13.4
Equation of Motion: Center of Mass
The equation of motion for a single particle can be expanded to encompass a system of particles consisting of n particles. For any arbitrarily chosen…
The equation of motion for a single particle can be extended to a system of particles. Consider a system with n numbers of particles.
The net force acting on an arbitrarily chosen particle is the sum of the net external and internal forces.
This equation of motion can be applied to any other particle from the system, and adding them all gives the equation of motion for the system of particles.
The internal forces between any two particles exist in collinear pairs that are equal in magnitude but opposite in direction. So, the summation of internal forces becomes zero.
Now, consider the center of mass G, which is expressed in terms of the position vectors of different particles from the system. Differentiating it twice with respect to time gives the equation of motion with respect to the center of mass of the system.
So, the net external forces acting on the system of particles are equivalent to the product of the system's total mass and the acceleration of its center of mass.
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Q1: How does the equation of motion extend from a single particle to a system of particles?
The equation of motion for a single particle can be extended to a system of n particles by summing the net forces acting on each particle. For any particle in the system, the net force equals the sum of external and internal forces. When equations are added for all particles, internal forces cancel because they exist as collinear pairs equal in magnitude but opposite in direction, leaving only external forces.
Q2: Why do internal forces cancel out in a system of particles?
Internal forces between any two particles in a system exist as collinear pairs that are equal in magnitude but opposite in direction, following Newton's third law. When summing forces across all particles, these paired forces cancel each other out, resulting in zero net internal force. This simplification allows the system's motion to be described entirely by external forces.
Q3: What role does the center of mass play in the equation of motion for a system?
The center of mass G is expressed using position vectors of all particles in the system. Differentiating this expression twice with respect to time yields the equation of motion for the center of mass. The net external forces acting on the system equal the product of the system's total mass and the acceleration of its center of mass, simplifying multi-particle dynamics.
Q4: How is the center of mass mathematically represented in a particle system?
The center of mass G is expressed in terms of position vectors of the different particles within the system. This mathematical representation allows tracking the collective motion of all particles as a single point. Differentiating the center of mass expression twice with respect to time provides the acceleration needed for the system's equation of motion.
Q5: What is the relationship between external forces and center of mass acceleration?
The net external forces acting on a system of particles are equivalent to the product of the system's total mass and the acceleration of its center of mass. This relationship shows that external forces alone determine how the center of mass moves, independent of internal particle interactions. This principle simplifies analyzing complex multi-particle systems by focusing on overall motion.
Q6: How do you derive the center of mass equation of motion from individual particle equations?
Start with the equation of motion for each particle, which includes both external and internal forces. Sum these equations across all particles in the system. Since internal forces cancel as equal and opposite pairs, only external forces remain. Expressing the result in terms of the center of mass position and differentiating twice yields the system's equation of motion.
Q7: Why is the center of mass concept useful for analyzing particle system dynamics?
The center of mass concept provides a simplified perspective for describing a system's overall motion without tracking individual particles. By reducing a multi-particle system to a single point mass, engineers can apply equations of motion rectangular coordinates and cylindrical coordinates more easily. This approach captures the system's dynamics while accounting for both internal interactions and external influences.