The mass of the pod is much less than the end-station of the space elevator. So the cable can be very thin at the far end. If the cable is tapering smoothly it weights 1/4, but it need not to do that, I think it tapers in some kind of exponential curve. So 1/16 might be quite reasonable assumption, but one needs recent college education to calculate that. On the other hand there are strong centrifugal forces, as the pod speeds up from standstill, which is also vividly described in Seveneves book.