Finding the Center of Mass of a Hemisphere

[ritwik06]

Homework Statement

Center of Mass of a hemisphere

http://www.goiit.com/templates/default/images/chapters/center_mass/image064.gif
http://www.goiit.com/templates/default/images/chapters/center_mass/image068.gif
Why is volume of elemental disc = $$Rd\theta (cos \theta) (\pi R^{2}cos^{2} \theta)$$ and not

$$Rd\theta (\pi R^{2}cos^{2}\theta)?$$

 

[gabbagabbahey]

Because the arc length is measured from the $\theta=0$ plane (yz plane) to the plane rotated through an angle $d\theta$, So the radius of the arc is the projection of R onto the $\theta=0$ plane which is $Rcos(\theta)$ and so the arc length is $Rcos(\theta)d\theta$.

 

[ritwik06]

Because the arc length is measured from the $\theta=0$ plane (yz plane) to the plane rotated through an angle $d\theta$, So the radius of the arc is the projection of R onto the $\theta=0$ plane which is $Rcos(\theta)$ and so the arc length is $Rcos(\theta)d\theta$.

I don't get it. Could you please take some of ur precious time out, just to draw a very rough sketch in paint? Please!