Substituting differentials in physics integrals.

AI Thread Summary
The discussion centers on the derivation of rotational kinetic energy from translational kinetic energy using differential substitutions. The original poster starts with the kinetic energy formula and attempts to substitute differentials, raising questions about the validity of their substitutions. They specifically inquire whether additional terms should be included in the differential equations due to the variability of velocity and mass. Responses clarify that the integration process does not require the product rule and that the substitutions made were appropriate for the context. The conversation emphasizes the importance of understanding variable changes in integration without overcomplicating the process.
subsonicman
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Today I tried to show that rotational kinetic energy was equivalent to standard translational kinetic energy.

So I started with kinetic energy, T = ∫dT. Then, because T=1/2mv^2, I substituted dT=1/2v^2dm and then because m=ρV, I substituted dm=ρdV. Then, after substituting v=ωr, I got the equation for rotational kinetic energy, 1/2Iω^2.

The problem I have is with the substituting differentials. Shouldn't dT=1/2v^2dm+vdvdm because both v and m are varying? Also, shouldn't dm=ρdV+Vdρ? I remember seeing this substitution made when calculating the mass of some shape from its density but I can't seem to justify it from the knowledge I have.

Any help would be appreciated.
 
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subsonicman said:
Today I tried to show that rotational kinetic energy was equivalent to standard translational kinetic energy.

So I started with kinetic energy, T = ∫dT. Then, because T=1/2mv^2, I substituted dT=1/2v^2dm and then because m=ρV, I substituted dm=ρdV. Then, after substituting v=ωr, I got the equation for rotational kinetic energy, 1/2Iω^2.

The problem I have is with the substituting differentials. Shouldn't dT=1/2v^2dm+vdvdm because both v and m are varying?
1. dT = (1/2)v^2.dm + mv.dv
2. what is dv/dm ?
 
You're confused. You're not doing an integration by parts. You're just doing a change of variable of integration. The product rule makes no sense here.
 
Yeah, I was being stupid. Thanks for the help!
 

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