The Centre of Mass of a system is that one point where any uniform force is acted upon the object. The Centre of Mass for complicated shapes can be defined as the unique position where the sum of the weighted position vectors of all the parts equals zero. In the case of a uniform disc, the Centre of Mass would lie on its Centre, while the Centre of Mass in the case of a ring lies on the Centre but there is not any material. It is possible that sometimes the Centre of Mass of an object does not lie over it. The Centre of Mass for the simple rigid objects having uniform density is located at the Centroid. In a system, the average position of all parts that are weighted according to their Masses is defined as the Centre of Mass. The idea of the center of mass (COM) is functional in analyzing the complicated movement of the system of objects, especially when two and more objects collide or an object explodes into fragments.Ī position that is defined relative to an object or a system of objects is known as the Centre of Mass of that object. This point is known as the center of mass of the system of particles.

The movement of this unique point is analogous to the movement of a single particle whose mass is equivalent to the sum of all independent particles of the system and the outcome of all the forces exerted on all the particles of the system by surrounding bodies (or) action of a field of force is exerted directly to that particle. Rather focus only on the dynamics of a unique point corresponding to that system. While studying the dynamics of the movement of the system of particles as a whole, do not be bothered about the dynamics of an individual particle of the system. The motion of a system can be described in terms of the motion of its center of mass. This point is therefore called the ‘center of mass’ of the system. When the system moves under an external force, then this point moves as if the entire mass of the system were concentrated at it, and also the external force was applied at it.

If the net (external) force on a system of particles is zero, the center of mass of the system will not accelerate (Newton's First Law applied to the center of mass.Every single physical system has associated itself with a certain point whose motion characterizes the motion of the whole system.

(Newton's Second Law applied to the center of mass).

The net (external) force on a system of particles equals the mass of the system times the acceleration of the system's center of mass.

The left side of the equation is the net force on the center of mass of the system if all of the mass of the system is located at the system's center of mass. Now, m 1a 1 is the net force on particle 1 (Newton's Second Law), m 2a 2 is the net force on particle 2, etc., so the right side is the net force on the system of n particles. Does this equation have anything to say? Well, multiplying both sides by M gives: Where a cm is the acceleration of the center of mass of the system, and a 1 is the acceleration of particle 1, etc. In the same way, we can derive an equation for the acceleration of the center of mass, a cm:

The system behaves as if all of its mass were located at the system's center of mass.The total momentum of the system equals the total mass of the system times the velocity of the system's center of mass (Mv cm).In other words, the equations above are saying that: Looking at the left side of the equation, Mv cm would be the momentum of the center of mass of the system - if we assume that all of the mass of the system is located at the system's center of mass. On the right side of this equation, m 1v 1 is the momentum of particle 1, m 2v 2 is the momentum of particle 2, etc., so the right side of the equation is the total momentum of the system of particles. Where v 1 is the velocity of the first particle, v 2 is the velocity of the second particle, etc., and M is the total mass of the system.Īside from being very handy when dealing with both inelastic and elastic collisions, what is this equation trying to tell us? Rearanging the last result gives: Starting with the center of mass equation, it is easy to show that the velocity of the center of mass of a system of n particles, v cm, is: Where x cm is the position of the center of mass, m 1 is the mass of the first particle, x 1 is the position of the first particle, etc.

The animation at right will hopefully convince you that the center of mass of a system of n particles (of constant mass) is given by:

Mass formula physics