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

Answer 1

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

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