질량관성모멘트 Moment of inertia

Moment

$$dM=dF\times r\cdots(1)$$

 

Here, \(F\) is

$$F=ma\cdots(2)$$

 

And \(\vec{a}\) is

$$l=r\theta$$

$$\frac{dl}{dt}=r\frac{d\theta}{dt}+\theta\frac{dr}{dt}$$

$$\vec{v}=\frac{dl}{dt}=r\frac{d\theta}{dt}=r\omega~~(r=const)$$

$$\vec{v}=r\omega$$

 

$$\frac{d\vec{v}}{dt}=r\frac{d\omega}{dt}+\omega\frac{dr}{dt}$$

$$\vec{a}=\frac{d\vec{v}}{dt}=r\frac{d\omega}{dt}=r\alpha~~(r=const)$$

$$\vec{a}=r\alpha\cdots(3)$$

 

Substitute \(\vec{a}\) of eqaution (2) with eqaution (3)

$$dF=dm\times(r\alpha)\cdots(4)$$

 

Then substitute \(dF\) of equation (1) with equation (4)

$$dM=dm\times(r \alpha)r$$

$$dM=\alpha r^{2} dm\cdots(5)$$

 

Integrate equation (5)

$$M=\int{\alpha r^{2}dm}=\alpha\int{r^{2}dm}=\alpha I$$

 

Here. I is 'moment of inertia'