Animated Solution for Physics - System of Particles: A uniform rectangular thin sheet $ABCD$ of mass $M$ has length $a$ and breadth $b$, as shown in the figure. If the shaded portion $HBGO$ is cut-off, the coordinates of the centre of mass of the remaining portion will be

Finding the Center of Mass of a Cut-Out Sheet

Imagine trying to balance a perfectly rectangular tray on your finger. Easy, right? The balance point is right in the middle. But what if someone takes a bite out of the corner? Suddenly, the tray tips. The balance point—the center of mass—has shifted. In this problem, we'll explore a brilliant trick called the Negative Mass Concept to find exactly where that new balance point hides.

Instead of breaking the weird L-shaped remaining sheet into smaller rectangles and doing tedious math, we use a clever shortcut. We imagine the original, complete rectangle and pretend we are adding a piece of "negative mass" exactly where the cut-out is.

Mathematically, the center of mass formula becomes: where $A_1$ is the area of the full sheet, and $A_2$ is the area of the cut-out.

Let's look at the full rectangle $ABCD$. Its dimensions are $a$ and $b$. Area, $A_1=ab$

Because it's uniform, its center of mass is right at the geometric center:

Now, let's focus on the piece that was removed, the rectangle $HBGO$. Its length is $a/2$ and its breadth is $b/2$. Area, $A_2=\frac{a}{2}\times \frac{b}{2}=\frac{ab}{4}$

Where is the center of this cut-out? It lies exactly in the middle of its own boundaries. Since it spans from $x=a/2$ to $x=a$, its x-center is:

Similarly, its y-center is:

Now, we plug these into our negative mass formula for the X-coordinate:

Let's simplify the numerator by factoring out $ab$:

Finding a common denominator for the top:

Because the rectangle and the cut-out are perfectly symmetrical in their proportions, the Y-coordinate calculation is identical!

So, the new center of mass is at $\left(\frac{5a}{12},\frac{5b}{12}\right)$.

Does this make physical sense? The original center was at $0.5a$. The new center is at $0.416a$. It shifted to the left and down! Since we removed mass from the top-right, the balance point naturally retreated to the bottom-left to compensate. Physics is beautiful!