Animated Solution for Physics - System of Particles: Three point particles of masses 1.0 kg, 1.5 kg and 2.5 kg are placed at three corners of a right angle triangle of sides 4.0 cm, 3.0 cm and 5.0 cm as shown in the figure. The centre of mass of the system is at a point

Finding the center of mass of a discrete system of particles is like finding the perfect balancing point of a tray holding different weights. Let's embark on a journey to locate this exact point for our triangular setup.

Imagine you are looking at this right-angled triangle. We have three masses: $1.0\text{ kg}$, $1.5\text{ kg}$, and $2.5\text{ kg}$ sitting at the vertices. To make our calculations incredibly elegant and simple, we need to establish a coordinate system.

The smartest move is to place the $1.0\text{ kg}$ mass exactly at the origin, $(0,0)$. Because it's a right-angled triangle, this naturally aligns the other two masses along the axes. The $1.5\text{ kg}$ mass sits on the x-axis at a distance of $3.0\text{ cm}$, giving it coordinates $(3,0)$. The $2.5\text{ kg}$ mass sits on the y-axis at a height of $4.0\text{ cm}$, giving it coordinates $(0,4)$.

Now, we need to find the center of mass. Let's start with the x-coordinate. The formula is simply the weighted average of the x-positions:

Let's carefully substitute our values. Notice how the $1.0\text{ kg}$ and $2.5\text{ kg}$ masses have an x-coordinate of zero? They completely vanish from the numerator! Only the $1.5\text{ kg}$ mass at $x=3$ contributes.

Doing the math, the numerator becomes $1.5\times 3=4.5$. The total mass is $1.0+1.5+2.5=5.0\text{ kg}$. Dividing $4.5$ by $5.0$ gives us exactly $0.9\text{ cm}$.

Halfway there! Now, let's tackle the y-coordinate. We use the exact same weighted average concept, but this time we focus purely on the vertical positions.

Substituting the values, we see a similar trick. The masses on the x-axis have a y-coordinate of zero. So, only the $2.5\text{ kg}$ mass at a height of $4\text{ cm}$ matters for the numerator.

Let's compute this. $2.5\times 4$ is a perfect $10$. Divide that $10$ by our total mass of $5.0$, and we get exactly $2.0\text{ cm}$.

So, what's our final destination? The center of mass is located at the coordinates $(0.9,2.0)$.

In physical terms, relative to our chosen origin, that is $0.9\text{ cm}$ to the right and $2.0\text{ cm}$ above the $1.0\text{ kg}$ mass. This perfectly matches option (d).

Food for thought: What if we had placed the origin at the $2.5\text{ kg}$ mass instead? The absolute coordinates would change, but the physical location of the center of mass relative to the triangle would remain exactly the same. Try it out!