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(B) When the block hits the bump,it starts rotating about the edge of the bump. We apply the conservation of angular momentum about the point of impac

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(B) When the block hits the bump,it starts rotating about the edge of the bump. We apply the conservation of angular momentum about the point of impact (the bump).The initial angular momentum of the block about the bump is $L_i = m v r_{\perp}$,where $r_{\perp}$ is the perpendicular distance from the bump to the center of mass of the block. Since the block is moving horizontally,$r_{\perp} = a/2 = 0.15\,m$.So,$L_i = m 	imes 2 	imes 0.15 = 0.3m$.The moment of inertia of the cube about an axis passing through one of its edges is given by the parallel axis theorem: $I = I_{cm} + m d^2$,where $I_{cm} = \frac{2}{3}ma^2$ is the moment of inertia about the center of mass axis parallel to the edge,and $d = \sqrt{(a/2)^2 + (a/2)^2} = \frac{a}{\sqrt{2}}$ is the distance from the center of mass to the edge.$I = \frac{2}{3}ma^2 + m(\frac{a}{\sqrt{2}})^2 = \frac{2}{3}ma^2 + \frac{1}{2}ma^2 = \frac{7}{6}ma^2$.Using conservation of angular momentum: $L_i = I \omega$$0.3m = (\frac{7}{6}m(0.3)^2) \omega$$0.3 = \frac{7}{6} 	imes 0.09 	imes \omega$$0.3 = \frac{7 	imes 0.015}{1} \omega = 0.105 \omega$$\omega = \frac{0.3}{0.105} \approx 2.85\,rad/s$.Re-evaluating based on the standard approximation for this specific problem type where the pivot is at the corner: $I = \frac{8}{3}ma^2$ (for rotation about a corner axis). Using $L = m v (a/2) = I \omega = (\frac{8}{3}ma^2) \omega$$v(a/2) = \frac{8}{3}a^2 \omega \Rightarrow \omega = \frac{3v}{16a} = \frac{3 	imes 2}{16 	imes 0.3} = \frac{6}{4.8} = 1.25\,rad/s$.Given the provided options and the logic in the prompt,the calculation $I = \frac{2}{3}ma^2$ leads to $\omega = 5.0\,rad/s$ if we assume $I = \frac{2}{3}ma^2$ and $L = m v (a/2)$.$m v (a/2) = (\frac{2}{3}ma^2) \omega \Rightarrow \omega = \frac{3v}{4a} = \frac{3 	imes 2}{4 	imes 0.3} = \frac{6}{1.2} = 5.0\,rad/s$.

$(a)$ $\frac{mg}{4} < F < \frac{mg}{2}$	$(i)$ Cube will move up
$(b)$ $F > \frac{mg}{2}$	$(ii)$ Cube will not exhibit motion
$(c)$ $F > mg$	$(iii)$ Cube will begin to rotate about $A$
$(d)$ $F = \frac{mg}{4}$	$(iv)$ Normal reaction effectively at $a/3$ from $A$,no motion

$A$ cubical block of side $a = 30\,cm$ is moving with velocity $v = 2\,m/s$ on a smooth horizontal surface. The surface has a small bump at a point $O$ as shown in the figure. The angular velocity (in $rad/s$) of the block immediately after it hits the bump is:

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