A block of mass $200 \mathrm{g}$ is kept stationary on a smooth inclined plane by applying a minimum horizontal force $F=\sqrt{x} \mathrm{N}$ as shown in figure.

The value of $x=\_\_\_\_\_$.

In the arrangement shown in figure $a_1,a_2,a_3$ and$a{ }_{4 }$  are the accelerations of masses $m_1,m_2,m_3$ and $m_4$ respectively. Which of the following relation is true for this arrangement?

A system of $10$ balls each of mass $2 \mathrm{kg}$ are connected via massless and unstretchable string. The system is allowed to slip over the edge of a smooth table as shown in figure. Tension on the string between the $7^{\mathrm{th} }$ and $8^{\mathrm{th} }$ ball is _____  $\mathrm{N}$ when $6^{\mathrm{th} }$ ball just leaves the table.

A system of two blocks of masses $m=2 \mathrm{kg}$ and $M=8 \mathrm{kg}$ is placed on a smooth table as shown in figure. The coefficient of static friction between two blocks is $0.5$. The maximum horizontal force $F$ that can be applied to the block of mass $M$ so that the blocks move together will be $\left(g=9.8 \mathrm{m} {\mathrm{s}}^{-2}\right)$

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

(Given, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)