$A$ spherical ball of mass $m=5\,kg$ rests between two planes which make angles of $30^{\circ}$ and $45^{\circ}$ respectively with the horizontal. The system is in equilibrium. Find the normal forces exerted on the ball by each of the planes. The planes are smooth.

One end of a horizontal uniform beam of weight $W$ and length $L$ is hinged on a vertical wall at point $O$ and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point $Q$,at a height $L$ above the hinge at point $O$. $A$ block of weight $\alpha W$ is attached at the point $P$ of the beam,as shown in the figure. The rope can sustain a maximum tension of $(2 \sqrt{2}) W$. Which of the following statement$(s)$ is(are) correct?

$A$ mass $m$ is suspended by two strings $T_1$ and $T_2$ as shown in the figure. Calculate the tensions $T_1$ and $T_2$.

Can a stationary object remain stationary if multiple external forces act upon it?

$A$ 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 $g = 10 \, m/s^2$)

The resultant of two forces,one double the other in magnitude,is perpendicular to the smaller of the two forces. The angle between the two forces is ........ $^o$