A rod $(AB)$ is attached to a fixed point $(C)$ using a light rope $(AC)$. The other end of the rod $(B)$ is sitting on ice with negligible friction and the system is in stationary position. Which of the following can be the equilibrium configuration of this system?

A uniform metal plate shaped like a triangle $A B C$ has a mass of $540 \,g$. The length of the sides $A B, B C$ and $C A$ 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 horizontal?

A ring is formed by joining two uniform semi circular rings $ABC$ and $ADC$. Mass of $ABC$ is thrice of that of $ADC$. If the ring is hinged to a fixed support ,at $A$, it can rotate freely in a vertical plane. Find the value of $tan\,\theta$, where $\theta$ is the angle made. by the line $AC$ with the vertical in equilibrium

One end of a horizontal uniform beam of weight $W$ and length $L$ is hinged on a vertical wall at point $O$ and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point $Q$, at a height $L$ above the hinge at point $O$. A block of weight $\alpha W$ is attached at the point $P$ of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of $(2 \sqrt{2}) W$. Which of the following statement($s$) is(are) correct ?

$(A)$ The vertical component of reaction force at $O$ does not depend on $\alpha$

$(B)$ The horizontal component of reaction force at $O$ is equal to $W$ for $\alpha=0.5$

$(C)$ The tension in the rope is $2 W$ for $\alpha=0.5$

$(D)$ The rope breaks if $\alpha>1.5$

$A$ uniform ladder of length $5m$ is placed against the wall as shown in the figure. If coefficient of friction $\mu$ is the same for both the walls, what is the minimum value of $\mu$ for it not to slip?