$A$ block of $\sqrt{3} \, kg$ 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 $30^{\circ}$ with the wall. The tension $T$ is $........... \, N$ : (Given $g = 10 \, ms^{-2}$)

$A$ smooth cylinder of mass $m$ and radius $R$ is resting on two corner edges $A$ and $B$ as shown in the figure. The relation between the normal reactions at the edges $A$ and $B$ is

$A$ ladder rests against a frictionless vertical wall,with its upper end $6 \, m$ above the ground and the lower end $4 \, m$ away from the wall. The weight of the ladder is $500 \, N$ and its center of gravity ($C$.$G$.) is at $1/3$ of its length from the lower end. The reaction force from the wall will be (in Newton):

$A$ $1 \,kg$ mass is suspended from the ceiling by a rope. $A$ horizontal force $F$ is applied at the midpoint of the rope so that the upper part of the rope makes an angle of $45^{\circ}$ with respect to the vertical axis as shown in the figure. The magnitude of $F$ is:

The following forces start acting on a particle at rest at the origin of the coordinate system simultaneously: ${\vec F_1} = -4\hat i - 5\hat j + 5\hat k$,${\vec F_2} = 5\hat i + 8\hat j + 6\hat k$,${\vec F_3} = -3\hat i + 4\hat j - 7\hat k$,and ${\vec F_4} = 2\hat i - 3\hat j - 2\hat k$. The particle will move:

$A$ small sphere is hung by a string fixed to a wall as shown in the figure. The sphere is pushed from the wall by a stick. The forces acting on the sphere are as shown. Then,identify the incorrect expression.