Centre of Mass Formula
Now, it is clear that bodies that are uniform and symmetrical have their Centre of masses at their centroid. But for bodies that are not symmetrical and uniform, the answer is not that simple. The Centre of mass for such bodies can be anywhere. To work out the Centre 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 of mass M, consisting of a set of masses “mi“, each at position ri, then as the total mass seems to concentrate at the centre of mass of the body then:
Mrcm = m1r1 + m2r2 + ….+ mnrn
Thus, the formula for the location of the Centre of mass rcm is given as:
[Tex]\bold{r_{cm} =\frac{ m_1r_1 + m_2r_2 + …+m_nr_n}{M}} [/Tex]
Where,
- M = ∑mi, which is the total mass of the body.
The above technique uses vector arithmetic. To avoid vector arithmetic, we can find out the Centre of mass of the body along the x-axis and y-axis respectively. Formulas for this case are given below:
[Tex]\bold{x_{cm} = \frac{ m_1x_1 + m_2x_2 + …}{M}} [/Tex]
[Tex]\bold{y_{cm} = \frac{ m_1y_1 + m_2y_2 + …}{M}} [/Tex]
Centre of Mass of a Body Having Continuous Mass Distribution
For objects that have non-uniform structures, their centre of mass is calculated by distributing the body into infinitely many small rectangles and then finding the centre of mass of individual rectangles and then integrating them to find the centre of mass of the object.
Suppose the object has mass m and its infinitely small part has mass dm, then the coordinate of its centre of mass is calculated as,
- xcm = ∫(x.dm)/∫dm
- ycm = ∫(y.dm)/∫dm
- zcm = ∫(z.dm)/∫dm
Centre of Mass
Centre of Mass is the point of anybody where all the mass of the body is concentrated. For the sake of convenience in Newtonian Physics, we take the body as the point object where all its mass is concentrated at the centre of mass of the body. The centre of mass of the body is a point that can be on the body such as in the case of a Circular sheet, rectangle sheet, sphere, etc, and can also be outside the body such as in the case of a ring, hollow sphere, etc.
In this article, we’ll explore the basic fundamentals of the centre of mass, including its definition, methods, and formula to calculate it. We’ll also discuss some real-world examples of the centre of mass to help you see its practical applications. So, let’s start learning about the fascinating world of the centre of mass and its role in the physics of motion.