질량관성모멘트 Moment of inertia

Moment

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

 

Here, $F$ is

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

 

And $\overrightarrow{a}$ is

$$l=r\theta$$

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

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

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

 

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

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

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

 

Substitute $\overrightarrow{a}$ 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'