Question

AB is a uniform rod of mass $1kg$ and length $1m$. Rod is placed on a smooth horizontal table in xy plane along $x-$axis. Initially center of mass $C$ of the rod is at origin as shown. A constant force $\overrightarrow{F}=1\hat{j}$ Newton starts acting on the rod at end A. Choose correct options

• A. $C$ will move on $y-$axis
• B. after a finite time rod will be parallel to $y-$axis momentarily.
• C. rod undergoes oscillatory motion about point A after some time.
• D. angular velocity of rod will be proportional to $\sqrt{\sin \theta }$ where $\theta$ is angle made by rod with $x-$axis, $(0\le \theta <\pi /2)$

Answer 1

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

angular velocity of rod will be proportional to $\sqrt{\sin \theta }$ where $\theta$ is angle made by rod with $x-$axis, $(0\le \theta <\pi /2)$

$T_C=I_c\alpha$
$\left(\frac{1}{2}\cos \theta \right)1=\frac{1(1{)}^2}{12}\frac{d^2\theta }{d{t}^2}$
${\int }_0^{\omega }\omega d\omega =6{\int }_0^{\theta }\cos \theta d\theta$
$\frac{{\omega }^2}{2}=6\sin \theta$
$\omega \propto \sqrt{\sin \theta }$
$T_c=\frac{1}{2}\cos \theta$ is positive as $\theta$ increases as $t$ increases. So rod will be parallel to $y-$axis in finite time.
Center of mass moves on $y$-axis because the force on centre of mass is only on y direction.
the rod cannot perform oscillations about point A as C always stays on $y$-axis.