How can I understand this mathematical formula?

Question

I was trying to understand this formula for the center of mass, given by: $$ \frac{m_1x_1 + m_2x_2}{m_1 + m_2}$$ Can you please explain this to me step by step of what it really means and how this equation can give me the center of mass.

Answer 1

The center of mass for an object, or group of objects, is the point about which the entire mass of the system is equally distributed.

The formula given here is for the center of mass $$\text{center of mass}= \frac{\text{sum of all (position $\times$ mass)}}{\text{sum of all masses}}$$

The concept of the center of mass is that of an average of the masses factored by their distances from a reference point.

If $m_1$ and $m_2$ be two masses and position of $m_1$ and $m_2$ from origin O be $x_1$ and $x_2$ respectively.

Again let $x_{cm}$ be the position of the center of mass then $x_{cm}=\frac{m_1 x_1 +m_2 x_2}{m_1 + m_2}$

Answer 2

${m_1 \over m_1+m_2}$ is the fraction of weight at $x_1$ and ${m_2 \over m_1+m_2}$ is the fraction of weight at $x_2$, so the balance point will be at $x^*={m_1 \over m_1+m_2} x_1 + {m_2 \over m_1+m_2} x_2$.

Answer 3

Here is a method working from a definition you may be using, or a feature of the centre of mass, that it acts as the balance point for a collection of masses.

So you are working on a line and you have masses $m_1$ and $m_2$ placed at points $x_1$ and $x_2$. Suppose $x_1\lt x_2$, then we are looking for the centre of mass $c$, a point between $x_1$ and $x_2$ where $$m_1(c-x_1)=m_2(x_2-c)$$

This gives $$(m_1+m_2)c=m_1x_1+m_2x_2$$ or $$c=\frac {m_1x_1+m_2x_2}{m_1+m_2}$$

You can check some things. It is dimensionally correct, for example - the units of the answer are units of length. It is found by dividing through by the total mass and is homogeneous in the individual masses and positions (no point or mass is special). Increasing $m_1$ the mass at $x_1$ puts the centre of mass closer to $x_1$. If there is no mass at $x_2$ the centre of mass is at $x_1$. So it agrees with a physical understanding of the situation.