MadSci Network: Physics
Query:

Re: Tangential Velocity and Force of gravity effects

Date: Fri Aug 28 17:13:48 1998
Posted By: Avra Brahma, Grad student, Aerospace Engineering, OSU
Area of science: Physics
ID: 903544384.Ph
Message:

very interesting question!

the first thing you have to be careful of when you are asking this question
is that you should specify where the object is ( the latitude of the
object). that is because the tangential velocity of an object in circular
motion depends on it's distance from the axis of rotation ( the radius).
at  the equator, this is the greatest, while at the poles this is zero. so
an object at the poles will have no tangential velocity.

next we come to the question that when the earth stops rotating, why
shouldn't the object stop too? well that is exactly what Newton's concept
of inertia is... a body tends to MAINTAIN it's STATE OF MOTION (rest or
moving) unless an external force acts on it. so if we assume that our
object is unrestrained, it will keep moving even though the earth below it
stops. this is also the reason why you fall forward when your car stops
suddenly. the upper part of your body keeps moving while your lower part
stops with the car.

so now that we have established that it keeps on moving with the tangential
velocity it had just before the earth stopped, let's try to analyse what
kind of orbit it goes into.

the reason why objects orbit the earth ( or any other heavenly body) is the
gravitational attraction between the object and earth. if you leave an
object without any motion in earth's gravitational field, it will fall down
to earth because of the attraction. if it has a velocity pointing away from
the earth's centre ( at least tangential) it will tend to go away from the
earth. it is the balance of these two tendencies that determines the shape
of the orbit. in other words the shape of the orbit depends on the velocity
of the object.

for an object to "escape" form the earth, i.e., achieve a parabolic orbit
relative to the earth, a critical velocity called the escape velocity is
needed. this velocity depends inversely on the distance of the object from
the earth's centre ( you can get the exact formula in any physics book).

from the above considerations, we might be tempted to say that the object
will go into the kind of orbit dictated by it's tangential velocity. but
before jumping to that conclusion, let us first consider the dynamics of
the object while the earth is still rotating. to make an object execute
circular motion, we need a CENTRIPETAL force. what supplies this force?
gravity. but the surface of the earth exerts an oppositely directed force
on the object ( the weight it feels). so the difference between the gravity
force and the ground reaction must equal the centripetal force. what
changes when the earth stops rotating? NOTHING ! gravity still acts, there
is still ground reaction and the object still has the same tangential
velocity due to inertia. so IT WILL CONTINUE to trace the SAME path in
space that it was doing before...it slides on the surface of the earth ! in
reality , friction will slow it down an bring it to a stop. then the ground
reaction will become equal to the gravity which means that the object will
become heavier.


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