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Center of Gravity

The point where the entire weight of an object appears to act, a fundamental concept in statics calculations.
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The statement of the theorem

Let B⊂R3\mathcal{B} \subset \mathbb{R}^3 be a continuous body with a spatially varying mass density ρ(r⃗)\rho(\vec{r}). The total mass MM of the body is defined by the volume integral:\n\nM=∭Bρ(r⃗)dVM = \iiint_{\mathcal{B}} \rho(\vec{r}) dV\n\nThe position vector of the Center of Gravity, r⃗cg=(xcg,ycg,zcg)\vec{r}_{cg} = (x_{cg}, y_{cg}, z_{cg}), is defined as the weighted average of the position over the volume, where the weighting function is the density ρ(r⃗)\rho(\vec{r}). Specifically, the coordinates are given by:\n\nxcg=1M∭Bxρ(r⃗)dVx_{cg} = \frac{1}{M} \iiint_{\mathcal{B}} x \rho(\vec{r}) dV\nycg=1M∭Byρ(r⃗)dVy_{cg} = \frac{1}{M} \iiint_{\mathcal{B}} y \rho(\vec{r}) dV\nzcg=1M∭Bzρ(r⃗)dVz_{cg} = \frac{1}{M} \iiint_{\mathcal{B}} z \rho(\vec{r}) dV\n\nIn vector form, the Center of Gravity is:\n\nr⃗cg=1M∭Br⃗ρ(r⃗)dV\vec{r}_{cg} = \frac{1}{M} \iiint_{\mathcal{B}} \vec{r} \rho(\vec{r}) dV