Determine the magnitude and direction of the effective value of g

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Homework Statement

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Determine the magnitude and direction of the effective value of g at a latitude of 45 degrees on the Earth. Assume the Earth is a rotating sphere.

Homework Equations

a = v²/r

The Attempt at a Solution

This question requires a vector diagram no doubt.
What I know is that there is gravity force towards to center of the Earth and normal force exactly opposite to it, and also centripetal force along the x axis. But centripetal force here confuses me as I don't know how to draw it. In my early problems with circular uniform motion, centripetal force was always component of some other force -like normal force, tension force, or friction force. I don't recall drawing an F_centripetal in the vector diagram for questions like these:

But in this gravity question, I think I need to show that there is an F_centripetal along the x axis? Or is it nothing more than a component of gravity force?

 

[gneill]

An object on the surface of the Earth moves in a circle about the axis of rotation of the Earth, so there must be something providing the centripetal force. You're right that it is a portion of the gravitational force that is co-opted for this purpose. The net weight of objects are decreased accordingly, and the net direction for the weight vector is deflected a small amount away from "straight down" to the center of the Earth when the object in question is not located on the equator.

If you choose as your frame of reference the rotating Earth (a non-inertial frame of reference since it's rotating) it allows you to speak of a centrifugal pseudo force without getting too many vocal objections from physicists :) In that case you can draw a free-body diagram for an object and pencil in the gravitational acceleration and centrifugal acceleration. Sum them to find the net acceleration. That net acceleration is what produces the effective weight of the object.