13.4
Equation of Motion: Center of Mass
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.
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 particle within this system, the net force acting upon it is the aggregate of both internal and external forces. Extending this principle to all particles within the system results in the equation of motion for the entire assembly.
Internal forces between any pair of particles manifest as collinear pairs of equal magnitude but opposite directions, leading to their summation equating to zero. Now, introduce a center of mass G expressed in terms of position vectors of the various particles. Differentiating this expression twice concerning time yields the equation of motion relative to the center of mass of the entire system.
Consequently, the net external forces influencing the system of particles translate to the product of the system's overall mass and the acceleration of its center of mass. This comprehensive formulation captures the dynamics of a multi-particle system, considering both internal interactions and external influences. The center of mass concept provides a helpful perspective, simplifying the description of the system's motion in relation to its overall characteristics.
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.
- Centre of Mass of a System of Particles - The combined location of n particles in a system.
- Motion of the Center of Mass - The overall movement of the system's center of mass.
- Equation of Motion - Mathematical description of the particle system's net movement.
- Mass Center Equation - Formula expressing the position and motion of the system's center of mass.
- Central Force Problem - A force challenge that is determined by the system's center of mass.
- Define Centre of Mass – Understand and explain its role and calculation in system of particles (e.g., centre of mass of a system of particles).
- Contrast Internal and External Forces – Explain their impacts on motion (e.g., equation of motion).
- Explore Formula Development – Understanding derivation and use cases of equations (e.g., mass center equation).
- Explain Central Force Problem – Discuss its role and solution approach in a system of particles.
- Apply Principle in Context – Illustrate how the motion of center of mass impacts real-world scenarios.
- What is the concept of the centre of mass of a system of particles and how is it calculated?
- How does the equation of motion account for both internal and external forces within a particle system?
- What is the central force problem and how does it relate to the system's centre of mass?
From Chapter 13:
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