201: Physics Lecture 12
More on Energy Conservation of energy
3/27/08
Physics 201, UW-Madison
1
A force is conservative if.... the work done by the force on an object is free enterprise the work done by the force on an object is independent of the starting and end points the work done by the force on an object is independent of the path the work done by the force in moving an object through an arbitrary closed path is positive. W NET = W1 + W2 + . . .+ Wn ! ! ! ! ! ! = Fi!r1 + Fi!r2 + ..... + Fi!rN ! ! ! ! = Fi( !r1 + !r2 + ... + !rN ) ! ! = Fi!rtot Wg = -mg y = F!y
m r 2 r 1
Question
mg j
y
r 3
Depends only on y, not on path taken!
3/27/08 Physics 201, UW-Madison
r n
2
Conservation of Energy
If only conservative forces are present, the total kinetic plus potential energy of a system is conserved i.e. the total "mechanical energy" is conserved. E=K+U
E = K + U = W + U using K = W = W + (-W) = 0 using U = -W
E = K + U is constant!!!
Both K and U can change, but E = K + U remains constant. But, if non-conservative forces act, then energy can be dissipated in other forms (thermal,sound)
3/27/08
Physics 201, UW-Madison
3
Non-conservative the Forces:
If work done does not depend on the path taken, the force is said to be conservative. If the work done does depend on the path taken, the force is said to be non-conservative. An example of a non-conservative force is friction. When pushing a box across the floor, the amount of work that is done by friction depends on the path taken. Work done is proportional to the length of the path!
3/27/08
Physics 201, UW-Madison
4
Spring pulls on mass : with friction
Now suppose there is a coefficient of friction between the block and the floor The total work done on the block is now the sum of the work done by the spring WS (same as before) and the work done by friction Wf. Wf = f r = - mg d
.
r m stretched position (at rest) d m vr
3/27/08 Physics 201, UW-Madison
f = mg
relaxed position i
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Spring pulls on mass : with friction
Again use Wnet = WS + Wf = K Wf = -mg d
1 WS = kd 2 2
1 !K = mvr 2 2
vr = k 2 d ! 2gd m
1 2 1 kd ! mgd = mvr 2 2 2
r
m stretched position (at rest) d m vr
3/27/08 Physics 201, UW-Madison
f = mg
relaxed position i
6
Elephant in the way Problem
3/27/08
Physics 201, UW-Madison
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