Center of Mass of a Solid and a Hollow Cone -

Here is the center of mass of a solid cone and a hollow cone.

Center of Mass of Solid Cone

$x_{cm} = 0$
$y_{cm} = \cfrac{h}{4}$

$y_{cm} = \cfrac{\int y dm}{\int dm}$

$dm = \pi (h-y)^2 \cfrac{R^2}{h^2} dy \rho$

$\int dm = \cfrac{1}{3} \pi R^2 h \rho$

Solving we get
$y_{cm} = \cfrac{h}{4}$

As measured from the bottom, $y_{cm} = \cfrac{1}{4} \cfrac{R^2 H^2 – r^2 h^2}{R^2H – r^2 h}$

$(m+M) \cfrac{H}{4} = m \cfrac{h}{4} + M y_{cm}$

$m=\cfrac{1}{3}\pi r^2 h \rho$

$M = \cfrac{1}{3} \pi \rho (R^2H – r^2 h)$

Solving we get
$y_{cm} = \cfrac{1}{4} \cfrac{R^2 H^2 – r^2 h^2}{R^2H – r^2 h}$

To continue to explore center of mass of other commonly encountered shapes, click here…