Question

Column I gives the bodies which are released one by one from the top of an inclined plane. Assume each body rolls purely.

Answer 1

for $A$ and $B$

$a=\frac{mg\sin \theta -\mu mg\cos \vartheta }{m}\alpha =\frac{2\mu mg\cos \theta }{m{R}^2}RR\alpha =a\Rightarrow \sin \theta =3\mu \cos \theta \Rightarrow a=\frac{2}{3}g\sin \theta$

For $C$ and $D$

$a=\frac{mg\sin \theta -\mu mg\cos \vartheta }{m}\alpha =\frac{4\mu mg\cos \theta }{m{R}^2}RR\alpha =a\Rightarrow \sin \theta =2\mu \cos \theta \Rightarrow a=\frac{1}{2}g\sin \theta a_{sc}>a_{cs}\Rightarrow t_{solid}<t_{shell}KE=\frac{1}{2}m{v}^2\sqrt{1+\frac{K^2}{R^2}}=mgh$

Higher $K$ lower is $V$

Therefore has higher rotational energy and solid cylinder has transitional

$\therefore \left(A\right)and\left(B\right)\Rightarrow 2,3\therefore \left(B\right)and\left(D\right)\Rightarrow 1,4$