How Long Would a Day Be If the Earth Rotated Fast Enough for Equatorial Objects to Be Weightless?

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To determine how long a day would be if the Earth rotated fast enough for objects at the Equator to be weightless, one must equate gravitational acceleration with the centripetal acceleration experienced by an object in circular motion. This involves calculating the necessary rotational speed to achieve this balance. The discussion also touches on the related question of how fast a satellite must travel to maintain a stable orbit just above the Earth's surface, emphasizing the same principle of matching gravitational force with centrifugal force. The key takeaway is that both scenarios rely on understanding the relationship between gravitational acceleration and centripetal acceleration in a rotating frame.
Theelectricchild
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Hi everyone I am a student in a mechanics course.

We just had a brief intro into gravitation and Newtons synthesis and I was wondering how I can solve this question...

How long would a day be if the Earth were rotating so fast that objects at the Equator were weightless?

The reading on the section is short, thanks for your help.
 
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The same question in different words:

How fast would a satellite have to travel to stay in orbit just a few centimeters above the Earth's surface?

It's all a matter of acceleration due to gravity matching centripetal acceleration--or, in my non-inertial frame, force of gravity matching centrifugal force.

P
 

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