Substituting differentials in physics integrals.

[subsonicman]

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.

 

[Simon Bridge]

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 ?

 

[dauto]

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.

 

[subsonicman]

Yeah, I was being stupid. Thanks for the help!