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

[lukea125]

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.

 

[tiny-tim]

welcome to pf!

hi lukea125! welcome to pf!

(have a theta: θ and an omega: ω )

I also thought I could express theta as angular velocity multiplied by time.

no

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

so square-root it, separate the variables, and integrate ​

 

[lukea125]

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