Question

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

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

According to Impulse momentum theorem,

$△L=Jl$

$L-L_0=Jl$

$L_0=0\Rightarrow L=Jl$(Angular Momentum of rod after impact)

$L=I\omega =Jl\Rightarrow \omega =\frac{Jl}{I}=\frac{Jl}{m{l}^2}\times 3=\frac{3Jl}{m{l}^2}=\frac{3J}{ml}$

$KineticEnergy=\frac{1}{2}I{\omega }^2=\frac{3J^2}{2m}$

$V_P=R_P\omega \Rightarrow V_{midpoint}=\frac{l}{2}\times \frac{3J}{Ml}=\frac{3J}{2M}$