What is the gravitational force experienced by an item in the center of the earth? (Suppose the planet is exactly spherical.)

 Gravity is measured by the acceleration that freely falling objects experience. The acceleration of gravity at the Earth's surface is approximately 9.8 meters (32 feet) per second per second. Thus, an object's speed rises by around 9.8 meters per second for every second it is in free fall.

Because a body at the center of the earth experiences equals gravitational force from all sides, the vector sum of all of these forces equals zero. 

As a result, an object at the center of the earth experiences no force.