Defying Gravity! - 4-sided

This is an extension of this problem .

Now, imagine four surfaces of equal length making a square in a controlled environment, with same parameters (the gravity can be switched to be perpendicular to any of the surfaces). Now, say a ball is dropped from the midpoint of one of the sides of the square, falling in direction perpendicular to the adjacent side of the square. After a certain time period $t_1$ , the direction of the acceleration due to gravity is switched in a cyclic manner, to the next adjacent surface. Then, after time $t_2$ , the direction is switched in the same fashion. Then after every time period $T$ , the direction of gravity is changed. The periodic changes occur in such a way that the ball visits the midpoints of all four surfaces. If the surfaces are $1 km.$ long, and the acceleration due to gravity is $10 ms^{-2}$ , find $\lfloor t_2-t_1\rfloor$ .

Also, derive the expression for $T$ , in terms of $k$ and $g$ , which are length of surface and acceleration due to gravity respectively.