Week 3: Work and Energy
Summary
•The Work-Kinetic Energy Theorem in words is “The work done by the total force acting on an object between two points equals the change in its kinetic energy.” As is frequently the case, though, this is more usefully learned in terms of its algebraic forms:
W (x1 → x2) = |
Zx1 |
Fxdx = 2 mv22 − |
2 mv12 = |
K |
(225) |
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x2 |
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in one dimension or |
Zx1 |
F~ · dℓ~ = 2 mv22 − |
2 mv12 = |
K |
(226) |
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W (~x1 → ~x2) = |
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x2 |
1 |
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in two or more dimensions, where the integral in the latter is along some specific path between the two endpoints.
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• A Conservative Force F c is one where the integral: |
(227) |
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W (~x1 |
→ ~x2) = |
Zx1 |
2 |
F~ c · dℓ~ |
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x |
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does not depend on the particular path taken between ~x1 and ~x2. In that case going from ~x1 to ~x2 by one path and coming back by another forms a loop (a closed curve containing both points). We must do the same amount of positive work going one way as we do negative the other way and therefore we can write the condition as:
I
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~ |
(228) |
F c · dℓ = 0 |
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C
for all closed curves C.
Note Well: If you have no idea what the dot-product in these equations is or how to evaluate it, if you don’t know what an integral along a curve is, it might be a good time to go over the former in the online math review and pay close attention to the pictures below that explain it in context. Don’t worry about this – it’s all part of what you need to learn in the course, and I don’t expect that you have a particularly good grasp of it yet, but it is definitely something to work on!
•Potential Energy is the negative work done by a conservative force (only) moving between two points. The reason that we bother defining it is because for known, conservative force rules, we can do the work integral once and for all for the functional form of the force and obtain an answer that is (within a constant) the same for all problems! We can then simplify the WorkKinetic Energy Theorem for problems involving those conservative forces, changing them into
energy conservation problems (see below). Algebraically:
Z
~ ~
U (~x) = − F c · dℓ + U0 (229)
141
142 |
Week 3: Work and Energy |
where the integral is the indefinite integral of the force and U0 is an arbitrary constant of integration (that may be set by some convention though it doesn’t really have to be, be wary) or else the change in the potential energy is:
Z ~x1
U (~x0 → ~x1) = − |
~ |
~ |
(230) |
F c · dℓ |
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~x0
(independent of the choice of path between the points).
•The Law of Conservation of Mechanical Energy states that if no non-conservative forces are acting, the sum of the potential and kinetic energies of an object are constant as the object moves around:
Ei = U0 + K0 = Uf + Kf = Ef |
(231) |
where U0 = U (~x0), K0 = 12 mv02 etc.
•The Generalized Non-Conservative Work-Mechanical Energy Theorem states that if both conservative and non-conservative forces are acting on an object (particle), the work done by the non-conservative forces (e.g. friction, drag) equals the change in the total mechanical
energy:
Z ~x1
Wnc = |
~ |
~ |
+ K0) |
(232) |
F nc · dℓ = Emech = (Uf + Kf ) − (U0 |
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~x0
In general, recall, the work done by non-conservative forces depends on the path taken, so the left hand side of this must be explicitly evaluated for a particular path while the right hand side depends only on the values of the functions at the end points of that path.
Note well: This is a theorem only if one considers the external forces acting on a particle. When one considers systems of particles or objects with many “internal” degrees of freedom, things are not this simple because there can be non-conservative internal forces that (for example) can add or remove macroscopic mechanical energy to/from the system and turn it into microscopic mechanical energy, for example chemical energy or “heat”. Correctly treating energy at this level of detail requires us to formulate thermodynamics and is beyond the scope of the current course, although it requires a good understanding of its concepts to get started.
• Power is the work performed per unit time by a force:
P = |
dW |
(233) |
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dt |
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In many mechanics problems, power is most easily evaluated by means of:
P = dt ³F~ · dℓ~´ |
= F~ · |
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(234) |
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dt = F~ · ~v |
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d |
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dℓ |
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•An object is in force equilibrium when its potential energy function is at a minimum or maximum. This is because the other way to write the definition of potential energy is:
Fx = − |
dU |
(235) |
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dx |
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so that if |
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dU |
= 0 |
(236) |
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dx |
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then Fx = 0, the condition for force equilibrium in one dimension.
For advanced students: In more than one dimension, the force is the negative gradient of the potential energy:
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∂U |
∂U |
∂U |
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F = − U = − |
∂x |
xˆ − |
∂y |
yˆ − |
∂z |
zˆ |
(237) |
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(where ∂x∂ stands for the partial derivative with respect to x, the derivative of the function one takes pretending the other coordinates are constant.