$A$ $1 \,kg$ mass is suspended from the ceiling by a rope. $A$ horizontal force $F$ is applied at the midpoint of the rope so that the upper part of the rope makes an angle of $45^{\circ}$ with respect to the vertical axis as shown in the figure. The magnitude of $F$ is:

If more than one force acts on a particle,state the condition for the equilibrium of the particle.

$A$ mass of $5\, kg$ is suspended by a rope of length $2\, m$ from a ceiling. $A$ force of $50\, N$ in the horizontal direction is applied at the mid-point of the rope. The angle made by the rope with the vertical,in equilibrium is ........ $^o$

When three forces of $50 \, N$,$30 \, N$,and $15 \, N$ act on a body,then the body is

$A$ mass of $10 \, kg$ is suspended vertically by a rope of length $5 \, m$ from the roof. $A$ force of $30 \, N$ is applied at the middle point of the rope in the horizontal direction. The angle made by the upper half of the rope with the vertical is $\theta = \tan^{-1}(x \times 10^{-1})$. The value of $x$ is ................
(Given $g = 10 \, m/s^2$)

$A$ block of mass $8 \ kg$ is suspended by a rope of length $3 \ m$ from the ceiling. $A$ force of $40 \ N$ is applied horizontally to the block. Then the angle that the rope makes with the vertical in equilibrium is (acceleration due to gravity $= 10 \ m \ s^{-2}$,neglect the mass of the rope).