Math problem

One day my brain came up with this question that could potentially create world peace. Unfortunately, I couldn't figure out the answer. Until recently. Let's see if you can figure it out. Please send your answers and solutions to

Problem #1
Assume that the Earth is a perfect sphere with a radius of 6378 km. If a rope is stretched around the Earth (i.e. the length of the rope is the same as Earth's circumference), how much longer does the rope need to be if you pinch the rope and lift it up 2 meters from the surface?
Note: the shape of the rope as seen form the side of the Earth will then be "tear drop" shaped.

Problem #2
Same scenario as in problem #1, but the question is: if the rope that is tightly around the Earth is extended 1 meter, how high from the Earth's surface can you lift the rope? Note: trial-and-error with problem #1's expression is not allowed.