Animated Solution for Physics - System of Particles: Four particles $A,B,C$ and $D$ with masses $m_A=m$, $m_B=2m$, $m_C=3m$ and $m_D=4m$ are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles (in ${\text{ms}}^{-2}$) is

Imagine four distinct particles situated at the corners of a square. We are given their masses as $m_A=m$, $m_B=2m$, $m_C=3m$, and $m_D=4m$.

Each particle is experiencing an acceleration of the exact same magnitude, $a$, but they are all moving in completely different directions. To make sense of this chaotic motion, we must anchor our perspective using a coordinate system. Let's assign vector notation to each particle's acceleration based on the standard X-Y axes:

Particle A is accelerating to the left: ${\mathbf{a}}_A=-a\hat{i}$ Particle B is accelerating upwards: ${\mathbf{a}}_B=a\hat{j}$ Particle C is accelerating to the right: ${\mathbf{a}}_C=a\hat{i}$ Particle D is accelerating downwards: ${\mathbf{a}}_D=-a\hat{j}$

When dealing with a system of discrete particles, the acceleration of the centre of mass (${\mathbf{a}}_{CM}$) is simply the mass-weighted average of the individual accelerations. The master formula is:

Now, we carefully substitute our known masses and their corresponding acceleration vectors into the formula.

The denominator is simply the total mass of the system, which adds up to $10m$. For the numerator, we expand the terms:

To simplify, we group the $\hat{i}$ components together and the $\hat{j}$ components together.

For the $\hat{i}$ direction: $3ma\hat{i}-ma\hat{i}=2ma\hat{i}$ For the $\hat{j}$ direction: $2ma\hat{j}-4ma\hat{j}=-2ma\hat{j}$

Putting it all back together, we get:

By factoring out $2ma$ and canceling the mass $m$, we arrive at our final elegant expression:

This result tells us a beautiful physical truth: despite the individual particles pulling in all four cardinal directions, the heavy mass of particle D ($4m$) pulling down and particle C ($3m$) pulling right dominates the system, causing the centre of mass to accelerate diagonally into the fourth quadrant.