A mass of $10 \mathrm{kg}$ is suspended by a rope of length $4 \mathrm{m}$, from the ceiling. A force $F$ is applied horizontally at the mid-point of the rope such that the top half of the rope makes an angle of $45°$ with the vertical. Then, $F$ equals (Take, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ and assume the rope to be massless)

A block of mass $5 \mathrm{kg}$ is $\left(\mathrm{i}\right)$ pushed in case $\left(A\right)$ and $\left(\mathrm{ii}\right)$ pulled in case $\left(B\right)$, by a force $F=20 \mathrm{N}$, making an angle of $30°$ with the horizontal, as shown in the figures. The coefficient of friction between the block, the floor is $\mu =0.2$. The difference between the accelerations of the block, in case $\left(B\right)$ and case $\left(A\right)$ will be (Take, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ )

When a maximum force of $3 \mathrm{N}$ is applied on a body kept on rough inclined plane of shown in the figure, then body remains stationary. The maximum external force up the inclined plane that does not move the block is $12 \mathrm{N}$. The coefficient of static friction between the block and the plane is [Take, $g=10 {\mathrm{ms}}^{-2}$ ]

Three blocks are connected by massless strings on a frictionless inclined plane of $30°$ as shown in the figure. A force of $104 \mathrm{N}$ is applied upward along the incline to mass $m_3$ causing an upward motion of the blocks. What is the acceleration of the blocks? (Assume, acceleration due to gravity, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)