- #1
Haths
- 33
- 0
I feel like a div. Quite simply I've forgotten how to use integration to calculate a moment of inertia (MOI). Ok I want to calculate the MOI of say a solid disk, about an axis perpendicular to the plane of the disk, running through its center of mass.
I remember that for any point particle in the disk its MOI is going to be:
mr2
I can show this because if we consider a point mass at some distance r from a point of rotation, it will still subscribe to F=ma in the tangential direction. Knowing that:
T= I [alpha]
Where T is the torque, I the MOI, and alpha the angular acceleration.
T=Fr
Therefore:
T=mar
As
[alpha] = a / r
mar=I a/r
I=mr2
Hence I'm certain of that. Thus I can appreciate that for a solid disk I am looking at a summation of the MOI's between the axis and the edge of the disk. Hence I need an integration factor such that;
IT= Integr0{ dm/dr r2 } dr
I think. Yes? Hence I need an expression for the rate of change of mass with respect to radial displacement. Hence I need an area integral, and this is where I'm left blank, because I have no idea how to express the circle's mass with respect to increasing radius. Unless...
I make my integral:
IT= Integ2PI0{ dm/dr r2 } d[theta]
Where dm/dr = r2PI
Therefore dm = 1/3 r3PI |r0 hence:
IT= Integ2PI0{ 1/3 r3 PI r2 } d[theta]
= 1/3 PI r5 [theta] |2PI0
= 2/3 PI2 r5
Which I'm certain is wrong. I'm not asking why it's wrong. But how should I construct my dm/dr and integrate that through with reasoning why that's a good method. Because I have a feeling I'm forgetting something fundamental in this part of the calculation.
Cheers,
Haths
I remember that for any point particle in the disk its MOI is going to be:
mr2
I can show this because if we consider a point mass at some distance r from a point of rotation, it will still subscribe to F=ma in the tangential direction. Knowing that:
T= I [alpha]
Where T is the torque, I the MOI, and alpha the angular acceleration.
T=Fr
Therefore:
T=mar
As
[alpha] = a / r
mar=I a/r
I=mr2
Hence I'm certain of that. Thus I can appreciate that for a solid disk I am looking at a summation of the MOI's between the axis and the edge of the disk. Hence I need an integration factor such that;
IT= Integr0{ dm/dr r2 } dr
I think. Yes? Hence I need an expression for the rate of change of mass with respect to radial displacement. Hence I need an area integral, and this is where I'm left blank, because I have no idea how to express the circle's mass with respect to increasing radius. Unless...
I make my integral:
IT= Integ2PI0{ dm/dr r2 } d[theta]
Where dm/dr = r2PI
Therefore dm = 1/3 r3PI |r0 hence:
IT= Integ2PI0{ 1/3 r3 PI r2 } d[theta]
= 1/3 PI r5 [theta] |2PI0
= 2/3 PI2 r5
Which I'm certain is wrong. I'm not asking why it's wrong. But how should I construct my dm/dr and integrate that through with reasoning why that's a good method. Because I have a feeling I'm forgetting something fundamental in this part of the calculation.
Cheers,
Haths