Systems of Particles
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Particle Systems (2 particles & more): |
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In Newtonian mechanics an object is considered like a particle. If we have a system of particles
then the mechanics of the system can be considered in such a way that the force which we apply
to the system would be applied at the centre of mass of the system. It implies that the system acts
as a whole and like a single object. Now, let us consider the following example.
Consider the collection of two balls. If no external force is acting on the collection of balls, their
center of mass (CM) is moving with a constant velocity in accordance with Newton's second law,
i.e Fext =0.
Thus, the acceleration of the CM is also zero.
That is acm = o
Or vcm = constant
For a system of two particles which is situated on x-axis, the
position of centre of mass is given by,
Then the velocity of the CM is given by, vCM = (m1v1+m2v2)/(m1+m2)
If there are more than two
particles in a system then the formula for the position of CM can be extended as given below:
Then the velocity of the center of mass will be given by,
vcm = (m1v1 + m2v2+ m3v3+……)/(m1+m2+m3+……)
If the external force acting over the system is zero then the velocity of the
CM is constant. Hence, the
CM of a system of particle moves with constant velocity if the total external force acting over the
system is zero.
Center of Mass of Solid Objects:
The solid object can then be considered as a system of particles. Then, the center of mass R of a
system of particles is defined as the average of their positions (ri),
weighted by their masses mi:
For a continuous distribution with mass density (ρ(r)) and total
mass M, the sum can be replaced
by an integral and can be written as:
If an object has uniform density then its center of mass is the same as the centroid of its shape.
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