Animated Solution for Physics - System of Particles: The position vector of the centre of mass ${\mathbf{r}}_{cm}$ of an asymmetric uniform bar of negligible area of cross-section as shown in figure is

Imagine you are looking at a strangely shaped, asymmetric bar. At first glance, finding its center of mass might seem like a daunting task requiring complex integration. But let's take a breath and look closer.

This continuous bar can be elegantly sliced into three distinct, uniform straight segments. By doing this, we transform a complex continuous body problem into a much simpler discrete particle problem.

For any uniform symmetric body, like a straight rod, we know that its center of mass lies exactly at its geometric midpoint. This is a powerful realization! It means we can replace each of our three segments with a single "point mass" located at its respective center.

The master equation for the position vector of the center of mass of a discrete system is:

Let's map out our three new "point masses":

Segment 1: This is the top horizontal part. It spans from $x=0$ to $x=2L$ at a height of $y=L$. Its mass is given as $2m$. Its geometric center $C_1$ is exactly in the middle, so $C_1=(L,L)$.

Segment 2: This is the vertical drop. It spans from $y=L$ down to $y=0$ at $x=2L$. Its length is $L$, so its mass is $m$. Its geometric center $C_2$ is halfway down, giving us $C_2=(2L,\frac{L}{2})$.

Segment 3: This is the bottom horizontal part. It spans from $x=2L$ to $x=3L$ along the x-axis ($y=0$). Its length is $L$, so its mass is $m$. Its geometric center $C_3$ is in the middle, at $C_3=(\frac{5L}{2},0)$.

Now, we simply plug these coordinates into our center of mass formula. Let's start with the X-coordinate:

Substituting our values:

Exactly in the same way, let's calculate the Y-coordinate:

Finally, we combine our calculated X and Y coordinates to form the complete position vector of the center of mass for the entire asymmetric bar:

By breaking the complex shape into simpler parts, we easily navigated to the correct answer!