Consider the situation shown in the figure. $A$ uniform rod of length $L$ can rotate freely about the hinge $A$ in a vertical plane. Pulleys and string are light and frictionless. If the rod remains horizontal at rest when the system is released,then the mass of the rod is:

$A$ uniform metal plate shaped like a triangle $ABC$ has a mass of $540 \,g$. The lengths of the sides $AB, BC$,and $CA$ are $3 \,cm, 5 \,cm$,and $4 \,cm$,respectively. The plate is pivoted freely about the point $A$. What mass must be added to a vertex so that the plate can hang with the long edge $(BC)$ horizontal?

Can a body remain in partial equilibrium? Explain with an illustration.

$A$ metre scale is supported on a wedge at its centre of gravity. $A$ body of weight '$W$' is suspended from the $20 \text{ cm}$ mark and another weight of $25 \text{ g-wt}$ is suspended from the $74 \text{ cm}$ mark to balance it, and the metre scale remains perfectly horizontal. Neglecting the weight of the metre scale, the weight of the body is: (in $\text{ g-wt}$)

$A$ uniform cylinder of mass $M$ and radius $R$ is to be pulled over a step of height $a$ $(a < R)$ by applying a force $F$ at its centre $'O'$ perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of $F$ required is

Write the condition for rotational equilibrium and translational equilibrium.