Discuss the variation of 'g' with angle of latitude.

Acceleration due to gravity changes with the angle of latitude due to shape and rotation of the earth about its own axis.
Variation of g due to shape:

Earth is not a perfect sphere and is flat at the pole, bulges out at the equator. Therefore, the polar radius is smaller than the equatorial radius.
Acceleration due to gravity is inversely proportional to the square of the radius of the earth, therefore the value of g increases as one moves from equator to pole. 
Variation of g due to rotation:

Earth rotates about its own axis as a result of which the objects on the earth experience the centrifugal force.
This centrifugal force decreases the acceleration due to gravity and magnitude of change in the value of g due to rotation of the earth depends on the value of the angle of latitude. 
Consider the earth to be a homogenous sphere of mass M and radius R.
The earth rotates about the polar axis.
Let, ω be the angular velocity of rotation of the earth.
All the objects at rest on earth also revolve about its polar axis with same angular velocity ω.
Consider a body of mass M placed at a point P on the earth at an angle of latitude λ.
                
If the earth were at rest, then the body would have been attracted towards the center of the earth with force mg.
Due to rotation of the earth, the body describes a circle of radius OP = r =  with center at O.
In rotating frame of the earth, body experiences pseudo force =  along  
Now the apparent weight of the body is resultant of force mg directed towards the center and force  along  
Applying parallelogram of vector addition, we get 

     
       
As   therefore can be neglected. 
Therefore,
        
                   [using binomial exp]
     
Substituting  we get 
 is the g.