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Horizontal force at the midpoint of a rope holding a hanging mass: angle of the rope or the force

The same question, asked 2 times: 27 Jun 2022*, 9 Apr 2024. Every wording below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    27 Jun 2022

    111
    A mass of 10 kg10\text{ }kg is suspended vertically by a rope of length 5 m5\text{ }m from the roof. A force of 30 N30\text{ }N is applied at the middle point of rope in horizontal direction. The angle made by upper half of the rope with vertical is θ=tan⁡−1(x×10−1)\theta=\tan^{- 1}\left( x \times10^{- 1} \right). The value of xx is (Given g=10 m/s2g = 10\text{ }m/s^{2} )

  2. Question 2

    9 Apr 2024

    108
    A 1 kg1\text{ }kg mass is suspended from the ceiling by a rope. A horizontal force ' FF ' is applied at the mid point of the rope so that the rope makes an angle of 45∘45^{\circ} with respect to the vertical axis as shown in figure. The magnitude of FF is:

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