# Finding the Center of Mass for a Hemisphere and Right Cone

• rahul.mishra
In summary, the conversation discusses finding the center of mass of a remaining portion after a hemisphere and a right cone are intersected. The approach involves considering the symmetry and calculating the center of mass using the mass and density of the hemisphere and the remaining portion. The source provides a different result due to an incorrect assumption about the cone being hollow. The correct location of the center of mass of the cone is at R/8.

#### rahul.mishra

Suppose there's a hemisphere of radius R (say) and a right cone of same radius R but ht. R/2 is scooped out of it then i have to find the center of mass of the remaining part.

Here's how i approached...

clearly by symmetry, Xcm = 0

Now, Let M be the mass of the hemisphere so,

Density per unit volume, ρ = M/(2/3.π .r3) x 1/3.π.r2.(r/2) = M/4

Now, Ycm of remaining portion = {M(3R/8) - M/4(R/6)}/{M-M/4} = 4R/9

Thus, C.M of the remaining portion = (0,4R/9)

But the result given by the source is 11R/24 from base...!

Now where am i wrong?

The center of mass of the cone is at R/8.
R*(3/8 - 1/4*1/8)/(1-1/4)=11/24 R

Thanks a lot...
my fault was actually i assumed the cone to be hollow but it is solid...!
so h/4 not h/3 is the location of its center of mass from base... got it now..!

## 1. What is the center of mass for a hemisphere?

The center of mass for a hemisphere is located at the geometric center, which is the midpoint of the radius and the center of the base.

## 2. How do you find the center of mass for a hemisphere?

To find the center of mass for a hemisphere, you can use the formula: x = 0, y = 0, z = 3r/8, where r is the radius of the hemisphere.

## 3. What is the center of mass for a right cone?

The center of mass for a right cone is located at the geometric center, which is the midpoint of the height and the center of the base.

## 4. How do you find the center of mass for a right cone?

To find the center of mass for a right cone, you can use the formula: x = 0, y = 0, z = h/4, where h is the height of the cone.

## 5. Can the center of mass be located outside of the object?

No, the center of mass for a hemisphere or a right cone will always be located within the object as it is the point where the mass is evenly distributed in all directions.