Understanding Centre of Mass for High School Graduates

[tomz]

I have just finished high school. When I read Paul's note (an online math book), in the section of centre of mass, it says the coordinate of the centre of mass in any axis can be found by divide the moment of that axis by the mass..Then it gives this weird formula for calculating y coordinate of the CM.

y=Mx/M=(1/2)∫(f(x)^2)dx from a to b, where f(x) is a curve, the region of uniform density is bound by f(x) from a to b, x axis, x=a and x=b. I used to see a very standard formula that I can understand, that is y=∫(g(x)^2)dy from c to d, where g(x) is the inverse function of f(x) and c,d are ends of y value.

I know moment of inertia, but i don't know why the formula given by the book will work as well, in a mathematical way, i try to associate this with the equivalence of the 2 ways of integration, shell's method and disk method, that may work, But I cannot understand this in a physical way...

Can anyone help me out?

Thanks

 

[meldraft]

I think that you have confused some formulas:

centre of mass: $x_c=\frac{\int \int x dxdy}{\int \int dxdy}=\frac{\int \int x dxdy}{Area}$

second moment of area: $I_{xx}=\int \int y^2 dxdy$

moment of inertia: $I=\int r^2 dm$

Try each of them in basic shapes where you already know the answer and then you can convince yourself of why they work. Keep in mind that by definition the moment of inertia is $I=r^2 m$

 

[tomz]

I think that you have confused some formulas:

centre of mass: $x_c=\frac{\int \int x dxdy}{\int \int dxdy}=\frac{\int \int x dxdy}{Area}$

second moment of area: $I_{xx}=\int \int y^2 dxdy$

moment of inertia: $I=\int r^2 dm$

Try each of them in basic shapes where you already know the answer and then you can convince yourself of why they work. Keep in mind that by definition the moment of inertia is $I=r^2 m$

Got it, thanks~