How to find the center of mass

The center of mass of an object is the point at which the object can be balanced. Mathematically, it is the point at which the torques from the mass elements of an object sum to zero. The center of mass is useful because problems can often be simplified by treating a collection of masses as one mass at their common center of mass. The weight of the object then acts through this point.

The position of the center of mass of a collection of masses is given by:

Sum of clockwise moments about the center of mass = Sum of anti-clockwise moments about the center of mass

or, if all the masses were instead placed at the center of mass then they must give the same resultant turning moment about any point in the system as all of the individual moments added together.

\[\vec{r}_{cm}=\dfrac{\sum_i m_i\vec{r}_i}{\sum_i m_i}\]

The center of mass of an object with a uniform density can often be found without calculation, but by instead just looking at the symmetry of the object. For a rod of uniform density, it is intuitive that the center of mass will be halfway along its length. We are equating the mass of the pieces either side of the point of the balance (in this case the tip of the wedge). This is the point at which the weight of the rod acts.

The most fundamental symmetry transformation for determining the position of the center of mass is rotational symmetry. If a massive object or collection of objects has rotational symmetry about a single point then that point will be the center of mass of the collection of objects.

Diagram showing the lines of reflectional symmetry (mirror lines – in grey) on a star.

An object or collection of objects with more than one line of reflectional symmetry will have its center of mass at the intersection of those lines. If the object only has one line of symmetry then the center of mass will be at some point along that line. If an object has a line of reflectional symmetry, then a mirror can be placed along that line and the “missing” half of the image will appear in the mirror.

Reflectional symmetry about a line or plane is less fundamental than rotational. If there are at least 2 lines of symmetry to form a point (which is the center of mass), then rotational symmetry about that point will also exist. Lines of reflectional symmetry are often more useful to think about than the point of rotation when determining the center of mass, as it is often a distance along with one of these lines that are used to describe the position of the center of mass. All the star’s lines of symmetry go through one point in the center, so this is the center of mass.

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