The center of the earth is the inner core.
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No, the centre of the Earth is called the Earth's core.
you can find center of earth by using only the formulas
No, it is a point on the earth's surface, and thus cannot be the centre as well.
Yes, the axis of the Earth passes through its centre.
at the center of the earth, simple pendulmn has not any gravitational force(if we thought,the earth is an etended object) so at the center the gravitational acceleation is about 'zero' and that's why pendulumn's time period is 'infinite'.