It's not that easy. The formula to compute the difference (delta s) of the length (s) of the rope at the bottom and at the top (h) of the sticks is:

delta s = s * h / R

where R is the radius of the earth. On a distance of 5 km and a stick height of 1 m you get on the globe a difference of the rope length of 0.78 mm. It is impossible to measure this difference with a 5 km long rope because there are no ropes that are stiff enough.

But there is a simpler way to get the shape of the earth.

How far is the distance from the north pole to the equator? e = 10008 km.
How do we compute the circumference of the equator on the flat earth and on the globe?

FE: U = 2 * e * pi = 2 * 10008 km * 3.14159 = 62882 km
GE: U = 2 * R * pi = 2 * 6371 km * 3.14159 = 40030 km

How big is the measured circumference of the earth? 40030 km!

So I come to the conclusion that the earth is a globe!