Force and gravitation

Question

Mass m resting on earth(of mass M and radius R) travels along with earth rotation at surface velocity V around earth center. So force towards earth center is = centripetal force = m.V.V/R but calculating other way, the force towards earth center is g(gravitational) = G.m.M/R.R. Why both answers are not same?

Answer 1

$v^2/r$ is the acceleration an object needs to remain in circular motion about a point. $g$ is the acceleration the Earth exerts on the object (using the force of gravity). The two values are different in this way, so there is no mistake, nothing has gone wrong.

What this tells you is that the Earth exerts a far greater acceleration on this object that it needs to to keep the object in circular motion. As a result, if there were nothing stopping it, then the object would spiral inwards, towards the centre of the Earth. We know this doesn't happen, due to the Earth's crust stopping the object from falling further.

Answer 2

You can only equate gravitational acceleration to centripetal acceleration when the accelerated object isn't in contact with a surface. For example, your comparison would make sense for the Moon's orbit around the Earth. (You may have also heard of dark matter, needed to explain there being enough gravity to keep stars in the galaxy despite the centripetal acceleration at the edge.) But your kilogram also feels a reaction force from the ground.