Classical Mechanics Lecture 9
Today's Concepts and Examples:a) Energy and Frictionb) Potential energy & force
Midterm 2 will be held on March 13. Covers units 4-9
Unit 9 Homework DueThursday March 12 11:30 PM.No extension!
Mechanics Lecture 9, Slide 1
Homework 8. Awesome Job!
Average = 92%
Mechanics Lecture 8, Slide 2
Practice Exams
Phys 1500 Exams https://utah.instructure.com/courses/320947/files- Spring 2013: http://www.physics.utah.edu/~springer/phys1500/exams/MidtermExam2.pdf- Solutions: http://www.physics.utah.edu/~springer/phys1500/exams/MidtermExam2Soln.pdf- Long Sample: https://utah.instructure.com/courses/320947/files/45779670/download?wrap=1
Phys 2210 Exams- Practice : http://www.physics.utah.edu/~woolf/2210_Jui/rev2.pdf- Spring 2015: http://www.physics.utah.edu/~woolf/2210_Jui/ex2.pdf
Mechanics Lecture 8, Slide 3
http://www.physics.utah.edu/%7Espringer/phys1500/exams/MidtermExam2.pdfhttp://www.physics.utah.edu/%7Espringer/phys1500/exams/MidtermExam2Soln.pdfhttps://utah.instructure.com/courses/320947/files/45779670/download?wrap=1http://www.physics.utah.edu/%7Ewoolf/2210_Jui/rev2.pdfhttp://www.physics.utah.edu/%7Ewoolf/2210_Jui/ex2.pdf
Main Points
Mechanics Lecture 8, Slide 4
Main Points
Mechanics Lecture 8, Slide 5
Incline with Friction: Work-Kinetic Energy
Using Work-Kinetic Energy Theorem
Same result as using Newton’s Law
Mechanics Lecture 8, Slide 6
H
N1
mg
N2
mg
µmg
must be negative
m
Work by Friction :Wfriction
Checkpoint
What is the total macroscopic work done on the block by all forces during this process?
A) mgH B) –mgH C) µk mgD D) 0
Mechanics Lecture 9, Slide 8
D
m
H
0000
=⇒=∆
===∆
tot
f
i
tot
WKKK
WK
Mechanics Lecture 8, Slide 8
Mechanics Lecture 8, Slide 9
Force from Potential Energy:1D
Mechanics Lecture 8, Slide 10
Force from Potential Energy in 3-d
Gradient operator
Mechanics Lecture 8, Slide 11
Potential Energy vs. Force
dxxdUxF )()( −=
Mechanics Lecture 9, Slide 12
Potential Energy vs. Force
2
21)( kxxU −=
dxxdUxF )()( −= kx−=
Mechanics Lecture 9, Slide 13
Potential Energy vs. Force
Mechanics Lecture 9, Slide 14
Potential Energy vs. Force
Mechanics Lecture 9, Slide 15
Demo
Potential Energy vs. Force
dxxdUxF )()( −=
Mechanics Lecture 9, Slide 16
Equilibrium
Mechanics Lecture 8, Slide 17
Equilibrium points
Mechanics Lecture 8, Slide 18
Equilibrium points
Mechanics Lecture 8, Slide 19
Equilibrium points
Mechanics Lecture 8, Slide 20
Block on Incline
1cos xTxdTWtension ∆=⋅= ∫ θ
mxTv
xTWvvmK
xTWWWWW
f
netif
tensiontensionnormalfrictionnet
1
122
1
cos2
cos)(21
cos
∆=
∆==−=∆
∆==++=
θ
θ
θ
Mechanics Lecture 8, Slide 21
Block on Incline
θµ cosxmgxdfW kkfriction ∆−=⋅= ∫
θsinxmgxdWWgravity ∆−=⋅= ∫
Mechanics Lecture 8, Slide 22
Block on Incline
θµθθ
θµθθθ
θ
cossincos
cossincoscos
cos2;0
12
2221
1
1
2
mgmgTxTx
xmgxmgxTxTxTK
mxTvv
KWWWW
xTxdTW
k
k
if
frictiongravitytensionnet
tension
−−∆−
=∆
∆−∆−∆=∆−∆−=∆
∆==
∆=++=
∆=⋅= ∫
Mechanics Lecture 8, Slide 23
Block on Incline
?0 =⇒
Energy Conservation Problems in general
)()( tUtKUKUKE ffiimechanical +=+=+=
For systems with only conservative forces acting
0=∆ mechanicalE
Emechanical is a constant
Mechanics Lecture 8, Slide 25
Gravitational Potential Energy
rEr
Err
−
Mr
Mrr
−
Mechanics Lecture 8, Slide 26
Gravitational Potential Problems
gravitymechanical hUhmvUKE )()(21 2 +=+=
conservation of mechanical energy can be used to “easily” solve problems.
Define coordinates: where is U=0?
M
M
E
Etotal
M
MMMoon
E
EEEarth
rrmGM
rrmGMrU
rmGMrU
rmGMrU
−−
−−=
−=
−=
)(
)(
)(
0)( →−=r
mGMrU E as ∞→r
rEr
Err
−
Mr
Mrr
−
Add potential energy from each source.
Mechanics Lecture 8, Slide 27
Trip to the moon
)2
1(21
21
21
222
2
iE
E
E
E
Ei
E
E
Ei
Ef
f
E
E
Ei
ffii
vGMRR
RGMv
GM
RmGMmv
mGMR
RmGM
RmGMmv
UKUK
−=
−
−=
−
−=
−=−
+=+
f
Ef
f
E
Ei
ii
RmGMU
KR
mGMU
mvK
−=
=
−=
=
0
21 2
Mechanics Lecture 8, Slide 28
Trip to the moon
%02.0000204.0
)01659.0)(01232.0()()(
)()()(
)(
)(
==
=⇒
≈
−=
−=
−−=
−=
EE
EM
EF
FMEE
EM
E
E
FME
M
EE
EM
FME
MFM
F
EFE
RURU
RR
RdMRM
RM
RdM
RURU
RdmGMRU
RmGMRU Can ignore effect
of moon for this problem at level of precision for SmartPhysics
Mechanics Lecture 8, Slide 29
Trip to the moon
( )
ME
M
E
ME
m
E
Ei
fME
m
f
E
ME
m
E
Ei
ddmGMcmGMb
dmGM
RmGMmva
bdxcbadaxcxbxbdaxadxcxxdbxdax
xdxcxxdb
xx
xdc
xb
xdxd
xdc
xba
RdmGM
RmGM
dmGM
RmGMmv
=−=−=
−−=
=−−++−
=−+−−
+−=−
−+−
=
−
+−−
=−
+=
−−−=−−
2
2
2
2
21
0)(0
)()()(
)()()(
21
…or you can practice solving the quadratic equation with many terms!!!
Mechanics Lecture 8, Slide 30
Trip to the moon
M
m
ME
Ef
ff
EM
m
E
Ei
ii
RmGM
dmGMU
mvK
dmGM
RmGMU
mvK
−−=
=
−−=
=
→
→
2
2
21
21 Can NOT ignore
effect of moon for this problem since the rocket is AT the moon in the end !!!!
McentersME
EcentersEM
RddRdd
−=−=
→
→
Mechanics Lecture 8, Slide 31
Trip to the moon
−−+−=
++−−=
++−−=
−−=−−
+=+
→→
→→
→→
→→
)1(21
22221
212
21
21
2
2222
2
22
ME
Em
ME
E
EME
Em
Ei
Eif
Mi
m
MEi
E
EMi
m
Ei
Eif
M
m
ME
E
EM
m
E
Eif
M
m
ME
Ef
EM
m
E
Ei
ffii
RMRM
dR
dMRM
RvGMvv
RvGM
dvGM
dvGM
RvGMvv
RGM
dGM
dGM
RGMvv
RmGM
dmGMmv
dmGM
RmGMmv
UKUK
Mechanics Lecture 8, Slide 32
Trip to the moon
020608.12
0453.0
0167.0
000207.0
)1(21
2
2
=
=
=
=
−−+−=
→
→
→→
Ei
E
ME
Em
ME
E
EME
Em
ME
Em
ME
E
EME
Em
Ei
Eif
RvGM
RMRM
dR
dMRM
RMRM
dR
dMRM
RvGMvv
Mechanics Lecture 8, Slide 33
Trip to the moon
xx
Mechanics Lecture 8, Slide 34
Block on Incline 2
1cos xTxdTWtension ∆=⋅= ∫ θ
11 xmgxdfW kkfriction ∆−=⋅= ∫ µ
Mechanics Lecture 8, Slide 35
Block on Incline 2
mxmgxTv
xmgxTWvvmK
xmgxTWWWWWW
kf
knetif
ktensionfrictiontensionnormalfrictionnet
)cos(2
cos)(21
cos
111
11122
111
∆−∆=
∆−∆==−=∆
∆−∆=+=++=
µθ
µθ
µθ
Mechanics Lecture 8, Slide 36
Block on Incline 2
θµθθ
θµθθθ
θ
cossincos
cossincoscos
cos2;0
12
2221
1
1
2
mgmgTxTx
xmgxmgxTxTxTK
mxTvv
KWWWW
xTxdTW
k
k
if
frictiongravitytensionnet
tension
−−∆−
=∆
∆−∆−∆=∆−∆−=∆
∆==
∆=++=
∆=⋅= ∫
Mechanics Lecture 8, Slide 37
Block on Incline 2
2sin xmgxdWWgravity ∆=⋅= ∫ θ
Mechanics Lecture 8, Slide 38
Clicker Question A.B.C.
D.
0% 0%0%0%
Suppose the potential energy of some object U as a function of xlooks like the plot shown below.
Where is the force on the object zero?A) (a) B) (b) C) (c) D) (d)
U(x)
x
(a) (b) (c) (d) dxxdUxF )()( −=
Mechanics Lecture 8, Slide 39
Clicker Question A.B.C.
D.
0% 0%0%0%
Suppose the potential energy of some object U as a function of xlooks like the plot shown below.
Where is the force on the object in the +x direction?A) To the left of (b) B) To the right of (b) C) Nowhere
U(x)
x
(a) (b) (c) (d) dxxdUxF )()( −=
Mechanics Lecture 8, Slide 40
Clicker Question A.B.C.
D.
0% 0%0%0%
Suppose the potential energy of some object U as a function of xlooks like the plot shown below.
Where is the force on the object biggest in the –x direction?A) (a) B) (b) C) (c) D) (d)
U(x)
x
(a) (b) (c) (d) dxxdUxF )()( −=
Mechanics Lecture 8, Slide 41
Classical Mechanics �Lecture 9Homework 8. Awesome Job!Practice ExamsMain PointsMain PointsIncline with Friction: Work-Kinetic EnergyWork by Friction :Wfriction
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