Question

In the given figure, an $L$-shaped frame is free to rotate in a vertical plane about a horizontal axis passing through a smooth hinge $O$. Each side of the frame has a length $L$ and mass $M$. The frame is allowed to fall, with one side horizontal and the other vertical.


Angular acceleration of the frame just after it is allowed to fall is:

• A. $\frac{4g}{3L}$
  
• B. $\frac{3g}{2L}$
  
  
• C. $\frac{9g}{10L}$
  
  
• D. $\frac{g}{2L}$
  
  

Answer 1

The correct option is C $\frac{9g}{10L}$



Let the angular acceleration just after the release is $\alpha$.
From Netwon's law: ${\tau }_{net}=I\alpha$
$\Rightarrow \alpha =\frac{{\tau }_O}{I_O}$
Net torque about O:
${\tau }_O=\frac{MgL}{2}+MgL=\frac{3MgL}{2}$

$I_O=I_{OA}+I_{AB}$
$I_{OA}=\frac{M{L}^2}{3}$


Using parallel axis theorem:

$I_{AB}\left(\text{about O}\right)=[\frac{M{L}^2}{12}+M(\frac{\sqrt{5}L}{2}{)}^2]=\frac{4M{L}^2}{3}$
$\therefore I_O=\frac{5M{L}^2}{3}$

$\Rightarrow \alpha =\frac{\frac{3MgL}{2}}{\frac{5M{L}^2}{3}}$

$\therefore \text{angular acceleration}\alpha =\frac{9g}{10L}$