# Substituting differentials in physics integrals.

• subsonicman
In summary, the conversation was about the equivalence of rotational and translational kinetic energy. The speaker started with the equation T = ∫dT and substituted for T=1/2mv^2 and m=ρV to get the equation for rotational kinetic energy, 1/2Iω^2. The issue was raised about the substitutions of differentials, but it was clarified that it was just a change of variable and not an integration by parts.
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.

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!

I would first commend you on your attempt to show the equivalence between rotational and translational kinetic energy. This is an important concept in physics and it is great that you are exploring it further.

In terms of your concerns about the substitutions in the integral, I can understand your confusion. It is important to note that the differential notation in physics can often be misleading and requires careful interpretation.

In this case, when you substitute dT=1/2v^2dm, you are essentially saying that a small change in kinetic energy (dT) is equal to 1/2 times the square of the velocity (v^2) times a small change in mass (dm). This is a valid substitution because we are assuming that the velocity is constant and only the mass is changing.

Similarly, when you substitute dm=ρdV, you are saying that a small change in mass (dm) is equal to the density (ρ) times a small change in volume (dV). Again, this is a valid substitution because we are assuming that the density is constant and only the volume is changing.

To address your concern about the varying variables, it is important to remember that when we are dealing with differentials, we are taking infinitesimally small changes. So even though both v and m may be varying, in the infinitesimal limit, we can assume them to be constant.

In the case of the mass substitution, it is important to note that the density (ρ) is also a function of volume (V). So when we take the derivative of this function, we get dρ/dV, which is why we only use dV and not Vdρ.

I hope this helps clarify your doubts. Remember, in physics, it is important to carefully interpret the notation and assumptions made in order to make valid substitutions. Keep up the good work in exploring and understanding these concepts!

## 1. What is the purpose of substituting differentials in physics integrals?

The purpose of substituting differentials in physics integrals is to simplify the integral and make it easier to solve. This technique is commonly used when the integrand involves a complex expression or contains multiple variables.

## 2. How do you substitute differentials in a physics integral?

To substitute differentials in a physics integral, you need to identify a variable that can be expressed in terms of another variable in the integrand. Then, you can use the chain rule to replace the original variable with the new variable in the integral.

## 3. What are the benefits of substituting differentials in physics integrals?

Substituting differentials in physics integrals can help to simplify the integral and make it easier to solve. It can also help to reveal patterns or relationships between variables in the integrand, making it easier to understand the physical meaning of the integral.

## 4. Are there any limitations to substituting differentials in physics integrals?

Yes, there are limitations to substituting differentials in physics integrals. This technique is only applicable when there is a variable in the integrand that can be expressed in terms of another variable. It may also not be useful for integrals with certain types of functions, such as trigonometric functions or logarithmic functions.

## 5. Can substituting differentials change the result of a physics integral?

Yes, substituting differentials can change the result of a physics integral. However, it should not change the final numerical value of the integral. Instead, it may change the form of the integral, making it easier to solve or revealing different physical interpretations of the integral.

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