Animated Solution for Physics - System of Particles: A circular plate of uniform thickness has a diameter of $56\text{ cm}$. A circular portion of diameter $42\text{ cm}$ is removed from one edge of the plate as shown in figure. Find the position of the centre of mass of the remaining portion.

Imagine a perfectly uniform circular plate. Its center of mass is exactly at its geometric center, a beautiful consequence of symmetry. But what happens when we take a bite out of it? By removing a circular portion from one edge, we disrupt this perfect balance. The right side becomes lighter, and intuitively, we know the center of mass must shift to the left to compensate. But how do we calculate exactly where this new balance point lies?

The "Negative Mass" Trick

Finding the center of mass of a crescent-like shape directly using integration is a nightmare. Instead, physicists use a brilliant mathematical sleight of hand known as the principle of superposition.

We can imagine our complex shape as the combination of two simpler shapes: a complete, solid circular plate with a positive mass, and a smaller circular plate with a "negative mass" placed exactly where the hole is.

By adding these two together, the negative mass perfectly cancels out the positive mass in that region, leaving us with our exact shape! This allows us to use the standard center of mass formula for two point masses:

To make our math as elegant as possible, we must choose our coordinate system wisely. Let's place the origin $(0,0)$ exactly at the center of the original, large circular plate.

The problem states the large plate has a diameter of $56\text{ cm}$. Therefore, its radius is:

The portion we are removing has a diameter of $42\text{ cm}$, giving it a radius of:

Now, where is the center of this removed plate? We know it is cut from the edge. Since the right edge of the large plate is at $x=28\text{ cm}$, and the removed plate has a radius of $21\text{ cm}$, we simply step back from the edge. The center of the removed plate, let's call it $C_2$, is located at:

Since the plate has a uniform thickness and density, the mass of any section is directly proportional to its area. We don't need the actual density; it will cancel out in our equation.

The "mass" of the large plate is proportional to its area:

The "mass" of the removed plate is proportional to its area:

Now, we substitute our values into the center of mass formula. The center of the large plate is at the origin, so $x_1=0$.

Notice how $\pi$ appears in every single term? We can factor it out and cancel it completely. The first term in the numerator becomes zero.

Here is where we use a smart algebra trick to avoid tedious multiplication. The denominator is a difference of squares, $a^2-b^2=(a-b)(a+b)$.

Substitute this beautifully simplified denominator back into our equation:

The $7$ in the numerator and denominator cancel out instantly. We are left with:

Dividing $441$ by $49$ gives us exactly $9$.

The Physical Interpretation: The negative sign is not just a mathematical artifact; it tells a physical story. Because we placed our origin at the center of the original plate, a negative $x$-coordinate means the center of mass has shifted $9\text{ cm}$ to the left. This perfectly matches our initial intuition: removing mass from the right side causes the balance point to shift to the left!