Why is moment of inertia defined with radius squared, not just radius?

[AmonRaMAsr]

This a question that has been haunting me for some time now. Regarding the rotational motion of rigid bodies why wasn't the moment of inertia defined as the integral sum elements of infinitesimal mass time the radius from the axis of rotation rather than the radius squared. In this case the equation would look something like this.
F=I*$$\alpha$$
and their would be no need for introducing the concept of the torque.
thnx in advance

 

[Studiot]

Hello and welcome to Physics Forums.

See post#8 here

https://www.physicsforums.com/showthread.php?t=498963

 

[santo35]

This a question that has been haunting me for some time now. Regarding the rotational motion of rigid bodies why wasn't the moment of inertia defined as the integral sum elements of infinitesimal mass time the radius from the axis of rotation rather than the radius squared. In this case the equation would look something like this.
F=I*$$\alpha$$
and their would be no need for introducing the concept of the torque.
thnx in advance

well, a good question. but it is a question not to be asked...first moment of inertia is a definition and definitions can't be put to questions...second you can't just consider one case to change a definition (just like you said it simplifies force eq)...for ex if you are done with the coulumb's law of electrostatics the value of constant is often written as 1/ 4 * pi *$$\epsilon$$0...in which all are constants...have you ever thought why coudnt they call an entire thing as a new constant instead of interducing a new constant $$\epsilon$$0..this is because that $$\epsilon$$0 is useful in many other places (like gauss law)...