Solve the problem.
-The shape and density of a thin shell are indicated below. Find the coordinates of the center of mass.
Shell: "nose" of the paraboloid $x ^ { 2 } + y ^ { 2 } = 2 z$ cut by the plane $z = 3$
Density: $\delta = \sqrt { x ^ { 2 } + y ^ { 2 } + 1 }$
A) $\left( 0,0 , \frac { 15 } { 8 } \right)$
B) $\left( 0,0 , \frac { 96 } { 11 } \right)$
C) $\left( 0,0 , \frac { 9 } { 11 } \right)$
D) $( 0,0,8 )$

Question

Quizplus · Accepted Answer

The center of mass of a symmetric object like the given paraboloid shell, which is symmetric about the z-axis and has a density that depends only on the distance from the z-axis, will lie on the z-axis itself, hence the x and y coordinates of the center of mass are 0. The calculation of the z-coordinate involves integrating over the volume of the shell, taking into account the given density. The symmetry and the nature of the density function simplify the problem, leading to the correct answer being the one that maintains this symmetry and accounts for the density distribution, which is option A.

Solve the Problem x2+y2=2zx ^ { 2 } + y ^ { 2 } = 2 zx2+y2=2z

Solve the Problem $x ^ { 2 } + y ^ { 2 } = 2 z$