How Can We Soar When the Floor Does No Work?
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It’s comparatively widespread on Physics Boards to see arguments which might be successfully much like the next:
After we bounce off the bottom, the bottom doesn’t transfer. Due to this, the power from the bottom on us does zero whole work. Because the power does no work, we can’t acquire any kinetic vitality. We subsequently can’t bounce off the bottom.
Now, the conclusion right here is clearly false. The world excessive bounce file is 2.45 meters, undoubtedly bigger than zero. So the place did the vitality come from? This Perception seeks to make clear this in a reasonably accessible manner.
An idealized instance
Earlier than leaping off into bipedal mammals competing within the excessive bounce, allow us to have a look at an idealized instance. This instance will assist us perceive what’s going on a bit of higher.
A mass ##m## has a perfect spring of size ##ell## and spring fixed ##okay## hooked up to it. The mass and spring are pressed in opposition to a set wall such that the spring is compressed by a distance ##D##, see the determine beneath.
In different phrases, the mass of the spring is zero, and the power at its ends is given by Hooke’s legislation. All of this happens in a horizontal airplane, which means that we don’t have to take care of gravity.
As soon as launched, the spring pushes the mass away from the wall. Much like the bounce off the bottom, the wall supplies no work. By the identical reasoning as in our instance argument, the mass can’t transfer away from the wall.
The place is the vitality?
So the place does the vitality come from? As a result of the spring will definitely launch the mass away from the wall. With the intention to reply this, allow us to first have a look at the method of compressing the spring. Specifically, we think about a small phase of the spring between the coordinates ##x_0## and ##x_0 + Delta x## when the spring is relaxed. The compressing power pushing on its ends is ##F = -kappa epsilon## in accordance with Hooke’s legislation (see the determine beneath). Right here ##epsilon## is the pressure and ##kappa = okay ell##.
Altering the pressure by ##depsilon##, the decrease finish of the string phase strikes by ##x_0 depsilon## and the higher by ##(x_0+Delta x)depsilon##. The entire work completed on the phase turns into $$dW = F x_0 depsilon – F (x_0+Delta x) depsilon = kappa epsilon Delta x , depsilon.$$ Integrating this from no pressure to a pressure ##epsilon_0## results in $$W = kappa Delta xint_0^{epsilon_0} epsilon,depsilon = frac{kappaepsilon_0^2}{2} Delta x.$$ That is the entire vitality saved within the spring phase at pressure ##epsilon_0##.
That the entire vitality saved within the spring is $$W = frac{kd^2}{2}$$, the place ##d## is the compression of the spring and ##okay = kappa/ell## is the spring fixed, follows straight from the above.
Vitality flux
The dialogue above suggests the concept that vitality can enter or exit an object and stay as inside vitality. This happens by forces performing on the thing performing work. A power ##vec F## performing on an object over a displacement ##dvec r## will do a complete work of ##vec F cdot dvec r##. Within the instance above, ##x_0 depsilon## replaces ##vec dr## for the decrease finish as that is the decrease finish’s displacement and we work in a single dimension. Equally, we have now a directed one-dimensional power ##F## as a substitute of the three-dimensional vector ##vec F##.
The amount ##F x_0 depsilon = – kappa epsilon x_0 depsilon## is, subsequently, a measure of the quantity of vitality flowing upward by the spring at place ##x_0## when the pressure modifications by ##depsilon##. When ##epsilon## is damaging, i.e., when the spring is compressed, vitality will movement upward if ##depsilon## is optimistic. In different phrases, when the spring is compressing vitality flows left within the spring and proper when it’s decompressing.
Launching the mass
The spring will decompress throughout the launch of the mass. The interior vitality saved within the spring then flows from the spring into the mass. Denoting the compression of the spring ##D(t)##, we discover that $$D(t) = D_0 cos(omega t)$$ with ##omega^2 = kappa/mell## throughout the launch, the place the preliminary compression is ##D_0##.
The launch time interval is ##0 leq t leq pi/2omega##. The pressure ##epsilon## is said to ##D## as ##epsilon = -D/ell##. We subsequently get hold of $$frac{depsilon}{dt} = -frac{D'(t)}{ell} = frac{D_0omega sin(omega t)}{ell}.$$
Consequently, the vitality flowing up by the spring at place ##x## is $$frac{dW}{dt} = -kappa epsilon x, depsilon = kappa frac{D_0cos(omega t)}{ell} x frac{D_0omega sin(omega t)}{ell} = frac{kappa D_0^2}{2ell^2} omega xsin(2omega t).$$
It’s pure that this grows linearly with ##x##. As all vitality launched from the spring flows into the mass, the vitality movement will get bigger the nearer to the mass we get.
Relation to the jumper
A jumper’s legs are certainly not a perfect spring. Nevertheless, the dialogue above does give some perception into the problem offered to start with:
- The higher physique will obtain web work from the legs very like the mass obtained web work from the spring throughout launch.
- The web work from the bottom is zero.
- The vitality is supplied from inside vitality saved within the jumper’s muscle mass. Simply because the vitality right here was supplied from inside vitality saved within the spring.
Some variations are additionally notable:
- Not like the spring, the jumper’s decrease physique could have non-zero kinetic vitality on the finish.
- Vitality will even be misplaced within the type of warmth because the effectivity of conversion of inside vitality to macroscopic kinetic vitality just isn’t 100%.
Whereas the bottom doesn’t do work on the jumper, the jumper’s momentum is supplied by the power from the bottom. This momentum is distributed all through the jumper’s physique by inside forces.
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