A wooden block of mass 8 kg slides down an inclined plane of inclination ${30}^o$ to the horizontal with constant acceleration $0.4\mathrm{ }m{s}^{-2}$ . The force of friction between the block and inclined plane is $\left(\mathrm{g}=10\mathrm{ }\mathrm{m}{\mathrm{s}}^{-2}\right)$

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:

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 $\mathrm{N}$)

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?

An inclined plane making an angle of $30°$ with the horizontal is placed in a uniform horizontal electric field $200\frac{\mathrm{N}}{\mathrm{C}}$ as shown in the figure. A body of mass $1 \mathrm{kg}$ and charge $5 \mathrm{mC}$ is allowed to slide down from rest at a height of $1 \mathrm{m}$. If the coefficient of friction is $0.2,$ find the time taken by the body to reach the bottom.
$\left[g=9.8 \mathrm{m} {\mathrm{s}}^{-2}; \sin 30°=\frac{1}{2}; \cos 30°=\frac{\sqrt{3}}{2}\right]$

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 student skates up a ramp that makes an angle $30°$ with the horizontal. He/she starts (as shown in the figure) at the bottom of the ramp with speed $v_0$ and wants to turn around over a semicircular path $xyz$ of radius $R$ during which he/she reaches a maximum height $h$ (at point $y$) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then ($g$ is the acceleration due to gravity)

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