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(a) Write the formulas similar to Equations 4 for the center of mass $ (\bar{x}, \bar{y}, \bar{z}) $ of a thin wire in the shape of a space curve $ C $ if the wire has density function $ \rho(x, y, z) $.

(b) Find the center of mass of a wire in the shape of the helix $ x = 2 \sin t $, $ y = 2 \cos t $, $ z = 3t $, $ 0 \leqslant t \leqslant 2\pi $, if the density is a constant $ k $.

a) $$

\overline{x}=\frac{1}{m} \int_{C} x \rho(x, y) d s

$$

and

$$

\overline{y}=\frac{1}{m} \int_{C} y \rho(x, y) d s

$$

and

$$

\overline{z}=\frac{1}{m} \int_{C} z \rho(x, y) d s

$$

b) The center of mass of the helix is $(0,0,3 \pi)$

Vector Calculus

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