Inside a smooth fixed hollow cylinder of radius $R$, three identical smooth cylinders of radius $\mathit{r}$ are placed with their axes parallel to axis of the fixed cylinder as shown. Then the equilibrium cannot exist unless $R<nr$. The value of $n$ is

The sum of the three vectors in the figure below is zero. The magnitude of $\overrightarrow{\mathrm{OC}}$ and $\overrightarrow{\mathrm{OB}}$ is,

The pulleys and strings shown in figure are smooth and of negligible mass. For the system to remain in equilibrium, the angle $\theta$ should be:

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?

A particle moving with velocity $\overrightarrow{V}$ is acted by three forces shown by the vector triangle $PQR$. The velocity of the particle will

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 $x=\_\_\_\_\_$.

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 $90 \mathrm{N}$ mass is hung on a rope tied between two poles as shown in the figure. The tension $T_1$ and $T_2$ in the two parts of the rope are (in $\mathrm{N}$)

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 $x=$_____ .

A block of $\sqrt{3} \mathrm{kg}$ is attached to a string whose other end is attached to the wall. An unknown force $F$ is applied so that the string makes an angle of $30°$ with the wall. The tension $T$ is : (Given g $=10 \mathrm{m} {\mathrm{s}}^{–2}$)