Question

A square plate of mass in & side length a can relate in vertical plane about a horizontal axis passing through $A$ and plate is released from rest at $1=0$ when it's diagonal $AC$ is horizontal. Force applied by axis on plate just after $t=0$ is

• A. $\frac{3mg}{4}$
• B. $\frac{mg}{2}$
• C. $\frac{mg}{4}$
• D. $\frac{3mg}{2}$

Answer 1

The correct option is C $\frac{mg}{4}$

$\begin{array}{c}\text{Let the Fora duu to the axis be}\left(F\right)\\ \Rightarrow \text{By constraint motion equation-}\\ mg-F=m{a}^{\prime }-1\\ \text{we know that moment of Inertia of a square plate}\\ \text{about its centre is}\\ I_c=\frac{m{a}^2}{6}a=\text{side length}\\ I^{\prime }=\frac{2}{3}m{a}^2\end{array}$

$\begin{array}{c}\Rightarrow \text{By Rotational motion equation -}\\ \begin{array}{cc}\mathrm{mg}\frac{a}{\sqrt{2}}& =\left(\frac{2}{3}m{a}^2\right)\cdot \alpha =\left(\frac{2}{3}m{a}^2\right)\frac{\sqrt{2}}{a}{a}^{\prime }\\ \Rightarrow & a^{\prime }=\frac{3g}{4}\end{array}-\left(2\right)\end{array}$

$\begin{array}{c}\Rightarrow \text{From eq}(1\text{and}2\text{we get}\\ F=\frac{mg}{4}\\ \text{Hence, option (C) is correct.}\end{array}$