Integration of an energy balance equation, with respect to time.

In summary, the conversation was about a person seeking help with integrating a gravitational potential energy equation involving velocity and angular velocity. They also discussed relating velocity to angular velocity and expressing theta as a function of time.
  • #1
lukea125
5
0
Hi Guys, I was hoping someone would be able to help me with this integration. It's been killing me haha.

I've got this energy balance equation:

mgh = 0.5m(v^2) + 0.5I(w^2) + T(theta)

Basically, It's a gravitational potential energy that is converted to a velocity of a falling object, a rotating flywheel and some frictional Torque (T). v is velocity, and w is angular velocity. I is the moment of inertia, which is known.

I need to integrate this expression with respect to time, t. I can relate velocity to angular velocity by: 0.0395w. I also thought I could express theta as angular velocity multiplied by time.

Any help would be greatly appreciated as I cannot get this out at all.
 
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  • #2
welcome to pf!

hi lukea125! welcome to pf! :smile:

(have a theta: θ and an omega: ω :wink:)
lukea125 said:
I also thought I could express theta as angular velocity multiplied by time.

no :redface:

you have (dθ/dt)2 as a function of θ …

so square-root it, separate the variables, and integrate :wink:​
 
  • #3
Thanks Tiny-Tim! I really appreciate it. I'll give it a go and see what I can get.
 

1. What is an energy balance equation?

An energy balance equation is a mathematical representation of the conservation of energy, which states that energy cannot be created or destroyed, only transferred or converted from one form to another. It takes into account all the energy inputs and outputs of a system to determine the overall energy balance.

2. Why is it important to integrate an energy balance equation with respect to time?

Integrating an energy balance equation with respect to time allows us to track the changes in energy over a period of time. This is important in understanding the dynamics of a system and how it responds to different inputs and outputs.

3. How is an energy balance equation integrated with respect to time?

An energy balance equation is integrated with respect to time by using calculus methods such as integration by parts or substitution. The specific method used depends on the complexity of the equation and the variables involved.

4. What are the applications of integrating an energy balance equation with respect to time?

Integrating an energy balance equation with respect to time has various applications in fields such as thermodynamics, fluid mechanics, and heat transfer. It is used to analyze and optimize energy systems, understand the behavior of natural processes, and predict the performance of engineering systems.

5. Are there any limitations or assumptions when integrating an energy balance equation with respect to time?

Yes, there are limitations and assumptions when integrating an energy balance equation with respect to time. Some common assumptions include steady-state conditions, negligible changes in potential and kinetic energy, and no heat transfer between the system and its surroundings. It is important to carefully consider these assumptions and their potential impact on the accuracy of the integrated equation.

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