Imagine yourself to be Krishna. You want to lift India on your finger tip. You have already sliced the country using a cosmic knife 1 km below the sea level. Where would you keep your finger? Remember the Indian plate needs to be balanced.
There is no easy answer this question. The estimates, however, are easier to make. One such estimate is below.
There are two markers on the map. The one in the state of Madhya Pradesh (MP) is the centroid of the country. If you draw a map on a paper, the middle point of the map is the centroid. But you lift the country at the centroid, it won’t be balanced. In case of India, the Himalayas are heavy and it will tilt the country to the north.
The center of mass of a country is a weighted centroid. Each pixel on the map below has a weight which is proportional to its elevation.
There are a few caveats to this answer. I am assuming uniform soil density which is not true. The sand in Ganges plains has less density than rocks of Western Ghats or the Himalayas. Also the boundaries of India and how they are mapped from the near spherical earth to a 2d paper.
The technical details can be found at this repository https://github.com/dilawar/map-india-center. This work can easily be extended to compute the center of mass of each state. I might do it in the future.