Question

A thin uniform rod of mass $m$ and length $l$ is free to rotate about its upper end. When it is at rest, it receives an impulse $J$ at its lowest point, normal to its length. Immediately after impact,

• A. the angular momentum of the rod is $J l$
• B. the angular velocity of the $\operatorname{rod}$ is $3 J / m /$
• C. the kinetic energy of the rod is $3 J ^{2} / 2 m$
• D. the linear velocity of the midpoint of the rod is $3 J / 2 m$

Answer 1

Answer: (A), (B), (C), (D)

Angular impact $J \ell=\Delta L =\frac{1}{3} m \ell^{2} \omega$
$\omega=\frac{3 J }{ m \ell}$
$K . E .=\frac{1}{2} \cdot \frac{1}{3} m \ell^{2} \cdot \frac{9 J ^{2}}{ m ^{2} \ell^{2}}=\frac{3 J ^{2}}{2 m }$
$V _{ L }=\omega \frac{\ell}{2}=\frac{3 J }{ m \ell} \times \frac{\ell}{2}=\frac{3 J }{ m }$