Uniform Circular Motion of a Spherical Earth

[Gabble1]

Homework Statement

In this question, the Earth is modeled as a uniform sphere of radius 6400km. Objects are released from points just above the Earth's surface at the equator and at the North Pole. Which will fall to the Earth with the greater acceleration and by how much?

Homework Equations

The answer in the book states that objects at the North Pole will fall with a greater acceleration of 0.0338 m/s^2 (3 s.f.) than objects at the equator.

The Attempt at a Solution

I first took a cross section of the Earth at the equator as an example of circular motion.
The centripetal acceleration due to the rotating Earth, a=(omega)^2 * r = (2pi/T)^2 * r = (7.27 * 10^(-5))^2 * 6400000 = 0.0338 m/s^2 <- Correct.

However, at the equator total centripetal acceleration = centripetal acceleration due to rotating Earth + acceleration due to gravity = (0.0338 + g) m/s^2

However, at the North Pole, an object is not rotating with circular motion, thus total centripetal acceleration would simply be acceleration due to gravity, ie. g m/s^2.

Due to this I answered that objects at the equator would fall with a greater acceleration of 0.0338 m/s^2, however this is the opposite answer to the answer given.

Could someone please point out where I am going wrong in my reasoning?

Cheers

 

[Bandersnatch]

Hi Gabble1,

Consider: is centripetal acceleration a real force that pulls an object in, or is it supplied by some other force pulling the object in?

 

[haruspex]

at the equator total centripetal acceleration = centripetal acceleration due to rotating Earth + acceleration due to gravity

You seem rather confused. There are not two different centripetal accelerations. There is gravitational acceleration (which is the actual acceleration, since there are no other forces), the centripetal acceleration (which would be the actual acceleration if the object were to stay at the same altitude while it circled the Earth), and the apparent acceleration(i.e. the second derivative of its altitude).
What do you think the relationship is between those three?