First Law for a Eulerian non-inertial system

[MathError]

Homework Statement

Write the First Law for an Eulerian-non inertial system

Relevant Equations

q+w_t=(h_2-h_1)+(c'_2^2-c'_1^2)/2+g(z_2-z_1)-(w^2*r_2^2-w^2*r_1^2)/2 (a)

Hello to everyone, I'm trying to demonstrate (a) starting from a Lagrangian system. However I have found some difficoulties so I hope you can help me.
First of all, I'm going to illustrate a demonstration I found of the First Law, for a Eulerian -inertial system:
q+w_t=(h_2-h_1)+(c_2^2-c_1^2)/2+g(z_2-z_1) (b)
applying the First Law for a Lagrangian system:
Q+W_e=ΔE_e+ΔE_g+ΔU=ΔE_t (c)

• E_c=kinetic energy
• E_g=gravitational potential energy
• U=internal energy
• E_t=total energy
• W_e=work of external forces (except weight which is still included in E_g)

Now I want to apply this method for a non inertial system, which rotates with the rotor. However in this case I have to consider all the forces I see from this system:
Q+W'_e=ΔE'_e+ΔE'_g+ΔU'=ΔE'_t
I can write this equation:

1. V'=V-V(OO')-W x R'
2. A'=A-A(OO')-W x (W x R')+dW/dt x R'+2*W x V'

In this case:

• O≡O'→V(OO')=A(OO')=0
• Stationary regime→dW/dt=0

So the equations become:

1. V'=V-W x R'
2. A'=A-W x (W x R')-2*W x V'

[Sorry but now I'm a bit confused because I have just realized I probably used a wrong expression for A' in my calculations. In this link
A'=A+W x (W x R')-2*W x V'
https://it.wikipedia.org/wiki/Inter...un'interazione,dei momenti reali, o effettivi. and that sign was one of my dubts. Anyway I'll continue so that you can give me your opinions]
From (2):
A'=A-W x (W x R')-2*W x V'
m*A'=m*A-m*W x (W x R')-m*2*W x V'
So I can see that:

• R'_e=m*A'=Risultant of real and apparent forces
• m*A=Resultant of real forces (This forces are the same I considered in the inertial system)
• -m*W x (W x R')-m*2*W x V'=resultant of apparent forces

Now I can calculate the works of R'_e as:
dW'_e=m*A'*dR'=m*A*dR'-m*[W x (W x R')]*dR'-m*2*[W x V']*dR'

• dR'=V'dt→[W x V']*dR'=0
• m*A*dR' is not equal to the work of forces in the inertial system because dR'=/=dR
• [W x (W x R')]*dR'=-w^2*r'*dr'

dW'_e=m*A*dR'+mw^2*r'*dr'
For the term (m*A*dR') I have done these considerations for this specific case: this term includes the work real forces which act on the system so that they consist in the forces of pressure (probably they include also the work of rotor but I'll consider it then). This forces, in this case, are parallel to V and perpendicular to (W x R') as we can see in this pic and so:
m*A*dR'=R*V'dt=m*A*(V-W x R')*dt=m*a*v*dt (equal to the inertial system)

W_apparent=∫m*w^2*r'*dr'=m*w^2*(r_2^-r_1^2)/2
In term of work/mass:
w_apparent=w^2*(r_2^-r_1^2)/2

dW_pressure=m*a*v*dt
in term of work/mass:
w_pressure= dW_pressure/(dt*m)=a*v=p_1*v_1-p_2*v_2
So, including the tecnique work (which is zero in this case) we obtain:
q+w'_t+p_1*v_1-p_2*v_2+w^2*(r_2^-r_1^2)/2=(u_2-u_1)+(c'_2^2-c'_1^2)/2+g(z_2-z_1)

q+w'_t=(h_2-h_1)+(c'_2^2-c'_1^2)/2+g(z_2-z_1) -w^2*(r_2^-r_1^2)/2
where z'Ξz and h'Ξh.

I don't think many of these steps are formal but I have tried to obtain it by myself. Also I don't speak english very well so I can't explain so clearly but I hope you can understand the steps and help me.
Thank you so much

 

[anathema]

for your help and attention.Hello! Thank you for sharing your demonstration and asking for feedback. It is clear that you have put a lot of effort into understanding and applying the First Law for a Lagrangian system, and your approach is definitely on the right track. Here are a few suggestions and comments that may help you refine your demonstration:

1. Clarify your notation: In your equations, you have used various symbols such as q, w_t, h, c, etc. It would be helpful to define these symbols at the beginning of your demonstration so that readers can follow along more easily.

2. Provide a clear explanation of your assumptions: In any scientific demonstration, it is important to clearly state the assumptions that are being made. This allows readers to understand the limitations of the demonstration and the context in which it applies. In your case, it would be helpful to explain the assumptions you are making about the system, such as its geometry, the forces acting on it, and the motion of its components.

3. Consider using vector notation: In some of your equations, you have used vector quantities such as R' and V'. It would be helpful to use vector notation consistently throughout your demonstration to avoid confusion.

4. Check your signs: As you have mentioned, you may have made a mistake with the sign of the term A' in your original calculations. It is important to carefully check all signs in your equations to ensure that they are consistent and accurate.

5. Use clear and concise language: In addition to the equations, it is important to provide clear and concise explanations of the concepts and steps involved in your demonstration. This will help readers understand your reasoning and follow your calculations.

Overall, your demonstration shows that you have a good understanding of the First Law and how it applies to a Lagrangian system. With a few refinements, it could be a very useful resource for others studying this topic. I hope these suggestions are helpful, and I wish you all the best in your research!