2 Poor man solution

Considering the Earth as an inertial frame of reference, we can apply Newton’s second law to the pendulum: ⃗P + ⃗T = m⃗Γ, with P⃗ the weight of the pendulum and m its mass, ⃗
T the tension in rod, and ⃗
 Γ the centripetal acceleration (see Fig. 1).


PIC

Figure 1: Schematic of the carousel and the pendulum, with forces applied to the pendulum.


We thus deduce :

T cosθ  =   mg,                                 (1)
T sin θ  =   m ω2(R + l sin θ),                   (2)
and consequently
        R-ω2-   lω2--
tan θ =   g  +  g  sinθ.
(3)

If we consider l sin θ as negligible compared to R, we get the 0th-order solution:

         R-ω2-
tan θ0 =   g .
(4)