Why can't we stop the Earth's rotation by walking in the opposite direction?

[Chewy0087]

Okay, this stems from a physics argument we had in class today;

everyone in the argument (my teacher included), seemed to think that we could stop the Earth's' rotation by walking in the opposite direction saying it was "common sense"

however I argued that surely it would have no effect as the same force you use walking forward with your step is acted back upon you as you put your foot back on the ground, due to conservation of momentum? I know that while you're in the process of stepping (between leaving and landing) the rotation is messed. (similiar to shooting a bullet - forces you backwards, but lands in a wall which pushes the wall backward) i hope you can understand what i mean however i drew a quick diagram;

excuse the poor diagram


so, who was right? even my teacher seemed baffled by my attempts to explain it (perhaps he needed time to think, or indeed, he's right). at the time i was sure but since then I've begun to doubt my claims.

 

[willem2]

I hope that wasn't a physics teacher.

you're quite right. If you're walking at constant speed there is no net force on you because of Newtons second law: F = ma, a is 0 so F is also 0. '
If there's no net force on you, you don't exert a net force on the Earth either (Newtons third law).

 

[D H]

The quality of teaching sinks to a new low.

You are correct; you're teacher is an idiot.

People can only walk so far in anyone direction. There are things called oceans that get in the way. Suppose that we built foot bridges that span the oceans so people could walk forever in one direction right along the equator. (Ignore that they will die eventually.) Suppose every person on the Earth partook in this "slow the Earth" campaign by walking "in the opposite direction" along this footpath. I would take "walking in the opposite direction" to mean walking to the west (the Earth's rotation gives us an eastward velocity with respect to inertial). So, what is going to happen to the Earth's rotation?

Answer: It will speed up! The increase in rotation rate will of course be incredibly small: I calculate a 4×10^-11 second decrease in the length of a day.

Why will it speed up? Before the start of this fruitless campaign, the peoples of the Earth were rotating with the Earth. The act of "walking in the opposite direction" decreases the total angular momentum of the people. This decrease has to be balanced by an increase in the Earth's rotation rate because angular momentum is a conserved quantity.


Edit:
The calculation.

Data:
Earth's radius: 6378 km
Walking speed: 1.34 m/s
World population: 6.79e9
Mean human mass: 50 kg (Mean adult mass is ~70 kg, but not all humans are adults)
Earth moment of inertia about the polar axis: 8.034e37 kg m²

Everyone walking opposite the Earth's rotation along the means that the human contribution to the angular momentum of the Earth+humans system will be reduced by

$$\Delta L_{\text{humans}} = -6.79*10^9\,*\,50\,\text{kg}\,*\, 6378\,\text{km}\,*\,1.34\,\text{m/s} = 2.9*10^{18}\, \text{m}^2\text{kg/s}$$

The Earth's angular momentum must have increased by this same amount to conserve angular momentum. The change in the Earth's angular velocity is

$$\Delta \omega_{\text{Earth}} = -\,\frac{\Delta L_{\text{humans}}} { I_{\text{Earth}}} = \frac{2.9*10^{18}\, \text{kg}\,\text{m}^2/\text{s}} {8.034*10^{37}\, \text{kg}\,\text{m}^2} = 3.6\*10^{-20}\,\text{s}^{-1}$$

In terms of length of day,

$$\Delta \text{LOD} = \frac{\Delta\omega_{\text{Earth}}}{\omega_{\text{Earth}}}\,86400\,\text{s} = 4.3*10^{-11}\,\text{s}$$

 

[Chewy0087]

Ahhh okay, thanks for that guys, in fairness perhaps what they meant was walk in the same direction as opposed to the opposite direction, still I'm not sure...

that explanation was great DH, much obliged sir

 

[D H]

Okay, if everyone walks eastward the Earth's rotation will slow down by just a tad, 43 nanoseconds -- while they are walking.

How fast would people have to "walk" to make the Earth stop rotating?

$$v_{\text{stop}} = \frac{8.034*10^{37}\, \text{kg}\,\text{m}^2\, * \,(2\pi/\text{day})} {6.79*10^9\,*\,50\,\text{kg}\,*\, 6378\,\text{km}} = 2.7*10^{15}\,\text{m/s}$$

Or 2.4×10¹¹ Earth escape velocity. Last I read humans cannot run quite that fast.

So, was this a physics teacher?

 

[fawk3s]

wait what..? did he like mean that the Earth's rotation would stop relative to you or that you could stop the Earth's rotation like that? don't quite follow.

 

[Chewy0087]

perhaps I have exaggerated on this a tad;

we were discussing fermi problems (me and my friend) and trying some of our own before my friend asked, "how many people would it take walking to stop the Earth's rotation?" cue debate, with more people (and my Physics teacher) then taking part. in fairness i can't really remember if my physics teacher said same or opposite direction, as my counter-argument was based on forces, as opposed to the more apparent angular momentum issue.

i'm positive that my friends were thinking in terms of forces etc and was therefore wrong however perhaps my teacher had the wrong end of the stick so as to speak and was therefore discussing angular momentum without us realising... in general is he a very clever man. (and talented teacher)

in response to fawk3s, the Earth's' rotation would stop completeley.

once again, thank you and much obliged for clearing this up for me!

 

[Dale]

Everyone walking opposite the Earth's rotation along the means that the human contribution to the angular momentum of the Earth+humans system will be reduced by

$$\Delta L_{\text{humans}} = -6.79*10^9\,*\,50\,\text{kg}\,*\, 6378\,\text{km}\,*\,1.34\,\text{m/s} = 2.9*10^{18}\, \text{m}^2\text{kg/s}$$

This is with the assumption that all of the humans are walking at the equator. This is a good idea since it would reduce the number of transoceanic bridges that would need to be built.

 

[D H]

This is with the assumption that all of the humans are walking at the equator. This is a good idea since it would reduce the number of transoceanic bridges that would need to be built.

That is exactly the assumption I made in post #3.