The cosmonauts who landed at the pole of a planet found that the force of gravity there is 0.01 of that on the Earth, while the duration of the day on the planet is the same as that on the Earth. It turned out besides that the force of gravity on the equator is zero. Determine the radius R of the planet.
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Sol. For a body of mass m resting on the equator of a planet of radius 
, which rotates at an
angular velocity 
, the equation of motion has the form
where N is the normal reaction of the planet surface, and 
is the free-fall acceleration on the planet. By hypothesis, the bodies on the equator are weightless, i.e. 
. Considering that 
, where T is the period of rotation of the planet about its axis (equal to the solar day), we obtain
Substituting the values
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