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 rope in horizontal direction. The angle made by upper half of the rope with vertical is $\theta=\tan ^{-1}\left(x \times 10^{-1}\right)$. The value of $x$ is ................

$\text { (Given } g =10 \,m / s ^{2} \text { ) }$

A uniform beam of weight $W$ is attached to a vertical wall by a hinge $H$ . The beam is held horizontal by a rope as shown below. Which one of the following best shows the direction of the reaction force $R$ at the hinge ?

There are four forces acting at a point $P$ produced by strings as shown in figure, point $P$ is at rest. The forces $F_1$ and $F_2$ are respectively

Two particles of mass $m$ each are tied at the ends of a light string of length $2a$ . The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance $'a'$ from the centre $P$ (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force $F$ . As a result, the particles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes $2x$ , is

A weight $M g$ is suspended from the middle of a rope whose ends are at the same level. The rope is no longer horizontal. The minimum tension required to completely straighten the rope is ......

Write important guidance for solving problems in mechanics.