Draw a picture by yourself. Let A be the angle between the cycloid and the vertical line, the pendulum length L, the ball mass M and the acceleration of gravity G, then

ma=m*g*sinA

When a is small (tends to 0), Sina is about equal to a.

m*a=m*g*A……( 1)

The differential equation corresponding to equation (1) is a second-order ordinary differential equation whose solution

s = c 1 * sin[sqrt(g/L)* A+B]+C2

(s represents the displacement from the central position C 1, C2, b, determined by the initial rod)

So this cycle

t = 2 * pi/(sqrt(g/L))= 2 * pi * sqrt(L/g)