A mass $m$ hangs with the help of a string wrapped around a pulley on a firctionless bearing. The pulley has mass $m$ and radius $R$. Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass $m$, if the string does not slip on the pulley, is:-

- A
$\frac{2}{3} g$

- B
$\frac{g}{3}$

- C
$\frac{3}{2} g$

- D
$g$

$3\; m$ long ladder wetghing $20 kg$ leans on a frictionless wall. Its feet rest on the floor $1\; m$ from the wall as shown in Figure Find the reaction forces of the wall and the floor.

$A$ rod of weight $w$ is supported by two parallel knife edges $A$ and $B$ and is in equilibrium in a horizontal position. The knives are at a distance $d$ from each other. The centre of mass of the rod is at a distance $x$ from $A$.

$A$ right triangular plate $ABC$ of mass $m$ is free to rotate in the vertical plane about a fixed horizontal axis through $A$. It is supported by a string such that the side $AB$ is horizontal. The reaction at the support $A$ is:

A $\sqrt{34}\,m$ long ladder weighing $10\,kg$ leans on a frictionless wall. Its feet rest on the floor $3\,m$ away from the wall as shown in the figure. If $F_{f}$ and $F_{w}$ are the reaction forces of the floor and the wall, then ratio of $F _{ a } / F _{f}$ will be:

(Use $\left.g=10\,m / s ^{2}\right)$

- [JEE MAIN 2022]

A uniform rod $AB$ of length $l$ and mass $m$ is free to rotate about point $A.$ The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about $A$ is $ml^2/3$, the initial angular acceleration of the rod will be

- [AIPMT 2007]