Question

A thin rod lies along the $x$-axis from $x=0$ to $x=2.7cm$. The linear density of the rod, $\lambda$, is given by the equation: $\lambda =1+e^x$ and has units of $g/cm$.
Where is the Center of Mass of the rod?

• A. At $x=0.55cm$
• B. At $x=1.35cm$
• C. At $x=1.50cm$
• D. At $x=1.70cm$
• E. At $x=1.81cm$

Answer 1

The correct option is E At $x=1.81cm$

Consider a small element $dx$ at a distance $x$ from the origin.
The center of mass is defined as $COM=\frac{{\int }_0^{2.7}x\left(\lambda dx\right)}{{\int }_0^{2.7}\lambda dx}$
Now, ${\int }_0^{2.7}x\left(\lambda dx\right)={\int }_0^{2.7}x(1+e^x)dx=|\frac{x^2}{2}{|}_0^{2.7}+|x{e}^x-e^x{|}_0^{2.7}=3.64+40.18-14.88+1=29.94$ and
${\int }_0^{2.7}\lambda dx={\int }_0^{2.7}(1+e^x)dx=|x+e^x{|}_0^{2.7}=2.7+14.88-1=16.58$
Thus, $COM=29.94/16.58=1.81cm$
Hence, the center of mass is at $x=1.81cm$