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

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SUMMARY

The forum discussion centers on integrating the energy balance equation: mgh = 0.5m(v^2) + 0.5I(w^2) + T(theta) with respect to time (t). The equation represents gravitational potential energy converted into kinetic energy of a falling object, rotational energy of a flywheel, and frictional torque. Key relationships include expressing velocity (v) in terms of angular velocity (w) as 0.0395w and relating angular displacement (theta) to angular velocity over time. Participants suggest separating variables and integrating to solve the equation.

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lukea125
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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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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:​
 
Thanks Tiny-Tim! I really appreciate it. I'll give it a go and see what I can get.
 

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