Unpacking equations - energy and torque
KEY EQUATIONS:
Energy conservation: Ki + Ui + W = Kf + Uf + Eth
K = 1/2 m v2 [sometimes called KE]
Ug = mgh
[sometimes called PE]
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2
Us = /2 k x
[sometimes labelled U]
W = F d cosFd
Torque:  = r F sinrF
As described in lecture, locomotion costs a lot of energy. But exactly what that energy
needs to be “spent” on depends a lot on how you are moving (e.g. walking vs. swimming)
and where you are doing it (e.g. walking across campus vs. swimming under Great Bay).
Look at the different things that are included in the equation:
Energy conservation: Ki + Ui + W = Kf + Uf + Eth
Left-hand side of equation:
Ki:
- what does this stand for?
- what is its value if locomotion begins from a standing start? what if you’re already
moving?
- if Ki is big, what does that mean?
- how can you calculate a value for Ki?
what do you need to know?
- what does the definition of (equation for) Ki tell you about what characteristics or
features of an animal will be important to its Ki: what matters most, and how can you
tell?
- If Kf > Ki, what must have happened in between?
Ui:
- what does this stand for?
- what different kinds or forms of energy might be included in Ui?
- how can you increase your Ui?
by performing or changing a behavior
by having a particular structure or morphology
- what characteristics or features of an animal will be important to its Ui?
- how can you tell whether to include Ug, Us, or both in a calculation: what does each of
them describe? Why can either (or both) count as part of Ui?
- how can you calculate a value for Ui?
Unpacking Equaitons
what do you need to know?
what does the definition of (equation for) Ui tell you about what factors are
important determinants of your Ui: what matters most, and how can you tell?
W:
- what does this stand for?
both WORK IN and WORK OUT…so how do you distinguish? (sign: + or -.
But I still wish we could put “work in” on the left side, and “work out” on the right
side….)
- what do you need W for in locomotion: that is, what do you need to “spend” W to
accomplish or overcome in order to get somewhere?
what equations or definitions do you know to quantify these costs?
what do those equations suggest about the costs of moving through air vs. water?
- what strategies or adaptations would help you minimize your W costs as you move?
structures, behaviors, environments….
(you may want to answer this separately for each part of your answer to the
previous question)
- how can you calculate a value for W?
what do you need to know?
- what does your answer to the previous question tell you about what factors are
important determinants of your W costs: what matters most, and how can you tell?
Right-hand side of equation:
Kf:
- what does this stand for?
- does it matter if locomotion begins from a standing start, vs. if you’re already moving?
- how can you increase your Kf?
in terms of behavior
in terms of energy investment: what can you “pay” to “buy” more Kf?
- what are some real examples of such transactions (energy transformations)?
- what characteristics or features of an animal will be important to its Kf?
- how can you calculate a value for Kf?
what do you need to know?
what does the definition of (equation for) Kf tell you about what factors are
important determinants of your Ki: what matters most, and how can you tell?
Uf:
-what does this stand for?
- what different kinds or forms of energy might be part of Uf?
- how can you increase your Uf?
by performing or changing a behavior
by having a particular structure or morphology
- what characteristics or features of an animal will be important to its Uf?
© J. A. Bolker and D. C. Meredith, UNH, 2009
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Unpacking Equaitons
- how can you tell whether to include Ug, Us, or both in a calculation: what does each of
them describe? Why are they both (potentially) part of Uf?
- how can you calculate a value for Uf?
what do you need to know?
what does the definition of (equation for) Uf tell you about what factors are
important determinants of your Uf: what matters most, and how can you tell?
Eth:
This is something you can measure empirically if you’re clever and careful (and working
with a smallish animal), but there isn’t a formula for it. So if we ask you to figure out
what it is in a particular case, how would you proceed?
- what does this stand for?
- where does it go?
- is it a useful form of energy for locomotion, in general?
- how can you calculate a value for Eth?
what do you need to know?
© J. A. Bolker and D. C. Meredith, UNH, 2009
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Unpacking Equaitons
Look at the different things that are included in the equation Torque:  = r F sinrF
What are the units of  ? Does this make sense? Why or why not?
Physicists usually think about rotational motion going around in full circles (wheels,
orbits, etc.), while biologists are more concerned with things pivoting back and forth
around hinges (e.g. elbows). But even in these biological cases, it really is rotational
motion – just not 360° worth. (epiphany for JAB!!)
Look at some diagrams or examples of rotational motion around a pivot or hinge,
and sketch in what the full circle would look like…remember, in real life the motion is
just an arc, or a small segment of that circle. (I find this really helps me figure out what
distance to use as r.)
To achieve a given amount of torque, you can use a lot of force or a long lever arm.
• What are some real-life examples of each strategy?
• Why (or under what circumstances) might it be preferable to maximize one or the
other?
Relate the terms in the equation to the notions of “force advantage” and “distance (or
speed) advantage” mentioned in the reader. What terms of the equation are being
maximized in each case? Why?
How can a lever “magnify” force or distance, and why is this useful – particularly for
those of us with muscles that can only shorten a little bit?
© J. A. Bolker and D. C. Meredith, UNH, 2009
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