On the Momentum Theorem for a Continuous System of Variable Mass

Abstract

Theorems of linear momentum are derived for a control system having an arbitrary motion in a continuous velocity field. The resulting equations are valid for a system of variable mass. By allowing the control system to be at rest, the usual equations of fluid mechanics are obtained as a special case; by allowing the control system to move with the fluid, the momentum equations for a system of constant mass are obtained as a special case. In this way, it is possible to discuss a common, but incorrect, conception of the momentum equation for a system of variable mass.

Remarks on Newton’s second law for variable mass systems

“…The reference to the 'spurious' (inertial) frame-dependent force,ṁ v, can be eliminated from Eqs. (1,3) by rewriting them in the form given by Eq. (2), with F being the total external force including…”

“…It should be made clear that our criticism to Refs. [3,4,5,6,7,8,9] is not about the derivations/calculations presented there but to the statement that Eq. (1) is not valid in general for variable mass systems.…”

Falling chains

“…Hence the theoretical tension (34) can be expected to hold untilx = 80/81, where T has almost doubled to 21.2M g.…”

“…III with the standard force equation of motion for a variable mass system 9,[33][34][35][36] . For the special case where no external forces act on these subchains, we show explicitly that the mass transfer is made up of an exoergic mass emission followed by an endoergic mass absorption when the transferred mass sticks inelastically to the receiving arm.…”

Equations of motion for the variable mass flow-variable exhaust velocity rocket

“…The equations of motion which apply to a variable mass system have been a subject of debate among physicists, resulting in no less than eight articles 1 2 -1 9 published inthe American Journal of Physics during the period 1964-1966 alone. The author finds the following two approaches to be equivalent and applicable to the variable mass problem considered herein.…”