Is this a correct derivation of Moment of Inertia?

[Marmoteer]

Hello everybody!
This is the derivation (for a single particle).
$$\tau = F_{\perp }r <br /> \ = ma_{\perp}r <br /> \ = \alpha mr^2 \\<br /> \text{if }\<br /> \tau = I\alpha<br /> \text{ where } I \text{ is resistance to acceleration then } \\<br /> I = mr^2$$
I'm curious what the problem with this is because I haven't seen it in any of my physics texts (serway and university at least). I also haven't seen it on the interwebs very much.
Thanks for reading.

 

[Andrew Mason]

Hello everybody!
This is the derivation (for a single particle).
$$\tau = F_{\perp }r <br /> \ = ma_{\perp}r <br /> \ = \alpha mr^2 \\<br /> \text{if }\<br /> \tau = I\alpha<br /> \text{ where } I \text{ is resistance to acceleration then } \\<br /> I = mr^2$$
I'm curious what the problem with this is because I haven't seen it in any of my physics texts (serway and university at least). I also haven't seen it on the interwebs very much.
Thanks for reading.

Moment of inertia refers to the ratio of the torque to angular acceleration of a rigid body about an axis. It is the inertial resistance to rotation that a rigid body has about a specific axis of rotation. A single particle is not a rigid body.

What you are doing is determining the ratio of torque to angular acceleration of a single point particle about an axis of rotation.

If you were to reduce a rigid body to a collection of single point particles of mass dm and add up all the dmr² terms where r is the distance from the point particle to the axis of rotation, you would end up with the moment of inertia of the rigid body.

AM