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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)

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