short version :7. conservation of energy
DESCRIPTION
Short Version :7. Conservation of Energy. 7.1. Conservative & Non-conservative Forces. F is conservative if. for every closed path C. W BA + W AB = 0. W AB = W BA. = W AB. W AB. i.e.,. is path-independent. W BA. W AB. - PowerPoint PPT PresentationTRANSCRIPT
Short Version : 7. Conservation of Energy
7.1. Conservative & Non-conservative Forces
F is conservative if
0C
d F r for every closed path C.
B
AB AW d F r is path-independenti.e.
,
0C
d F r
B
AB AW d F r is path-dependent
WAB WBA + WAB = 0 WAB = WBA = WAB
F is non-conservative if there is a closed path C such that
WBAWAB
Mathematica
Example: Work done on climber by gravity
Going up:
W1 = ( m g ) h
= m g h
Going down:
W2 = ( m g ) ( h)
= m g h
Round trip: W = W1 + W2 = 0
Horizontal displacement requires no work.
Gravity is conservative.
ˆm gF j
ˆh r j
ˆh r j
ˆm gF j
Example: Work done on trunk by friction
Going right:
W1 = ( m g ) L = m g L
Going left:
W2 = ( m g ) ( L) = m g L
Round trip: W = W1 + W2 = 2 m g L 0
Friction is non-conservative.
ˆL r iˆf m gF i
ˆL r i ˆf m gF i
GOT IT? 7.1.
If it takes the same amount of work to push a trunk across a rough floor as it
does to lift a weight to the same distance straight upward.
How do the amounts of work compare if the trunk & weight are moved along
curved paths between the same starting & end points?
Ans. Work is greater for the trunk.
7.2. Potential Energy
Conservative force:
Potential energy = stored work
= ( work done by force )
B
AB AB AU W d F r
Note: only difference of potential energy matters.
1-D case: 2
1
x
xU F x d x
Constant F: 2 1U F x x
Gravitational Potential Energy
U m g h
Horizontal component of path does not contribute.
Vertical lift: mgh
U mg y
m g
Elastic Potential Energy
2
1
x
xU F x d x
Ideal spring:
2
10
x
xk x x d x
2
1
20
1
2
x
x
k x x x
00U at x x
0
2
0 0
1
2
x
xU x k x x d x k x x
parabolic
x x0 x = x0 x x0
U is always positive
F k x 0k x x x0 = equilibrium position
2 22 1 0 2 1
1
2k x x x x x 2 2
2 0 1 0
1
2k x x x x
Let
Setting x0 = 0 : 21
2U x k x
21
2k x
7.3. Conservation of Mechanical Energy
netK W c ncW W ncU W
ncE K U W
Mechanical energy: E K U
Law of Conservation of Mechanical Energy:
0E K U if 0ncW ( no non-conservative forces )
constantE K U
Example 7.5. Spring & Gravity
A 50-g block is placed against a spring at the bottom of a frictionless slope.
The spring has k = 140 N/m and is compressed 11 cm.
When the block is released, how high up the slope does it rise?
Initial state: 20 0 0
1
2E U k x
Final state: E U m g h
20
2
k xh
m g
0E E
22
3 2
140 / 11 10
2 50 10 9.8 /
N m mh
kg m s
1.7 m
Example 7.6. Sliding Block
A block of mass m is launched from a spring of constant k that is compressed a distance x0.
The block then slides on a horizontal surface of frictional coefficient .
How far does the block slide before coming to rest?
Initial state: 20 0 0
1
2E U k x
Work done against friction:
nc fW f x
20
2
k xx
m g
m g x
Final state: 0E
Launch:
21 1
1
2E K m v
1 0 0E E Conservation of energy :
10 ncE W
7.4. Potential Energy Curves
Frictionless roller-coaster track
How fast must a car be coasting at
point A if it’s to reach point D?
A CE UCriterion:
21
2 A A Cm v m g h m g h
2A C Av g h h
turning points
potential barrier
potential well
Example 7.7. H2
Near the bottom of the potential well of H2, U = U0 + a ( x x0 )2 ,
where U0 = 0.760 aJ, a = 286 aJ / nm2 , x0 = 0.0741 nm. ( 1 aJ = 1018 J )
What range of atomic separation is allowed if the total energy is 0.717 aJ?
Turning points:
E U 2
0 0U a x x
00
E Ux x
a
2
0.717 0.760
286 /
aJ aJ
aJ nm
0.0123 nm
0.0864 0.0618x nm to nm0 0.074x nm
Force & Potential Energy
Force ~ slope of potential curve
U F r F x
( x along direction of F )
UF
x
0limx
UF
x
dU
d x
dU F r
Gaussian Gun
1 2
1 2U U U
1 1 2
K K U UK
i induced i iU M r H r
12U U
2 1U U
Assume fields of the induced dipoles negligible compared to that of the magnet.
3
1H
r
Video
1 2E K U U
12E K U
i iH H
2 1 2 1r r U U