Question

A rigid rod of length $l$ leans against a vertical wall ($y$-axis) as shown in figure. The other end of the rod is on the horizontal floor. Point $\text{A}$ is pushed downwards with constant velocity. Find the path of the COM of the rod is

• A. A parabola
• B. An ellipse
• C. A circle of radius $\frac{l}{2}$ and its center is at origin
• D. A circle of radius $\frac{l}{2}$ but its center isn't at origin

Answer 1

The correct option is C A circle of radius $\frac{l}{2}$ and its center is at origin

Let, coordinates of the COM be $(x,y)$ as shown in figure.


From the figure,

$y=\frac{l}{2}\sin \theta$

$\sin \theta =\frac{2y}{l}.......\left(1\right)$

And $x=\frac{l}{2}\cos \theta$

$\therefore \cos \theta =\frac{2x}{l}.......\left(2\right)$

Squaring and adding $\left(1\right)$ and $\left(2\right)$, we get,

$1=\frac{4y^2}{l^2}+\frac{4x^2}{l^2}$

$x^2+y^2={\left(\frac{l}{2}\right)}^2$

This is the equation of a circle, hence COM undergoes a circle of radius $\frac{l}{2}$ with its center at the origin.

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(C\right)$ is the correct answer.