Question

A trolley is falling freely on an inclined plane (∅). The angle of string of pendulum with the ceiling of trolley (α) is?

Answer 1

Let the string of pendulum OP and this will make an angle 0 with the vertical position we will balance the force in the vertical direction.

T cos 0 + mgsin ∅ . sin ∅ = mg
T cos 0 = mg - mgsin^2 ∅

T cos 0 = mg cos^0 ∅ - (1)

now we will balance the force in the horizontal direction

T sin 0 = mgsin ∅ cos ∅ - (2)

By equation (2) divided by equation (1)

tan 0 = tan ∅

= 0 = ∅

Answer 2

Given:
Let us assume that string of pendulum, makes an angle θ  with the vertical
Net force in vertical direction:
T cosθ +mgsinΦ.sinΦ=mg
Tcosθ=mg-mgsin²Φ
Tcosθ=mg(1-sin²θ)
T cosθ=mgcos²Φ -------(1) [ ∵ cos²Φ+sin²Φ=1)
Net force in horizontal direction:
Tsinθt=mgsinΦcosΦ -----(2)
divide equation (2) by  eqution (1)
T sinθ/T cosθ=mgsinΦcosΦ/mgcos²Φ
tanθ=tanΦ
⇒θ=Φ