Center of vass

The vass is to time (duration) as the mass is to space (length). As noted before here, the vass can be measured by a similar procedure as the mass. The mass and vass are inverses with opposite uses.

The center of mass is the point that two or more particles (point masses) are balanced (or one large mass is balanced). For two particle masses, m₁ and m₂ that are located at points x₁ and x₂, respectively:

Center of Mass (CM) = (m₁x₁ + m₂x₂)/(m₁ + m₂),

which is the weighted arithmetic mean with the masses as the weights. This is similar to the momentum, in which the velocity is weighted by the mass: mv.

The center of vass is the point in time that two or more particle vasses are balanced. For two particle vasses, n₁ and n₂ that are located at points in time t₁ and t₂, respectively:

Center of Vass (CE) = (n₁t₁ + n₂t₂)/(n₁ + n₂) = ((t₁/m₁ + t₂/m₂)/(1/m₁ + 1/m₂)),

which is the weighted arithmetic mean with the vasses as the weights. Compare the levamentum, in which the lenticity is weighted by the vass: nu.

In order to generalize this, let’s use the derivation of the center of mass, as in Knudsen and Hjorth’s Elements of Newtonian Mechanics, chapter 9. Start with the time position vector, T, to find the center of vass for a system of particle vasses:

N T_cv = Σ n_i t_i,

where n_i and t_i are the vass and the time position vector of the i^th particle vass, and N = Σ n_i is the total vass of the system. Then differentiate with respect to space length to get

N T_cv´ = Σ n_i t_i´= Σ q_i := Q,

where the total linear levamentum of the system is denoted Q. In other words, the total linear levamentum Q of a system of particle vass is the same as that of a particle vass with vass N moving the the lenticity of the center of vass. This is also stated as

Q = N u_cv,

where u_cv is the lenticity of the center of vass.