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Concept Version 5
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Center of Mass and Inertia

The center of mass for a rigid body can be expressed as a triple integral.

Learning Objective

  • Use multiple integrals to find the center of mass of a distribution of mass


Key Points

    • In the case of a single rigid body, the center of mass is fixed in relation to the body, and if the body has uniform density, it will be located at the centroid.
    • In the case of a system of particles $P_i, i = 1, \cdots , n$, each with mass $m_i$ that are located in space with coordinates $\mathbf{r}_i, i = 1, \cdots , n$, the coordinates $\mathbf{R}$ of the center of mass is given as $\mathbf{R} = \frac{1}{M} \sum_{i=1}^n m_i \mathbf{r}_i$.
    • If the mass distribution is continuous with the density $\rho (r)$ within a volume $V$, the center of mass is expressed as $\mathbf R = \frac 1M \int_V\rho(\mathbf{r}) \mathbf{r} dV$.

Terms

  • rigid body

    an idealized solid whose size and shape are fixed and remain unaltered when forces are applied; used in Newtonian mechanics to model real objects

  • centroid

    the point at the center of any shape, sometimes called the center of area or the center of volume


Full Text

Multiple integrals are used in many applications in physics and engineering. In this atom, we will see how center of mass can be calculated using multiple integrals.

The center of mass is the unique point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. In the case of a single rigid body, the center of mass is fixed in relation to the body, and if the body has uniform density, it will be located at the centroid. The center of mass may be located outside the physical body, as is sometimes the case for hollow or open-shaped objects, such as a horseshoe. In the case of a distribution of separate bodies, such as the planets of the Solar System, the center of mass may not correspond to the position of any individual member of the system.

Center of Mass

Two bodies orbiting around the center of mass inside one body

A System of Particles

In the case of a system of particles $P_i, i = 1, \cdots , n$, each with mass $m_i$ that are located in space with coordinates $\mathbf{r}_i, i = 1, \cdots , n$, the coordinates $\mathbf{R}$ of the center of mass satisfy the condition:

 $\displaystyle{\sum_{i=1}^n m_i(\mathbf{r}_i - \mathbf{R}) = 0}$

Solve this equation for $\mathbf{R}$ to obtain the formula:

 $\displaystyle{\mathbf{R} = \frac{1}{M} \sum_{i=1}^n m_i \mathbf{r}_i}$

where $M$ is the sum of the masses of all of the particles.

A Continuous Volume

If the mass distribution is continuous with the density $\rho (r)$ within a volume $V$, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass $\mathbf{R}$ is zero; that is:

 $\displaystyle{\int_V \rho(\mathbf{r})(\mathbf{r}-\mathbf{R})dV = 0}$

Solve this equation for the coordinates $\mathbf{R}$ to obtain:

$\displaystyle{\mathbf R = \frac 1M \int_V\rho(\mathbf{r}) \mathbf{r} dV}$

where $M$ is the total mass in the volume. The integral is over the three dimensional volume, so it is a triple integral.

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