homework and exercises - Why does gravity act at the center of mass?

Sorry if this is a trivial question.

Why does gravity act at the center of mass?

If we have a solid $E$, shouldn't gravity act on all the points $(x,y,z)$ in $E$? Why then when we do problems we only only consider the weight force from the center of mass?

Answer

Suppose I have a collection of n vectors $x_i\quad \forall i\in(1,n), i\in \mathbb{Z}$ such that the corresponding masses at each $x_i$ is $m_i$. This is your body $E$ and if the total mass of your body is $M$, then

$$M=\sum_{i=1}^{n}m_i$$ In that case, if $E$ is subjected to a uniform acceleration field $\vec{g}$, as specified in the answer above, then the net force acting on the body is $$F=\sum_{i=1}^{n}m_i \ddot{x}_i$$ But, the force on the entire body would be $F=Mg$. Let there be a point $X$ on the body such that I can say that $\ddot{X}=g$, Then I can write $F= M\ddot{X}=\sum_{i=1}^{n}m_i \ddot{x}_i$.

From this you can interpret that

$$\ddot{X}=\frac{\sum_{i=1}^{n}m_i \ddot{x}_i}{\sum_{i=1}^{n}m_i}$$ And the centre of mass is defined as $$\begin{equation}\label{com} x_{com}=\frac{\sum_{i=1}^{n}m_ix_i}{\sum_{i=1}^{n}m_i} \end{equation}$$ Since the body $E$ has constant mass, you can get the definition of center of mass above by simple integration.