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A uniform thin wooden plank AB of length $\mathrm{L}$ and mass $\mathrm{M}$ is kept on a table with its $\mathrm{B}$ end slightly outside the edge of the table. When an impulse $\mathrm{J}$ is given to the end $\mathrm{B}$, the plank moves up with centre of mass rising a distance 'h' from the surface of the table. Then-
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The correct answer is:
$\mathrm{J}^{2} / 2 \mathrm{M}^{2} \mathrm{~g} < \mathrm{h} < 9 \mathrm{~J}^{2} / 8 \mathrm{M}^{2} \mathrm{~g}$
Case-I
Considering angular momentum w.r.t. end A $\omega=$ Angular velocity just after impulse then
$$
(\omega)\left(\frac{\mathrm{ML}^{2}}{3}\right)=\mathrm{J}(\mathrm{L})
$$
$$
\omega=\frac{3 \mathrm{~J}}{\mathrm{ML}}
$$
$\Rightarrow$ velocity of $\mathrm{CM}$
$$
\mathrm{V}_{\mathrm{cm}_{1}}=\frac{\omega \mathrm{L}}{2}=\frac{3 \mathrm{~J}}{2 \mathrm{M}}......(1)
$$
Case-II
Apply conservation o moment, if $\mathrm{V}_{\mathrm{cm}_{2}}=$ Velocity of $\mathrm{CM}$ just after impulse then $\mathrm{M} \mathrm{V}_{\mathrm{cm}_{2}}=\mathrm{J}$
$$
\mathrm{V}_{\mathrm{cm}_{2}}=\frac{\mathrm{J}}{\mathrm{M}}......(2)
$$
comparing (1) \& (2)
$$
\mathrm{V}_{\mathrm{cm}_{2}} < \mathrm{V}_{\mathrm{cm}_{1}}
$$
$\Rightarrow$ velocity of CM just after impulse would be between above two extreme values
with (1), $\mathrm{Mgh}_{\max } < \frac{1}{2}(\mathrm{M})\left(\frac{3 \mathrm{~J}}{2 \mathrm{M}}\right)^{2}$
$$
\Rightarrow \quad \mathrm{h}_{\max } < \frac{9 \mathrm{~J}^{2}}{8 \mathrm{M}^{2} \mathrm{~g}}......(3)
$$
with (2), $\mathrm{Mgh}_{\max }>\frac{1}{2} \mathrm{M}\left(\frac{\mathrm{J}}{\mathrm{M}}\right)^{2}$
$$
\mathrm{h}_{\max }>\frac{\mathrm{J}^{2}}{2 \mathrm{M}^{2} \mathrm{~g}}......(4)
$$
Use (3) \& (4)
Hence $\frac{\mathrm{J}^{2}}{2 \mathrm{M}^{2} \mathrm{~g}} < \mathrm{h}_{\max } < \frac{9 \mathrm{~J}^{2}}{8 \mathrm{M}^{2} \mathrm{~g}}$
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