- #1
MHD93
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Why is dp/dt = F even if the mass is changing, can it be derived from Newton's second law?
Derivative of Momentum: Why is dp/dt = F?
- Context: Undergrad
- Thread starter MHD93
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- Derivative Momentum
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Discussion Overview
The discussion centers around the relationship between the derivative of momentum and force, specifically the expression dp/dt = F, and whether it can be derived from Newton's second law in the context of changing mass. The scope includes theoretical interpretations and mathematical formulations related to classical mechanics.
Discussion Character
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- Some participants assert that dp/dt = F is a direct representation of Newton's second law, emphasizing that force equals the rate of change of momentum.
- Others clarify that the formulation F = ma applies only when mass is constant, suggesting that the original formulation allows for changing mass.
- One participant points out that if mass is changing, the momentum must account for both the velocity change and the mass change, leading to the expression F = mdv/dt + vdm/dt.
- A later reply questions the derivation of dp/dt = F from Newton's second law, suggesting that it is a mathematical form of the law rather than a derivation.
Areas of Agreement / Disagreement
Participants express differing views on the applicability of Newton's second law in cases of changing mass, indicating that there is no consensus on whether dp/dt = F can be derived from the law itself.
Contextual Notes
The discussion highlights the conditions under which Newton's second law applies, particularly the distinction between constant and variable mass, but does not resolve the implications of these conditions.
- #2
tiny-tim
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Hi Mohammad! 
That is Newton's second law …
We can only use "F = ma" when the mass is constant.
(constant mass is the exception
)
Mohammad_93 said:
That is Newton's second law …
force = rate of change of momentum.
We can only use "F = ma" when the mass is constant.
(constant mass is the exception
)- #3
MHD93
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Newton's second law says that force is proportional to acceleration, meaning that mass is the proportionality constant
Isn't that true?
Isn't that true?
- #4
olivermsun
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It doesn't really matter how you write it. The original formulation is F = dp/dt, but if mass is constant you can take it outside the derivative. If not, then you will need to account for the momentum carried into or out of the system by the mass (change).
- #5
DrBloke
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Another may of looking at it, but reiterating what the others have said, is as follows:
p=mv
F=dp/dt (Newton's second law states force equals rate of change of momentum)
F=d(mv)/dt
=mdv/dt +vdm/dt (equation 1)
if the mass is constant then the rate of change of mass, dm/dt=0 so this becomes
F=mdv/dt
=ma (equation 2)
p=mv
F=dp/dt (Newton's second law states force equals rate of change of momentum)
F=d(mv)/dt
=mdv/dt +vdm/dt (equation 1)
if the mass is constant then the rate of change of mass, dm/dt=0 so this becomes
F=mdv/dt
=ma (equation 2)
- #6
dextercioby
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Mohammad_93 said:
As others may have stated, dp/dt = F_net IS the mathematical form of the 2nd law, thus it can't be derived from itself.
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