Moment of inertia of circular flywheel

[KiNGGeexD]

Hi I'm new to this forum so still getting used to it!

I have to find the moment of inertia of a circular flywheel which has radius, a and mass m! But also had area density (mass per unit area) of ρ0,

Where ρ0=ρa(1+r/a)

I know moment of inertia is the integral of a^2 dm and in order to get mass m I just multiply the above expression by area of the flywheel which I think is pi*a^2

I end up with a rather foolish answer when I have competed the intergratiom!
Any help would be great

Thanks guys

 

[Vineeth T]

Is the answer (9∏\rhoa⁵ /10) ?
If its' correct pls. reply so that i would post my solution.

 

[haruspex]

I know moment of inertia is the integral of a^2 dm and in order to get mass m I just multiply the above expression by area of the flywheel which I think is pi*a^2

Your process is not clear from that description. Please post your actual working.

 

[haruspex]

Is the answer (9∏\rhoa⁵ /10) ?
If its' correct pls. reply so that i would post my solution.

That's not how this forum works. We don't just post solutions. The idea is to provides hints and guide the thread originator to the solution.

 

[Vineeth T]

Oh! Sorry then.
The idea is to take a differential strip(a ring of thickness dr) at a general distance r, find its' M.I and then integrate it to get M.I of the whole disc(a flywheel).

 

[KiNGGeexD]

Yea I know sorry!
Well what I have done is used polar coordinates and defined my moment of inertia as of course

I= integral of r^2 dm

And my element of mass is just ρ0+[1+r/α] *2pi a^2 but I am having trouble with integrating the expression! Are my boundaries a -0 or r-0?

 

[haruspex]

Yea I know sorry!
Well what I have done is used polar coordinates and defined my moment of inertia as of course

I= integral of r^2 dm

And my element of mass is just ρ0+[1+r/α] *2pi a^2 but I am having trouble with integrating the expression! Are my boundaries a -0 or r-0?

Seems to me there's some confusion between variables and constants here.
In the OP you had r as a variable from 0 to a (constant); ρ₀ as a variable from ρ a to 2 ρ a, ρ being a constant. That's very unusual - I would have expected ρ and ρ₀ to have reversed roles. Assuming you had them crossed over, I'll write ρ(r) = ρ₀a(1+r/a). (Maybe you meant ρ_a, not ρ a, in the OP, but that's just a change to the constant ρ₀.)
You want to integrate wrt r. At radius r the density is ρ(r). To get the element of mass you need to multiply that by the element of area, i.e. the area of an annulus between radius r and radius r+dr. In your above post, you seem to have that as 2pi a^2, which is obviously wrong. Also, is "ρ0+[1+r/α]" a typo? Should it be ρ₀[1+r/α], or ρ₀a[1+r/α]?

 

[KiNGGeexD]

Why is 2pi a^2 obviously wrong? It is the area of the flywheel is it not?

 

[haruspex]

Why is 2pi a^2 obviously wrong? It is the area of the flywheel is it not?

Well, no, the area of the whole flywheel would be pi a², not 2pi a². But for the purposes of the integrand we want the area of an element between radius r and radius r+dr, not the whole flywheel, right? That is independent of a.