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System of Particles and Rotational Motion

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Physics Part I

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Class 10 Class 12
In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 Å (1 Å = 10–10 m). Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus. 


Distance between H and Cl atoms = 1.27 Å 

Mass of H atom = 

Mass of Cl atom = 35.5

Let the centre of mass of the system lie at a distance x from the Cl atom. 

Distance of the centre of mass from the H atom = (1.27 – x)

Let us assume that the centre of mass of the given molecule lies at the origin.

Therefore, we can have

fraction numerator left square bracket straight m space left parenthesis 1.27 space minus space straight x right parenthesis space plus space 35.5 space mx right square bracket over denominator left parenthesis space straight m space plus space 35.5 space straight m right parenthesis end fraction space equals space 0 
           m(1.27 – x) + 35.5mx =  0 

                              1.27 - x = -35.5

∴                                       x = fraction numerator negative 1.27 over denominator left parenthesis 35.5 space minus space 1 right parenthesis end fraction

                                           =  -0.037 Å 

Here, the negative sign indicates that the centre of mass lies at the left of the molecule.

Hence, the centre of mass of the HCl molecule lies 0.037Å from the Cl atom. 

Is it necessary for centre of mass to lie within the body?

No, centre of mass needs not to lie within the body. It is not necessary that the total mass of the system be actually present at the centre.

The position of the centre of mass is calculated using the usual Newtonian type of equations of motion. 

Define centre of mass.

Centre of mass of a body or a system of bodies is a point at which the entire mass of the body or system is supposed to be concentrated. 

What is the significance of defining the center of mass of a system?

The motion of n particle system can be reduced to one particle motion.

An equivalent single point object would enable us to discuss the overall motion of the system. 

Is it necessary that there should be matter at the centre of mass of system?

No, it is not necessary that there be matter at the centre of mass of the system.

For e.g., if two equal point masses are separated by certain distance, the centre of mass lies at the mid point of two point masses and there is no mass at that point.

What is the need of centre of mass?

Newton’s second law of motion is strictly applicable to point masses only. To apply the Newton's law of motion to rigid bodies, the concept of centre of mass is introduced.

The concept of centre of mass of a system enables us to discuss overall motion of the system by replacing the system by an equivalent single point object.