Question

A uniform rectangular plate of mass $m$ which is free to rotate about the smooth vertical hinge passing through the center and perpendicular to the plate, is lying on a smooth horizontal surface. A particle of mass $m$ moving with speed ${}^{\prime }{u}^{\prime }$ collides with the plate and sticks to it as shown in figure. The angular velocity of the plate after collision will be:

• A. $\frac{12}{5}\frac{u}{a}$
• B. $\frac{12}{19}\frac{u}{a}$
• C. $\frac{3}{2}\frac{u}{a}$
• D. $\frac{3}{5}\frac{u}{a}$

Answer 1

The correct option is D $\frac{3}{5}\frac{u}{a}$

$L_i=L_f$

$\therefore mu\left(a\right)=lw$

or $w=\frac{mua}{l}=\frac{mua}{m{\left(\frac{5u}{2}\right)}^2+\frac{{ma}^2}{12}+\frac{{m\left(2a\right)}^2}{12}}$

$=\frac{3u}{5a}$