Doubt about variable mass systems

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Discussion Overview

The discussion revolves around the concept of variable mass systems in physics, particularly focusing on the application of Newton's second law and the derivation of related formulas. Participants explore the implications of mass changes on system dynamics, questioning established reasoning and derivations found in textbooks.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant questions the reasoning in Halliday's treatment of variable mass systems, suggesting a potential flaw in the derivation of force equations.
  • Another participant asks why the velocity \( u \) of a mass element might change, indicating confusion about the implications of time tending to zero.
  • Some participants argue that if no external force acts on a mass element \( \Delta M \), its velocity would remain constant.
  • Several participants express confusion over the notations and definitions used in the discussion, particularly regarding the treatment of mass changes and relative velocities.
  • One participant suggests that multiple systems should be considered within the total mass \( M \), with some gaining and others losing mass, leading to different momentum equations.
  • Another participant highlights the inconsistency in the treatment of mass changes and the implications for derivations, particularly referencing objections from other threads.

Areas of Agreement / Disagreement

Participants do not reach a consensus; multiple competing views remain regarding the treatment of variable mass systems and the validity of different derivations. Confusion persists around the definitions and implications of mass changes.

Contextual Notes

Participants note limitations in the clarity of notations and justifications, which contribute to ongoing confusion. There is also mention of unresolved mathematical steps and the need for clearer definitions in the context of variable mass systems.

  • #31
Hak said:
OK, thanks. Were you able to see the posts in the thread I referred you to in post #26?
The OP in that thread is guilty of setting up the reader for confusion.
First the OP says "The rocket now has velocity v+dv and mass M+dM, where the change in mass dM is a negative quantity.". That's fine.
Then the OP says "If I'm allowed to change M+dM with M-dM and -dM with dM then Eq. 9-38 becomes
M * v = dM * U + (M - dM) * (v + dv)"
Equation 9.38 is
M * v = -dM * U + (M + dM) * (v + dv) ... (9-38)

Well, having defined dM as a negative quantity, one is not allowed to change dM to -dM but to -|dM| to avoid conflict with the definition plus confusion about the variables. Then Eq. 9-38 should be
M * v = |dM| * U + (M - |dM|) * (v + dv) ... (9-38a)
OP writes the equation
M * v = dM * U + (M - dM) * (v + dv) ... (9.38b)

Do you see the difference between equations (9.38a) and (9.38b)? They are the same but only if dM in (9.39b) is defined as a positive quantity which is in direct contradiction to its original definition. Is that what confused you?

The equation $$m~\frac{dv}{dt}=(u-v)\frac{dm}{dt}$$ gets around the issue whether ##dm## is positive or negative. The mass transfer to or from the system of interest 1 (e.g the rocket) and the secondary system 2 (e.g. the fuel) is such that ##dm_1+dm_2=0.## Study the derivation in the insight. Which one is positive and which one is negative is irrelevant. In the equation ##m## and ##dm## are ##m_1## and ##dm_1## with the subscripts dropped as explained in the derivation.
 
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