Question

where does the centre of gravity of the atmosphere of the earth lie

Answer 1

Let me help you this question.
The center of gravity of Earth lies at the center of Earth. Its because at that point, the mass of the Earth is equally distributed in all the directions, thus pulling equally in all directions. Hence the net pull is zero. 
As the distance from the centre decreases, the acceleration due to gravity also decreases.
 At the center the distance is zero, hence gravity is zero.
Hope it will help you. Thanks

Answer 2

The earth's atmosphere is about 10,000 km thick,but most of its bulk is contained in the first 11 km above the earth's surface. Since the earth and its atmosphere are roughly spherical,the centre of the earth is also the center of gravity of the earth's atmosphere

Hope it may helps u