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 rod is $3J/ml$
• 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)

The correct options are
A the angular momentum of the rod is $Jl$
B the angular velocity of the rod is $3J/ml$
C the linear velocity of the midpoint of the rod is $3J/2m$
D the kinetic energy of the rod is $3J^2/2m$

$Angularimpulse=JlAngularimpulse=changeinangularmomentum=I\omega -0\Rightarrow \frac{m{l}^2}{3}\omega =Jl\Rightarrow \omega =\frac{3J}{ml}KE=\frac{1}{2}I{\omega }^2=\frac{1}{2}\times \frac{m{l}^2}{3}\times {\left(\frac{3J}{ml}\right)}^2=\frac{3J^2}{2m}velocityofthemidpoint=\omega \frac{l}{2}=\frac{3J}{ml}\times \frac{l}{2}=\frac{3J}{2m}$