9.14:
Significance of Center of Mass
The center of mass for a collection of particles is expressed as the mass-weighted average position of all the particles.
Consider a two-particle system with known coordinates.
The x-coordinate of the center of mass's position is the ratio of the sum of the product of the particles' masses and their respective position coordinates to the system's total mass. Similarly, expressions for other coordinates can be written.
A time derivative of the position vector of the center of mass gives its velocity.
The sum of the individual masses of the particles is the system's total mass. The product of the mass and velocity of individual particles gives a particle's momentum. So, the total mass times the center of mass velocity equals the system's total momentum.
The rate of change of the system's total momentum gives the net force acting on the system. According to Newton's third law, the internal forces cancel out. So, the system's motion is dictated as if the net external force acts on the center of mass of the system.
9.14:
Significance of Center of Mass
The center of mass of an object is defined as the mass-weighted average position of all the particles that comprise the object. The significance of the center of mass of an object can be seen by looking at its dynamics. The time derivative of the center of mass gives its velocity, assuming that the object's mass remains constant over time. Furthermore, the total linear momentum of an object can be seen as the linear momentum of a single particle of the object's total mass moving with the velocity of the center of mass.
The forces acting on every particle of an object can include internal and external forces. Due to Newton's third law of motion, the internal force exerted by particle one on particle two will be equal and opposite to that of the force exerted by particle two on particle one. Thus, all the internal forces of an object cancel out. Therefore, the net force acting on an object is only the external force. The net external force on an object's motion can be viewed as if the net force is acting on the center of mass of the object. The center of mass of an object obeys Newton's second law of motion, such that the conservation of linear momentum of a system holds if no net force is acting on the object.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/1-introduction
- Fundamental of Physics, Halliday and Resnick, 8th edition, pages 274-275