e is always constant, but ke and pe can change if pe and ke change, they must change in such a way...
TRANSCRIPT
E is always constant, but KE and PE can change
If PE and KE change, they must change in such a way as to keep E constant
Example
Consider the 1D free-fall of an object of mass m from a height of y0=h
PEKEE
y m y0=h, t0=0, v0=0y<h, t, v>0
The initial energy of the system defines the total energy
mghPEKEE
KEmghPE
0 , 00
system
System – the collection of objects being study to the exclusion of all other objects in the surroundings, in this example, we consider the object of mass m only
Some time later, KE and PE have changed, but E has not
)(
v221
yhmg
mgymghKE
mgymmgh
PEKEE
Energy
y0h
E
PE=mgy
KE
What is KE, PE, and v when y=h/2?
2)
2( ,
2
hmg
hhmgKE
hmgmgyPE
J 90.422
J 9.80m) 00.1)( kg)(9.80 00.1( 13.3m) 00.1)(80.9(v
v2
2
2
sm
sm
sm
221
EhmgKEPE
mghEgh
mh
mgKE
For y=0 (just before the object hits the ground)?
sm
sm
221
43.4m) 00.1)(80.9(22vv
J 80.9 ,0
2
ghmghm
mghKEEmgyPE
Note: we have neglected air resistance and what happens when the object hits the ground.
Example
A hockey puck slides across the ice. Its speed slows from 45.00 m/s to 44.67 m/s after traveling a distance of 16.0 m. Determine the coefficient of kinetic friction between the ice and the puck.
Solution:
Given: v0=45.00 m/s, vf=44.67 m/s, x=16.0 m=s
Method: Use work-energy theorem
FN
fk
mg
s
mgssfsfsFW
mgFfmafF
mgFmgFF
kkk
kNkkxkx
NNy
180cos cos
,
0
094.0)0.16)(80.9(2
)00.4567.44(
2
)v(v
2vv
vv
2220
2
20
2
202
1221
0
gs
gs
mgsmm
KEKEW
fk
kf
kf
f
Conservative and Non-conservative Forces
Conservative Force: a force for which the work it does on an object does not depend on the path. Gravity is an example.
h
y m
A B C
A
mg s
mghW
mghh
mgsmgW
mghsFW
sin)sin()sin(
cos
mg mg
s s
B
C
Non-conservative Force - a force for which the work done depends on the path
- friction- air resistance
If both conservative and non-conservative forces act on an object, the work-energy theorem is modified
CfNC
fNCCtotal
WKEKEWKEKEWWW
0
0
For the case of gravity
)()(
)()(
00
00
00
00
0
PEKEPEKEPEPEKEKE
PEKEWmgymgyKEKEWyymgKEKEW
yymgWW
ff
ff
NC
ffNC
ffNC
fCg