From: Ronald Wong (ronwong@inreach.com)
Date: Thu Dec 16 1999 - 02:55:59 PST
Date: Thu, 16 Dec 1999 02:55:59 -0800 (PST) Message-Id: <l03102801b47dda73f384@[209.209.16.118]> From: Ronald Wong <ronwong@inreach.com> Subject: Re: Centripetal Force at Equator
David Porter posted the following:
>A problem asks why, if the earth spins at such a great speed, don't all of us
>fly off. After calculating the centripetal force due to gravity that is
>necessary to hold us down, one comes up with around 2.3 N for the equator.
>That explains any object, like a person, that is heavier than 2.3 N. It
>obviously doesn't explain why objects that weigh less don't fly off. This
>question came up with one of my students, but I didn't have an explanation.
>Is there anyone out there who does? Thanks, david.
David:
If your object (whose mass is apparently approx. 68.2 kg) is standing at
the equator, there are TWO forces acting on him/her and NEITHER of them is
the centripetal force referred to in your post.
One of them is due to the force of gravity that is pulling the person down
towards the center of the earth.
The other comes about because the earth is in the way and prevents the
person from descending to the center of the earth. As a result, the object
bears down on the surface of the earth and it reacts by pushing up on the
object. This upward force acting on the object is commonly called the
normal force (you could determine the magnitude of this second force by
simply placing the object on a bathroom scale and reading it off).
IF the earth wasn't rotating, then these two forces would be balanced and
the force of gravity (668 N in your case) would equal the normal force
(i.e. the bathroom scale reading would be the same as the force of gravity).
But the earth IS rotating (or so we have been told).
This means that the object is in a rotating frame of reference.
A rotating frame of reference is an example of an accelerated frame of
reference and IN an accelerated frame of reference fictitious forces come
into play (and you thought scientists only dealt with "real" things).
As is true with anything that is moving in a circular fashion, your object
is "falling" towards the center of the motion and just like your stomach
feels like something is lifting it up towards the ceiling of the elevator
as the elevator begins to go down (that something being a fictitious
force), so your object feels an outward, fictitious force as it falls
inwards towards the center of the earth. In this case, the fictitious force
is called the centrifugal force - your 2.3 N force.
__________________________________
In the object's frame of reference, this is a "real" force so it sees THREE
forces acting on it - not two. The centrifugal force and the normal force
acting away from the center of the earth and the gravitational force acting
towards the center of the earth.
Because the centrifugal force is less than the weight of the object, the
object will still bear down on the earth (that's why it doesn't "fly off")
and, as a result, the earth continues to push back up on the object.
Since the object isn't moving relative to the bathroom scale or the surface
of the earth these forces must be balanced. Thus the force of gravity (668
N) must be equal to the sum of the centrifugal force (2.3 N) and the
normal force. In order for this to be true, the normal force must be 665.7
N (approx. 666 N) - 2.3 N less than the object's weight.
Since the bathroom scale is a reflection of the magnitude of this normal
force it too reads 666 N and not 668 N. You "weigh" less at the equator
because of the tangential velocity due to the earth's spin. As you move
towards the poles, the tangential velocity gets less and the readings on
the bathroom scale begin to approach your weight. At the poles, the
readings would be the same as your weight.
The centrifugal force and the weight are proportional to the mass of the
object. So, no matter how small the mass is, the object will not "fly off".
____________________________________
In the frame of reference of the stars, there is NO fictitious force. There
are only two forces acting: gravity and the normal force. They are NOT
balanced.
Since the normal force is 666 N acting outwards, and the gravitational
force is 668 N acting inwards, there is an unbalanced force of 2.3 N acting
inwards (another way of explaining why the object doesn't "fly off"). This
unbalanced force is the centripetal force responsible for the circular
motion observed from the star's frame of reference.
Again, these forces are proportional to the mass of the object. So once
again, no matter how small (or how large) the mass is, the object will not
"fly off".
Notice that the centripetal force is NOT due to gravity. The centripetal
force is the consequence of TWO forces acting on the object - the normal
force and the gravitational force.
Enough said.
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