System of Particles
As we have now found the centre of mass of the object that has rigid structures but we also have to find the centre of mass of the objects that are not rigid and are formed by infinite particles that are free to move individually and are not fixed as in case of the uniform body. The particles in this type of body interact with each other through internal forces. In the case of motion of these particles, they move differently for different particles, but there is a point in this body where all the mass of the object can be assumed to be placed and this is called the centre of mass of the body.
Centre of Mass of a System of Particles
The derivation of the Centre of Mass of the System of Particles is explained below:
Suppose we have n particles of masses,
m1, m2, m3, ….., mn. And m1, m2, m3, ….., mn = M
Let their position vector be, [Tex]\vec{r_1},~\vec{r_2},~\vec{r_3},~…..~\vec{r_n} [/Tex]
Then the centre of mass of this system of particles with respect to the origin is,
[Tex]\vec{r_{cm}}~=~\frac{m_1\vec{r_1}~+~m_2\vec{r_2}~+~m_3\vec{r_3}~+…..~+m_n\vec{r_n}}{m_1~+~m_2~+~m_n~+….+~m_n} [/Tex]
We know that,
m1, m2, m3, ….., mn = M
[Tex]\vec{r_{cm}}~=~\frac{m_1\vec{r_1}~+~m_2\vec{r_2}~+~m_3\vec{r_3}~+…..~+m_n\vec{r_n}}{m_1~+~m_2~+~m_n~+….+~m_n} [/Tex]
Two particle system
For two particle system we take two particles of mass m1 and m2 and their position vector be, [Tex]\vec{r_1}~and~\vec{r_2} [/Tex]
Then the centre of mass of this system of particles with respect to the origin is,
[Tex]\vec{r_{cm}}~=~\frac{m_1\vec{r_1}~+~m_2\vec{r_2}}{m_1~+~m_2} [/Tex]
In the Cartesian coordinate system,
- Xcm = (m1x1 + m2x2)/(m1 + m2)
- Ycm = (m1y1 + m2y2)/(m1 + m2)
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.