From: Ronald Wong (ronwong@inreach.com)
Date: Thu Feb 27 2003 - 23:37:25 PST
Message-Id: <l03102801ba84206dcdbd@[209.209.19.188]> Date: Thu, 27 Feb 2003 23:37:25 -0800 From: Ronald Wong <ronwong@inreach.com> Subject: Re: Falling off the earth
Recently Jumki brought up the following:
>I think it's correct to say that the earth is an accelerating reference
>frame, which, because of rotation and revolution, causes me to move in (two)
>circles, one around the axis of the earth, the other, to a close enough
>approximation, around the sun
>
>So why, as a result of either or both of these circular accelerations, do I
>not feel as if I am falling off the earth, as a result of my own inertia?
>(This is how I feel when I am in a turning car or on a merry-go-round.)
As Paul pointed out, the centripetal acceleration due to our motion about
the sun is so small that it's negligible compared to the centripetal
acceleration due to our motion about the earth's axis of rotation.
In the latter case, two things are worth keeping in mind:
1. The centripetal acceleration depends on the latitude. It's zero at the
poles and maximum at the equator.
2. What your body is feeling when you stand on the earth's surface is the
force of the soles of your shoes pushing up against your feet as your feet
bear down on them.
Consider the case where the centripetal force is greatest - at the equator.
Let's say the earth is a sphere, rotating once every 24 hours and,
initially, perfectly smooth (NO friction!).
If we placed you on the surface, only two forces would act on you - the
gravitational force pulling you down towards the center of the earth and,
since your are bearing down on the earth, the earth pushing back on you
with a force equal to your weight. You would be standing on the surface of
the earth as it - amazingly enough - spun underneath your feet (no friction
- remember? By the way: This establishes the fact that gravity is NOT
responsible for our "going around". It just holds us against mother
earth).
Since we aren't moving about the earth's axis of rotation, there would be
no centripetal acceleration and no sense of "falling off".
In reality of course there is friction so we are swept along with the earth
as it spins about it's axis. We are part of a system that was set in motion
a long time ago when the earth was being formed and, because the system
has rotational inertia it, which includes us, continues to maintain that
rotational motion. There is no force responsible for it's (and our)
rotation. Frictional forces may slow it down (and we may choose to jump in
a plane and engage in motion that is over and above that supplied by the
earth's motion) but nothing is keeping the earth and the things attached to
it in motion.
Away from the poles, this movement brings with it a fictitious force (the
centripetal force) and with it, a fictitious acceleration (the centripetal
acceleration). With the earth's current rate of rotation, the centripetal
acceleration at the equator doesn't amount to much - about 0.03 m/s^2 away
from the center of the earth.
So, in reality, when we placed you on the surface at the equator, there are
still two forces acting on you - the gravitational force pulling you down
towards the center of the earth and, since your are bearing down on the
earth with a slightly smaller force due to the effect of your rotational
motion, the earth pushing back on you with a slightly smaller force than
your weight (the difference is about 0.3 %).
In both cases, you end up bearing down on the earth while it in turn pushes
back on the bottom of your feet. It's this similarity of experience in the
two cases that leads you to "not feel as if I am falling off the earth".
The reason is because you aren't.
Now if the earth spun faster and faster and the frictional forces continued
to keep you in one place on the equator, you would bear down on the earth
with a smaller and smaller force due to the greater centripetal
acceleration (if you dropped a ball, it would fall to the earth with a
smaller and smaller acceleration). Finally, you would reach a point where
the force would be zero. At this point, you would no longer be bearing down
on the earth (the centripetal acceleration would be about 10 m/s^2) and you
would indeed feel as if you were falling off the earth.
There is a problem with this scenario. For one thing, there can't be any
atmosphere (you would be traveling with a speed of around 8000 m/s and
would burn up in an atmosphere). For another, we wouldn't have much time to
get things done (our days would be less than an hour and a half in length)
Of course, in PHYSICS, anything is possible.
Cheers.
ron
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