Abstract
Abstract We study the dynamics of the center of mass (CM) of a set of particles when the number of particles in the system varies during a short interval of time, or when the value of the mass of one of the particles depends on time. We show that in both situations, while the population or mass of the system varies, the linear momentum of the CM differs from the total linear momentum of the system. During this time interval, the dynamics of the CM is driven by the external forces that act on the system plus transient forces that depend on the position and velocity of the CM. This dynamics is applied to study the time evolution of the vector position of the CM of a system of free particles moving on an air rail in which the mass of one the particles varies in a short time interval. We want to show it to students that with the knowledge and tools they have at an elementary Mechanics course they can formulate and answer questions which are not in the textbooks.
Highlights
The concept of the center of mass (CM) of a system with fixed number N ≥ 2 of particles with constant masses is commonly approached in introductory courses of Mechanics [1,2,3,4,5,6]
An important physical quantity associated to the CM is its linear momentum, that is equal to the total linear momentum of the system of particles
That is the equation of motion for the CM of S1, in terms of variables related to the center of mass itself (RC(1M), PC(1M) ), to the migrating L-th particle, to the forces acting on each particle of S1 due to the particles in S2 besides the external forces of S that act on the physical particles belonging to S1 during the whole movement, that is t ∈ (−∞, +∞)
Summary
The concept of the center of mass (CM) of a system with fixed number N ≥ 2 of particles with constant masses is commonly approached in introductory courses of Mechanics [1,2,3,4,5,6]. An important physical quantity associated to the CM is its linear momentum, that is equal to the total linear momentum of the system of particles. In the present paper we characterize the CM of a particle system in two different scenarios: a) when the number of particles in the system varies by one unit, and this transition happens in a very short time, but all particle masses are constant; b) when the number of particles in the system is fixed, but the mass of one of them varies in time. In subsection 2B we derive the dynamics of the CM of S1, including the transition time interval when the L-th particle moves from S1 to S2. In Appendix B we present a particular function that has the same properties as the transition function f (t)
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