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

A hanging mass $M$ is connected to a four times bigger mass by using a string pulley arrangement, as shown in the figure. The bigger mass is placed on a horizontal ice-slab and being pulled by $2\mathrm{Mg}$ force. In this situation, tension in the string is $\frac{x}{5}\mathrm{Mg}$ for $x=$_____. Neglect mass of the string and friction of the block (bigger mass) with ice slab.
(Given $g=$ acceleration due to gravity)

A uniform disc with mass $M=4 \mathrm{kg}$ and radius $R=10 \mathrm{cm}$ is mounted on a fixed horizontal axle as shown in figure. A block with mass $m=2 \mathrm{kg}$ hangs from a massless cord that is wrapped around the rim of the disc. During the fall of the block, the cord does not slip and there is no friction at the axle. The tension in the cord is _____ $\mathrm{N}$.

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

A block of metal weighing $2 \mathrm{kg}$ is resting on a frictionless plane (as shown in figure). It is struck by a jet releasing water at a rate of $1 \mathrm{kg} {\mathrm{s}}^{-1}$ and at a speed of $10{ \mathrm{m} \mathrm{s}}^{-1}$. Then, the initial acceleration of the block, in $\mathrm{m} {\mathrm{s}}^{-2}$, will be

A block of mass $40 \mathrm{kg}$ slides over a surface, when a mass of $4 \mathrm{kg}$ is suspended through an inextensible massless string passing over frictionless pulley as shown below.
The coefficient of kinetic friction between the surface and block is $0.02$. The acceleration of block is: (Given $g=10{ \mathrm{m} \mathrm{s}}^{-2}$.)

Four forces are acting at a point $P$ in equilibrium as shown in figure. The ratio of force $F_1$ to $F_2$ is $1:x$, where $x=$_____ .

For a free body diagram shown in the figure, the four forces are applied in the '$x$' and '$y$' directions. What additional force must be applied and at what angle with positive $x$-axis so that the net acceleration of body is zero?