- #1

subsonicman

- 21

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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.