How Can We Leap When the Floor Does No Work?

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

After we soar off the bottom, the bottom doesn’t transfer. Due to this, the power from the bottom on us does zero complete work. Because the power does no work, we can not achieve any kinetic power. We subsequently can not soar off the bottom.

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

An idealized instance

Earlier than leaping off into bipedal mammals competing within the excessive soar, allow us to take a look at an idealized instance. This instance will assist us perceive what’s going on just a little higher.

A mass ##m## has a really perfect spring of size ##ell## and spring fixed ##okay## hooked up to it. The mass and spring are pressed in opposition to a hard and fast wall such that the spring is compressed by a distance ##D##, see the determine under.

A mass ##m## hooked up to a spring of size ##ell## subsequent to a wall. When pressed in the 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 regulation. All of this happens in a horizontal airplane, that means that we don’t must take care of gravity.

As soon as launched, the spring pushes the mass away from the wall. Just like the soar 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. With a purpose to reply this, allow us to first take a look at the method of compressing the spring. Particularly, we think about 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 regulation (see the determine under). Right here ##epsilon## is the pressure and ##kappa = okay ell##.

To compress a spring section ##Delta x##, forces equal in magnitude however reverse in path act on each ends. The displacement on the high of the section (crimson) is bigger than that on the backside (blue). Subsequently, the higher power does optimistic work of a bigger magnitude than the destructive 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 whole work completed 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 whole power saved within the spring section at pressure ##epsilon_0##.

That the whole power 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.

Power flux

The dialogue above suggests the concept that power can enter or exit an object and stay as inside power. 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, now we have a directed one-dimensional power ##F## as an alternative 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 power flowing upward by the spring at place ##x_0## when the pressure modifications by ##depsilon##. When ##epsilon## is destructive, i.e., when the spring is compressed, power will circulate upward if ##depsilon## is optimistic. 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 throughout the launch of the mass. The interior 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## 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 power 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 power launched from the spring flows into the mass, the power circulate will get bigger the nearer to the mass we get.

Relation to the jumper

A jumper’s legs are under no circumstances a really perfect spring. Nevertheless, the dialogue above does give some perception into the difficulty offered to start with:

• The higher physique will obtain internet work from the legs very similar to the mass acquired internet work from the spring throughout launch.
• The online work from the bottom is zero.
• The power is offered from inside power saved within the jumper’s muscle groups. Simply because the power right here was offered from inside 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 inside power to macroscopic kinetic power shouldn’t be 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 inside forces.