Variable-mass system events. The following derivation is for a body that is gaining mass (accretion). A body of time-varying mass "m" moves at a velocity v at an initial time "t". In the same instant, a particle of mass dm moves with velocity u. The initial momentum can be written as Now at a time "t" + d"t", let both the main body and the particle accrete into a body of velocity v + dv. Thus the new momentum of the system can be written as Since d"m"dv is the product of two small values, it can be ignored, meaning during d"t"
Effective mass (spring–mass system) uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring-mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). This is because external acceleration does not affect the period of motion around the equilibrium point. We can find the effective mass of the spring by finding its kinetic energy. This requires adding all the mass elements' kinetic energy, and requires the following integral, where formula_8 is the velocity of mass element : Since the spring is uniform, formula_10, where formula_11 is the length of the spring.