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How Can We Bounce When the Floor Does No Work?


It’s comparatively frequent on Physics Boards to see arguments which might be successfully much like the next:

Once we leap 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 not acquire any kinetic power. We due to this fact can not leap off the bottom.

Now, the conclusion right here is clearly false. The world excessive leap report is 2.45 meters, undoubtedly bigger than zero. So the place did the power come from? This Perception seeks to make clear this in a reasonably accessible means.

An idealized instance

Earlier than leaping off into bipedal mammals competing within the excessive leap, allow us to take 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 really perfect spring of size ##ell## and spring fixed ##ok## 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 under.

spring of length

A mass ##m## hooked up to a spring of size ##ell## subsequent to a wall. When pressed in direction of the wall, the spring compresses by a distance ##D##.

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. Just like the leap off the bottom, the wall gives no work. By the identical reasoning as in our instance argument, the mass can not transfer away from the wall.

The place is the power?

So the place does the power come from? As a result of the spring will definitely launch the mass away from the wall. In an effort to reply this, allow us to first take a look at the method of compressing the spring. Specifically, we take into account a small section 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 under). Right here ##epsilon## is the pressure and ##kappa = ok ell##.

compress a spring segment

To compress a spring section ##Delta x##, forces equal in magnitude however reverse in route act on each ends. The displacement on the prime of the section (purple) is bigger than that on the backside (blue). Subsequently, the higher power does constructive work of a bigger magnitude than the adverse work of the decrease power.

Altering the pressure by ##depsilon##, the decrease finish of the string section strikes by ##x_0 depsilon## and the higher by ##(x_0+Delta x)depsilon##. The full work finished on the section 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 full power saved within the spring section at pressure ##epsilon_0##.

That the full power saved within the spring is $$W = frac{kd^2}{2}$$, the place ##d## is the compression of the spring and ##ok = kappa/ell## is the spring fixed, follows straight from the above.

Power flux

The dialogue above suggests the concept power can enter or exit an object and stay as inner power. This happens via forces performing on the article 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’ve got 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, due to this fact, a measure of the quantity of power flowing upward via the spring at place ##x_0## when the pressure adjustments by ##depsilon##. When ##epsilon## is adverse, i.e., when the spring is compressed, power will move upward if ##depsilon## is constructive. In different phrases, when the spring is compressing power flows left within the spring and proper when it’s decompressing.

Launching the mass

The spring will decompress through the launch of the mass. The inner power 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## through 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 due to this fact get hold of $$frac{depsilon}{dt} = -frac{D'(t)}{ell} = frac{D_0omega sin(omega t)}{ell}.$$

Consequently, the power flowing up via 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 power launched from the spring flows into the mass, the power move will get bigger the nearer to the mass we get.

Relation to the jumper

A jumper’s legs are not at all a really perfect spring. Nonetheless, the dialogue above does give some perception into the difficulty introduced 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 power is offered from inner power saved within the jumper’s muscle tissue. Simply because the power right here was offered from inner power saved within the spring.

Some variations are additionally notable:

  • Not like the spring, the jumper’s decrease physique could have non-zero kinetic power on the finish.
  • Power may even be misplaced within the type of warmth because the effectivity of conversion of inner power to macroscopic kinetic power isn’t 100%.

Whereas the bottom doesn’t do work on the jumper, the jumper’s momentum is offered by the power from the bottom. This momentum is distributed all through the jumper’s physique by inner forces.

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