classical mechanics lecture 8 today’s examples a) work and kinetic energy problems today's...
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Classical Mechanics Lecture 8
Today’s Examplesa) Work and Kinetic Energy Problems
Today's Concepts:a) Potential Energyb) Mechanical Energy
Mechanics Lecture 8, Slide 1
Midterm 2 will be held on March 13. Covers units 4-9
Main Points
Mechanics Lecture 7, Slide 2
Main Points
Mechanics Lecture 7, Slide 3
Main Points
Mechanics Lecture 7, Slide 4
Physics 211 Lecture 7, Slide 5
Work done by a Spring: Conservative Force!
Net Work over closed path = 0
Conservative Force
Work done by Gravitational Force
Mechanics Lecture 7, Slide 6
rr
mGMF E
gravity ˆ2
ldFW gravitygravity
Work done by Gravitational Force
Mechanics Lecture 7, Slide 7
drrrdrrdrdrr
rdrdrrr
mGMldFdW E
gravity
0ˆˆ)ˆˆ(ˆ
)ˆˆ(ˆ2
rr
mGMF E
gravity ˆ2
ldFW gravitygravity
Work done by Gravitational Force
Mechanics Lecture 7, Slide 8
)11
(
1
1221
221
2
2
1
rrmGMW
drr
mGMdWW
drr
mGMdW
E
r
r
E
E
Work done by Gravitational Force
Mechanics Lecture 7, Slide 9
Clicker QuestionA. B. C.
Mechanics Lecture 7, Slide 10
0% 0%0%
A) Case 1 B) Case 2 C) They are the same
In Case 1 we send an object from the surface of the earth to a height above the earth surface equal to one earth radius.
In Case 2 we start the same object a height of one earth radius above the surface of the earth and we send it infinitely far away.
In which case is the magnitude of the work done by the Earth’s gravity on the object biggest?
Case 1 Case 2
Mechanics Lecture 7, Slide 10
12
11
rrmGMW e
Case 1 Case 2
Case 1:
2RE
RE
Same!
Clicker Question Solution
Mechanics Lecture 7, Slide 11
E
e
EEe R
mGM
RRmGMW
2
1
2
1
Case 2:E
e
Ee R
mGM
RmGMW
22
11
Potential Energy
Mechanics Lecture 8, Slide 12
Checkpoint Clicker Question A. B. C. D.
Mechanics Lecture 8, Slide 13
0% 0%0%0%
In Case 1 we release an object from a height above the surface of the earth equal to 1 earth radius, and we measure its kinetic energy just before it hits the earth to be K1.
In Case 2 we release an object from a height above the surface of the earth equal to 2 earth radii, and we measure its kinetic energy just before it hits the earth to be K2.
Compare K1 and K2.
A) K2 = 2K1 B) K2 = 4K1 C) K2 = 4K1/3 D) K2 = 3K1/2
Case 1Case 2
wrong
Clicker QuestionA. B. C.
Mechanics Lecture 8, Slide 14
0% 0%0%
Usurface =
Usurface =
Usurface =
GMem0
GMem
RE
GMem
2RE
What is the potential energy of an object of mass m on the earths surface:
For gravity: +U0
Mechanics Lecture 8, Slide 14
A)
B)
C)
RE
r
mGMrU e)(
Clicker QuestionA. B. C.
Mechanics Lecture 8, Slide 15
0% 0%0%
A)
B)
C)
Case 1Case 2
RE
r
mGMrU e)(
E
e
R
mGMU 1
E
e
R
mGMU
21
E
e
R
mGMU
31
What is the potential energy of a object starting at the height of Case 1?
Clicker QuestionA. B. C.
Mechanics Lecture 8, Slide 16
0% 0%0%
What is the potential energy of a object starting at the height of Case 2?
A)
B)
C)
Case 1Case 2
RE
E
e
R
mGMU 2
E
e
R
mGMU
22
E
e
R
mGMU
32
r
mGMrU e)(
A)
B)
What is the change in potential in Case 1?
Mechanics Lecture 8, Slide 17
Case 1Case 2
RE
E
e
R
mGMU
21 E
e
R
mGMU
32 E
esurface R
mGMU
e
e
eeecase R
mGM
RRmGMU
2
11
2
11
e
e
eeecase R
mGM
RRmGMU
22
1
2
111
What is the change in potential in Case 2?
What is the change in potential in Case 2?
A)
B)
Mechanics Lecture 8, Slide 18
Case 1Case 2
RE
e
e
eeecase R
mGM
RRmGMU
3
21
3
12
E
e
R
mGMU
21 E
e
R
mGMU
32 E
esurface R
mGMU
e
e
eeecase R
mGM
RRmGMU
3
2
3
112
A) 2
B) 4
C) 4/3
D) 3/2
Case 1Case 2
Mechanics Lecture 8, Slide 19
What is the ratio
e
ecase R
mGMU
21 e
ecase R
mGMU
3
22
1
2
1
2
U
U
K
K
3
4
21
32
Work on Two Blocks
Mechanics Lecture 7, Slide 20
)( hmgWgravity gdmWgravity 2
xx
x
N ldNW0
0
xx
x
N idxjmgW0
0
ˆˆ
00ˆˆ NWij
Work on Two Blocks
Mechanics Lecture 7, Slide 21
KW
11, KWTension
x
WT
xTW
1
1
)( 2 hgmW 2221
220
2221 )(
2
1))((
2
1ff vmmvvmmK
22212 )(
2
1)( fvmmhgm
)(
2
21
221 mm
hgmvv ff
211
210
2111 )(
2
1)(
2
1ff vmvvmK
2111, 2
1ftension vmW
Work on Two Blocks
Mechanics Lecture 7, Slide 22
222
220
2222 )(
2
1)(
2
1ff vmvvmW
xx
x
Tension ldTW0
0
2,
jTT ˆ2
jdyld ˆ
02, TensionW
0)ˆˆ( TdyjjTdyldT
Work on Two Blocks 2
Mechanics Lecture 7, Slide 23
xFxdFWxx
x
0
0
00
0
xx
x
N xdNW
21
220
2
2
2
1
2
1
mm
Wv
mvvvmxFW
f
ff
Work on Two Blocks 2
Mechanics Lecture 7, Slide 24
222
1fvmW
x
WT
xTxFW
2111, 2
1fnet vmKW
211 2
1fnet vmW xTFxFW netnet )(1or
Block Sliding
Mechanics Lecture 7, Slide 25
2
2
10
0
0
0
xkxdxkxdFWxx
x
xx
x
spring
m
Wv
mvW
springf
fspring
2
2
1 2
Block Sliding
Mechanics Lecture 7, Slide 26
xmgxdimgxdFW k
xx
x
k
xx
x
frictionfriction
0
0
0
0
ˆ
)(2
1 20
2 vvmW ffriction
0
21
2 2
m
xkxmgv
springfrick
f
m
mvWv
friction
f
202
12
22
1springfrictionk xkxmg
mg
xkx
k
spring
friction
2
21
Block Sliding
Mechanics Lecture 7, Slide 27
mg
xkx
k
spring
friction
2
21
k
xmgx frictionk
spring
2
2
2
1xkWspring
Block Sliding 2
Mechanics Lecture 7, Slide 28
22
2
1
2
1fspring mvvmKW
22
2
1
2
1mvxkWspring
k
mvx f
2
Block Sliding 2
Mechanics Lecture 7, Slide 29
20
22
2
1
2
1vvmvmW fspring
mg
Wx
xmgW
k
friction
kfriction
Block Sliding 2
Mechanics Lecture 7, Slide 30
2patch
kfriction
xmgW
frictionspring WW
2)(2
1newspring xkW
2)(2
1
2 newpatch
k xkx
mg
k
xmgx patchk
new
2' patch
kpatchkfriction
xmgxmgW
2' kk
Block Sliding 2
Mechanics Lecture 7, Slide 31
roughkfriction xmgW
Potential & Mechanical Energy
Mechanics Lecture 8, Slide 32
Work is Path Independent for Conservative Forces
Mechanics Lecture 8, Slide 33
Work is Path Independent for Conservative Forces
Mechanics Lecture 8, Slide 34
Conservative Forces. Work is zero over closed path
Mechanics Lecture 8, Slide 35
Potential Energy
Mechanics Lecture 8, Slide 36
Gravitational Potential Energy
Mechanics Lecture 8, Slide 37
Mechanical Energy
Mechanics Lecture 8, Slide 38
Mechanical Energy
Mechanics Lecture 8, Slide 39
Conservation of Mechanical Energy
Mechanics Lecture 8, Slide 40
NCspringsgravity WWW
Relax. There is nothing new here
Mechanics Lecture 8, Slide 41
It’s just re-writing the work-KE theorem: everythingexcept gravityand springs
If other forces aren't doing work0E
NCWE NCWUK
NCspringsgravity WUUK
springsgravity UU totWK
Conservation of Mechanical Energy
Mechanics Lecture 8, Slide 42
http://www.statkraft.com/energy-sources/hydropower/pumped-storage-hydropower/
Pumped Storage Hydropower: Store energy by pumping water into upper reservoir at times when demand for electric power is low….Release water from upper reservoir to power turbines when needed ...
Energy “battery” using conservation of mechanical energy
Upper reservoir
Dh
Gravitational Potential Energy
Mechanics Lecture 8, Slide 43
Earth’s Escape Velocity
Mechanics Lecture 8, Slide 44
What is the escape velocity at the Schwarzchild radius of a black hole?
Potential Energy Function
Mechanics Lecture 8, Slide 45
U0 : can adjust potential by an arbitrary constant…that must be maintained for all calculations for the situation.
Finding the potential energy change:
Mechanics Lecture 8, Slide 46
Use formulas to find the magnitude
Check the sign by understanding the problem…
Clicker Question A. B. C.
Mechanics Lecture 8, Slide 47
0% 0%0%
11 KhMghmgW
122 22 KKhMghmgW
Spring Potential Energy: Conserved
Mechanics Lecture 8, Slide 48
Vertical Springs
Mechanics Lecture 8, Slide 49
Massless spring
Vertical Spring in Gravitational Field
Mechanics Lecture 8, Slide 50
Formula for potential energy of vertical spring in gravitational field has same form as long as displacement is measured w.r.t new equilibrium position!!!
Vertical Spring in Gravitational Field
Mechanics Lecture 8, Slide 51
Non-conservative Forces
Mechanics Lecture 8, Slide 52
Work performed by non-conservative forces depend on exact path.
Summary
Mechanics Lecture 8, Slide 53
Work done by any force other than gravity and springs will change E
Lecture 7Work – Kinetic Energy theoremtotalWK
Lecture 8For springs & gravity (conservative forces)
WU
Total Mechanical Energy E = Kinetic + Potential
UKE
NCWE
Summary
Mechanics Lecture 8, Slide 54
kx2
Spring Summary
Mechanics Lecture 8, Slide 55
x
M
Checkpoint
Mechanics Lecture 8, Slide 56
A. B. C. D.
Three balls of equal mass are fired simultaneously with equal speeds from the same height h above the ground. Ball 1 is fired straight up, ball 2 is fired straight down, and ball 3 is fired horizontally. Rank in order from largest to smallest their speeds v1, v2, and v3 just before each ball hits the ground.
A) v1 > v2 > v3
B) v3 > v2 > v1
C) v2 > v3 > v1
D) v1 = v2 = v3
1
2
3
h
87% correct
CheckPoint
Mechanics Lecture 8, Slide 57
A) v1 > v2 > v3
B) v3 > v2 > v1
C) v2 > v3 > v1
D) v1 = v2 = v3
1
2
3
h
They begin with the same height, so they have the same potential energy and change therein, resulting in the same change in kinetic energy and therefore the same speed when they hit the ground.
The total mechanical energy of all 3 masses are equal. When they reach the same height their kinetic energies must be equal, hence equal velocities.
0 UKE87% correct
Clicker Question
Mechanics Lecture 8, Slide 58
A. B. C.
0% 0%0%
Which of the following quantities are NOT the same for the three balls as they move from height h to the floor:
1
2
3
h
A) The change in their kinetic energiesB) The change in their potential energiesC) The time taken to hit the ground
Clicker CheckpointA. B. C.
Mechanics Lecture 8, Slide 59
0% 0%0%
A) B) C)
x
A box sliding on a horizontal frictionless surface runs into a fixed spring, compressing it a distance x1 from its relaxed position while momentarily coming to rest.
If the initial speed of the box were doubled, how far x2 would the spring compress?
12 2xx 12 2xx 12 4xx 90% correct
A) B) C)
21
2KE mv
21
2PE kx
CheckPoint
Mechanics Lecture 8, Slide 60
x
12 2xx 12 2xx 12 4xx
21
)2(
)(
)2(
)2()2(
)(
2
21
21
1
2
21
12
21
11
kvm
kvm
x
x
k
vmvvxx
k
mvvvxx
Mechanics Lecture 8, Slide 61
http://hyperphysics.phy-astr.gsu.edu/hbase/shm2.html#c4
Clicker Question A. B. C.
Mechanics Lecture 8, Slide 62
0% 0%0%
A) At xB) At oC) At y
A block attached to a spring is oscillating between point x (fully compressed) and point y (fully stretched). The spring is un-stretched at point o. At which point is the acceleration of the block zero?
x o y
Mechanics Lecture 8, Slide 63
A=0 at x=0!
http://hep.physics.indiana.edu/~rickv/SHO.html
Integrals as Area Under a curve
Mechanics Lecture 8, Slide 64http://hyperphysics.phy-astr.gsu.edu/hbase/integ.html
Homework
Mechanics Lecture 8, Slide 65
Average=85% Average=86%
Average=82% including 2 who did not do the homework
Example Test Problem
Mechanics Lecture 6, Slide 66
1. What is the work done by gravity during its slide to the bottom of the ramp?
2. What is the work done by friction during one pass through the rough spot?
3. What is the maximum distance the spring is compressed by the box?
4. What is the maximum height to which the box returns on the ramp?
Lecture Thoughts
Mechanics Lecture 8, Slide 67