Solve the problem.
-Find the center of mass of the hemisphere of constant density bounded $z = \sqrt { 36 - x ^ { 2 } - y ^ { 2 } }$ and the $x y - p l a n e$.
A) $\bar { x } = 0 , \bar { y } = 0 , \bar { z } = \frac { 9 } { 4 }$
B) $\bar { x } = 0 , \bar { y } = 0 , \bar { z } = 4$
C) $\bar { x } = 0 , \bar { y } = 0 , \bar { z } = \frac { 3 } { 2 }$
D) $\bar { x } = 0 , \bar { y } = 0 , \bar { z } = 2$

Question

Quizplus · Accepted Answer

The center of mass of a solid hemisphere can be found using symmetry and integration. Due to symmetry, $\bar{x}$ and $\bar{y}$ are 0 because the mass is uniformly distributed around the z-axis. The formula for the z-coordinate of the center of mass of a hemisphere is $\bar{z} = \frac{3R}{8}$ for a hemisphere of radius $R$. Given $R = 6$, $\bar{z} = \frac{3 \times 6}{8} = \frac{9}{4}$.

Solve the Problem z=36−x2−y2z = \sqrt { 36 - x ^ { 2 } - y ^ { 2 } }z=36−x2−y2​

Solve the Problem $z = \sqrt { 36 - x ^ { 2 } - y ^ { 2 } }$