Question

A thin rod lies along the $x$-axis from $x=0$ to $x=3cm$. The linear density of the rod, $\lambda$, is given by the equation: $\lambda =2+x^2$ and is in units of $g/cm$.
Where is the Center of Mass of the rod?

• A. At $x=0.51cm$
• B. At $x=1.50cm$
• C. At $x=1.95cm$
• D. At $x=2.08cm$
• E. At $x=2.93cm$

Answer 1

Answer: (C)

At $x=1.95cm$

Consider a small element $dx$ at a distance $x$ from the origin.
The center of mass is defined as $COM=\frac{{\int }_0^{3}x\left(\lambda dx\right)}{{\int }_0^3\lambda dx}$
Now, ${\int }_0^{3}x\left(\lambda dx\right)={\int }_0^{3}x(2+x^2)dx=|x^2+\frac{x^4}{4}{|}_0^3=9+20.25=29.25$ and
${\int }_0^3\lambda dx={\int }_0^3(2+x^2)dx=|2x+\frac{x^3}{3}{|}_0^3=6+9=15$
Thus, $COM=29.25/15=1.95cm$
Hence, the center of mass is at $x=1.95cm$