Center Of Mass

What is Center of Mass?

The Center of Mass or COM is the relative position of an object or a body which represents its mean position.  For any simple rigid object which has uniform density, the COM is located at the centroid.

The formula of COM

We can extend the formula to a system of particles.The equation can be applied individually to each axis,

\(X_{com}\) = \(\frac{∑_{i=0}^n~ m_i x_i }{M}\)

\(Y_{com}\) = \(\frac{∑_{i=0}^n~ m_i y_i}{M}\)

\(Z_{com}\) = \( \frac{∑_{i=0}^n~m_i z_i }{M}\)

The above formula can be used if we have point objects. But we have to take a different approach if we have to find the center of mass of an extended object like a rod. We have to consider a differential mass and its position and then integrate it over the entire length.

\(X_{com}\) = \(\frac{∫~x ~dm}{M}\)

\(Y_{com}\) = \(\frac{∫~y ~dm}{M} \)

\(Z_{com}\) = \(\frac{∫~z ~dm}{M} \)

Suppose we have a rod as shown in the figure and we have to find its COM.

Center of Mass

Let the total mass of the rod be \(M\) and the density is uniform. Also we assume that the breadth of the rod is negligible i.e. the COM lies on the x-axis. We consider a small dx at a distance from the origin. Therefore,

\(dm\) = \(\frac{M}{l}~ dx\)

Using the equation for finding COM,

\(X_{com}\) = \(\frac{∫~\frac{M}{l}~ dx ~.x}{M}\)

\(X_{com}\) = \( \frac{∫~ dx ~.x}{l}\)

Integrating it from \(0\) to \(l\) we get,

\(X_{com}\) = \(\frac{l}{2}\)

Using the above method we can find center of mass for any geometrical shape. You can try out for a semi circular ring or a triangle. So if a force is applied on that extended object it can be assumed to act through the center of mass and hence it can be converted to a point mass.

Center of gravity

Along with COM we also talk about center of gravity. Center of gravity is a point from where the entire mass of a body acts. When the external field is constant over the entire range of the body then the center of gravity coincides with COM.

How can we find the center of gravity for irregular bodies? The answer to this and many more interesting questions are here at BYJU’S.

Practise This Question

Eddy currents are produced when  [CBSE PMT 1993; AFMC 2002]