$$\text { Find the center of mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$$
-$$r ( t ) = 8 t i + 7 t j + 3 t ^ { 2 } k , 0 \leq t \leq 1 ; \delta ( x , y , z ) = \frac { x } { \sqrt { 113 + 12 z } }$$
A) $\left( \frac { 64 } { 3 } , \frac { 56 } { 3 } , 8 \right)$
B) $( 8,7,3 )$
C) $\left( \frac { 16 } { 3 } , \frac { 14 } { 3 } , \frac { 3 } { 2 } \right)$
D) $( 16,14,6 )$

Question

Quizplus · Accepted Answer

The center of mass of a wire in three dimensions is given by the coordinates $(\bar{x}, \bar{y}, \bar{z})$, where each coordinate is the weighted average of that coordinate over the length of the wire, with the weight being the density of the wire at each point. Given the parametric representation of the wire $r(t) = 8ti + 7tj + 3t^2k$ for $0 \leq t \leq 1$ and the density function $\delta(x, y, z) = \frac{x}{\sqrt{113 + 12z}}$, we need to integrate over the curve to find the weighted average positions. The correct answer must account for the specific form of the density function and the curve, leading to the calculation of the center of mass through integration, which results in the coordinates $\left( \frac { 16 } { 3 } , \frac { 14 } { 3 } , \frac { 3 } { 2 } \right)$.

Find the center of mass of the wire that lies along the curve r and has density δ. \text { Find the center of mass of the wire that lies along the curve } r \text { and has density } \delta \text {. } Find the center of mass of the wire that lies along the curve r and has density δ.

$\text { Find the center of mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$