PHY 113 A General Physics I 9-9:50 AM MWF Olin...
Transcript of PHY 113 A General Physics I 9-9:50 AM MWF Olin...
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 1
PHY 113 A General Physics I 9-9:50 AM MWF Olin 101
Plan for Lecture 13:
Chapter 8 -- Conservation of energy
1. Potential and kinetic energy for conservative forces
2. Energy and non-conservative forces
3. Power
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 2
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 3
Note: Because of PHY 114 exams on Sunday and Monday, the tutorials those nights will be moved from Olin 101 – check for signs – this week only.
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 4
( ) ( )( )
( ) ( ) EUmvUmv
UUdW
mvmvdW
ffii
iftotalfi
iftotalfi
f
i
f
i
=+=+⇒
−−=⋅=
−=⋅=
∫
∫
→
→
rr
rrrF
rF
r
r
r
r
22
22
21
21
:forces veconservatiFor
21
21
:oremenergy the kinetic- workGeneral
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 5
Example A block, initially at rest at a height h, slides down a frictionless incline. What is its final velocity?
h h=0.5m
i
f
( )
( ) 2
21 ;0 ;0 :
; ;0 :
fff
ii
mvEUvf
mghEmghUvi
==>
===
r
r
( )( )m/s 3.13
5.08.92
221 2
=
=
=
=
ghv
mghmv
f
f
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 6
Energy diagram
E
yi yf
K
U
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 7
Energy diagrams 22
21
21 kxmvE +=
2max2
1221
2max2
1
max
,0(0) :0 when :Note ,0
: when :Note
kxmvUx
kxEvxx
==
=
==
=
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 8
Example: Model potential energy function U(x) representing the attraction of two atoms
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 9
Example: Simplified version:
(Assume that the sandbag is massive enough to provide the tension in the rope and R=3m.) If the actor starts at vi=0 and θ=60o, what is vf?
( )( )( )m/s 5.4
5.0138.92221
21
)cos1(21
22
2
=
−==
=+=
−=+=
if
fff
ii
gyv
mvmgymvE
mgRmgymvE θ0
0
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 10
Example: Mass sliding on frictionless looping track
i
iclicker exercise: In order for the ball completes the loop at A, what must the value of h?
A. h=R B. h=2R C. h>2R D. Not enough
information.
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i Example: Mass sliding on frictionless looping track
( )RmgmvE
mghmvE
A
i
22121
2
2
+=
+=
mgRmv
Rvmmgn
A
A
21
21
:Aat on track stayingfor Condition
2
2
=⇒
−=−−0
0
( )
Rh
mgRmgRmgRRmgmvmghE A
25
252
212
21 2
=⇒
=+=+==
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 12
Another example; first without friction
Mass m1 (=0.2kg) slides horizontally on a frictionless table and is initially at rest. What is its velocity when it moves a distance ∆x=0.1m (and m2 (=0.3kg) falls ∆y=0.1m)?
iclicker exercise: What is the relationship of the final velocity of m1 and m2?
A. They are equal B. m2 is faster than m1. C. m1 is faster than m2.
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 13
Another example; first without friction
Mass m1 (=0.2kg) slides horizontally on a frictionless table and is initially at rest. What is its velocity when it moves a distance ∆x=0.1m (and m2 (=0.3kg) falls ∆y=0.1m)?
T
T
m2 g
21
2
21
21
221
21
mmxgmv
vmvmxTW
f
if
+∆
=
−=∆=0
xgmm
mmxTW
gmm
mmT
amgmTamT
∆+
=∆=
+=⇒
−=−=
21
21
21
21
22
1
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 14
Another example; now with friction
Mass m1 (=0.2kg) slides horizontally on a table with kinetic friction and is initially at rest. What is its velocity when it moves a distance ∆x=0.1m (and m2 (=0.3kg) falls ∆y=0.1m)?
T
T
m2 g
( ) 21
21 2
121
if vmvmxfTW −=∆−=
0
f
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 15
( ) xfmm
mxgmm
mmxfTW
fmm
mgmm
mmT
amgmTamfT
∆+
−∆+
=∆−=
++
+=⇒
−=−=−
21
1
21
21
21
2
21
21
22
1
( )
gmfmm
xfxgmv
vmxfTW
k
f
f
1
21
2
21
2221
µ=+
∆−∆=
=∆−=
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 16
iclicker exercise: Assume a mass m starts at rest at A and moves on the frictionless surface as shown. At what position is the speed the largest?
A. A B. B C. C D. none of these.
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 17
Power
Watt(s)
(Joules) : Units
: thatNote
: workof change of rate timeas Defined
≡=
⋅=⋅=
⋅=
≡
P
dtd
dtdW
ddWdt
dW P
vFrF
rF
9/28/2012 PHY 113 A Fall 2012 -- Lecture 13 18
( )( ) Watts106/3102 /3 ,102For
:motorby exertedPower
44
4
×=×=
=×=
=⋅=≡
smNPsmvNT
Tvdt
dW P vF