An elevator of mass $m$ is connected to a rope which pulls the elevator up with a maximum acceleration equal to $a$. If the maximum bearable tension in the rope is $T$,then for a safe journey the minimum diameter of the rope is $[g=$ acceleration due to gravity$]$

$A$ monkey of mass $m$ is climbing a rope hanging from the roof with acceleration $a$. The coefficient of static friction between the body of the monkey and the rope is $\mu$. Find the direction and value of the friction force on the monkey.

$A$ bar of mass $m$ resting on a smooth horizontal plane starts moving due to a constant force $F$. In the process of its rectilinear motion,the angle $\theta$ between the direction of this force and the horizontal varies as $\theta = k x$,where $k$ is a constant and $x$ is the distance traversed by the bar from its initial position. The velocity $(v)$ of the bar as a function of the angle $\theta$ is

Two masses of $5 \,kg$ and $3 \,kg$ are suspended with the help of massless inextensible strings as shown in the figure below. When the whole system is moving upwards with an acceleration of $2 \,m/s^2$, the value of $T_1$ is (use $g = 9.8 \,m/s^2$): (in $\,N$)

$A$ lift of mass $m$ is ascending with an acceleration $a$ $(a < g)$. The tension in the cable of the lift is $(g = \text{acceleration due to gravity})$

$A$ system of $10$ balls,each of mass $m = 2 \; kg$,is connected via a massless and unstretchable string. The system is allowed to slip over the edge of a smooth table as shown in the figure. Find the tension $T$ in the string between the $7^{th}$ and $8^{th}$ ball when the $6^{th}$ ball just leaves the table. (in $; N$)