Lecture 25 910/31/2012
22
2
1
2
1
:object rolling
ofenergy kinetic Total
CM
CMrollingtotal
MvI
KKK
CMvRdt
dR
dt
dsdt
d
: thatNote
2
2
22
2
2
1
2
1
2
1
CM
CM
CMrollingtotal
vMR
I
MvRR
I
KKK
PHY 113 A Fall 2012 -- Lecture 25 1010/31/2012
How to make objects rotate.
Define torque:
t = r x F
t = rF sin q
r
F
q
αarτFr
aF
Im
m
sinr
q
F sin q
Note: We will define and use the “vector cross product” next time. For now, we focus on the fact that the direction of the torque determines the direction of rotation.
PHY 113 A Fall 2012 -- Lecture 25 1110/31/2012
Vector cross product; right hand rule
sinBAC
BAC
ˆ ˆ ˆ ˆ ˆ ˆ 0
ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
i i j j k k
i j j i k
j k k j i
k i i k j
PHY 113 A Fall 2012 -- Lecture 25 1210/31/2012
Newton’s law for torque:
αω
τ Idtd
Itotal
F
fs
FfMRI
IMRFf
I
RfaRIaIRf
MafF
s
s
sCMCMs
CMs
312
21
2
2
cylinder, solid aFor
/1
1
/
Note that rolling motion is caused by the torque of friction:
PHY 113 A Fall 2012 -- Lecture 25 1310/31/2012
prv
rv
rarτFr
aF
dt
d
dt
md
dt
dmm
mTorque and Newton’s second law:
(constant) :Then
0 0 if : thatNote
:Define
L
Lτ
prL
prτFr
dt
d
dt
d
PHY 113 A Fall 2012 -- Lecture 25 1410/31/2012
Example of conservation of angular momentum:
wheelbf
wheelwheelbf
wheelibiwheelfbf
LL
LLL
LLLL
2
0
PHY 113 A Fall 2012 -- Lecture 25 1510/31/2012
Another example of conservation of angular momentum
mm
d1 d1
mm
d2 d2
I1=2md12 I2=2md2
2
I1w1=I2w2 w2=w1 I1/I2
w2=w1 (d1/d2)2
w1 w2
PHY 113 A Fall 2012 -- Lecture 25 1610/31/2012
Conditions for stable equilibrium
0 : torqueof Balance
0 :force of Balance
ii
ii
τ
F
PHY 113 A Fall 2012 -- Lecture 25 1710/31/2012
0)1( :Torques
0 :Forces
gxmmgM
gmgmgMn
cD
PcD
**X
Example:
PHY 113 A Fall 2012 -- Lecture 25 1810/31/2012
)frequency" natural" -- s(determine ω
)φω(cos)φω(cos ω)φωcos(
:equation thesatisfies guessthat Condition
2
22
2
m
k
tAm
ktA
dt
tAd
constantsunknown are and and where)cos()(
:form thehas )(for solution that Guess
:system spring-massfor law sNewton'
2
2
2
2
AtAtx
tx
xm
k
dt
xd
kxFdt
xdmma
Simple harmonic motion:
PHY 113 A Fall 2012 -- Lecture 25 1910/31/2012
Simple harmonic motion:
Note that:
xmk
dt
xd
dt
xdmkxF
2
2
2
2
mk
tAtx ω);φω(cos)(
φ)ωcos(ω)(
)φωsin(ω)(
2
tAdtdv
ta
tAdtdx
tv
Conveniently evaluated in radians
Constants
Summary --
PHY 113 A Fall 2012 -- Lecture 25 2010/31/2012
Energy associated with simple harmonic motion
2222
2
22222
22
2
1)cos()sin(
2
1
But
)cos(2
1)sin(
2
12
1
2
1
:Energy
kAttkAE
m
k
tkAtAmE
kxmvE
PHY 113 A Fall 2012 -- Lecture 25 2110/31/2012
Simple harmonic motion for a pendulum:
Q
L ) (since sinsin
αsinτ
22
2
2
2
mLILg
ImgL
dt
d
dt
d-IImgL
Approximation for small :Q
Lg
ω );φωcos()
:Solution
sin
2
2
tA(t
Lg
dt
d
PHY 113 A Fall 2012 -- Lecture 25 2210/31/2012
The notion of resonance:
Suppose F=-kx+F0 sin(Wt)
According to Newton:
)sin(
:)ous"inhomogene("equation alDifferenti
)sin(
02
2
2
2
0
tmF
xmk
dtxd
dtxd
mtFkx
)sin(ω
/)sin(
/
/)(
:Solution
220
20 t
mFt
mk
mFtx
PHY 113 A Fall 2012 -- Lecture 25 2310/31/2012
Universal law of gravitation Newton (with help from Galileo, Kepler, etc.) 1687
212
122112
ˆ
r
mGm rF
2
211
kg
mN 10674.6
G
PHY 113 A Fall 2012 -- Lecture 25 2410/31/2012
Review: Gravitational force of the Earth
RE m
2226
2411
2
2
m/s8.9m/s)1037.6(
1098.51067.6
E
E
E
E
R
GMg
R
mGMF
Note: Earth’s gravity acts as a point mass located at the Earth’s center.
PHY 113 A Fall 2012 -- Lecture 25 2510/31/2012
Example: Satellite in circular Earth orbit
onous)(geosynchr
11053.8
1083.35 If4
6
daysT
mh
E
E
E
EE
GM
hRT
r
mGMr
Tm
r
Trm
r
vm
hRrMm
3
2
222
2
2/2
For
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