Center of Mass

A lot of problems can be simplified if it is assumed that the mass of the object is located at one particular point. If the correct position is chosen, then equations of forces and motion behave the same way as they behave if applied when mass is spread out. This special location is termed the Center of Mass.

Its position is defined relative to an object or the system of objects whose Center of mass is to be calculated. Usually for uniform shapes, it’s their centroid. For the shapes that are symmetrical and uniform, their Center of mass is located at their centroid. For a ring, its Center of mass lies inside the ring, which means the Center of mass of a body doesn’t need to lie in the body itself. 

Finding the Center of Mass

Now, it is clear that bodies that are uniform and symmetrical have their Center of masses at their centroid. But for bodies that are not symmetrical and uniform, the answer is not that simple. The Center of mass for such bodies can be anywhere. To work out the Center of mass of a complex object. A weighted average of the locations of each mass of the body is taken. 

Let’s say there is a body consisting of a set of masses “m_i“, each at position r_i, the location of the Center of mass r_cm is given by the formula below. 

Mr_cm = m₁r₁ + m₂r₂ + …. 

⇒ r_cm = [Tex]\frac{ m_1r_1 + m_2r_2 + …}{M}[/Tex]

In this case, M =[Tex] \sum m_i     [/Tex], which is the total mass of the body. 

The above technique uses vector arithmetic. To avoid vector arithmetic, we can find out the Center of mass of the body along the x-axis and y-axis respectively. Formulas for this case are given below: 

cm = [Tex]\frac{ m_1x_1 + m_2x_2 + …}{M}[/Tex]

y_cm = [Tex]\frac{ m_1y_1 + m_2y_2 + …}{M}[/Tex]

Center of Mass is an important property of any rigid body system. Usually, these systems contain more than one particle. It becomes essential to analyze these systems as a whole. To perform calculations of mechanics, these bodies must be considered as a single-point mass. The Center of mass denotes such a point. Often the mechanical systems move in a transitory or a rotatory manner. In that case, the Center of mass also moves and acquires some velocity and acceleration. Let’s see how to calculate these metrics for such systems in detail.