Tru Physics

Center of Mass

—

by

Tru Physics
in

The center of mass is a concept in physics that describes the average position of an object or system of objects. It is an important concept because it allows us to simplify the description of a system by replacing it with a single point. This single point represents the average location of all the mass in the system.

Definition

The center of mass of a system is the point at which the mass of the system can be considered to be concentrated. It is the point about which the system would balance if suspended from that point. The center of mass of an object or system is a geometric property that can be determined mathematically.

Calculating the Center of Mass

The center of mass of a system of objects/particles can be calculated using the following formula:

x_{com} = \dfrac{m_1x_1 + m_2x_2 + ... + m_nx_n}{m_1 + m_2 + ... + m_n}

y_{com} = \dfrac{m_1y_1 + m_2y_2 + … + m_ny_n}{m_1 + m_2 + … + m_n}

z_{com} = \dfrac{m_1z_1 + m_2z_2 + ... + m_nz_n}{m_1 + m_2 + ... + m_n}

where m_1, m_2, …, m_n are the masses of the individual particles in the system, and x_1, y_1, z_1, …, x_n, y_n, z_n are their corresponding coordinates.

More generally, we can express the center of mass (COM) coordinate with vector notation:

\vec{r}_{com} = \dfrac{m_1 \vec{r}_1 + m_2 \vec{r}_2 + ... + m_n \vec{r}_n}{m_1 + m_2 + ... + m_n}

where \vec{r}_{com} is the position vector of the center of mass.

In some cases, the center of mass may lie outside of the physical boundaries of the object or system. For example, the center of mass of a donut would be located in the hole of the donut.

The Center of Mass of Continuous Objects

For continuous objects (like the donut mentioned above), the center of mass can be calculated using integrals:

x_{com} = \frac{1}{M} \int x \mathrm{d} m

y_{com} = \frac{1}{M} \int y \mathrm{d} m

z_{com} = \frac{1}{M} \int z \mathrm{d} m

where M is the total mass of the object, and the integrals are taken over the entire object.

More generally, we can write:

\vec{R}_{com} = \frac{1}{M} \iiint_Q \rho (\vec{r}) \vec{r} \mathrm{d} V

where \vec{R}_{com} is the vector that points to the center of mass, \vec{r} is the set of vectors that point to each coordinate within the object (Q), \rho (\vec{r}) is the mass density, and V is the volume.

For objects with symmetry, the center of mass will lie on the axis of symmetry. For example, the center of mass of a sphere is located at its geometric center.

Want to study more? Visit our Index here!

Do you prefer video lectures over reading a webpage? Follow us on YouTube to stay updated with the latest video content!

Comments

Have something to add? Leave a comment!