Question

(a) Write the formulas for the center of mass (¯x, y,¯ z¯) of a thin wire in the shape of a

space curve C if the wire has density rho(x, y, z) function .
(b) Find the center of mass of a wire in the shape of the helix ~r(t) =< 2 sin t, 2 cos t, 3t >
if the density is constant K. And. (0, 0, 3π)

Answer 1

(a) The center of mass for a thin wire in the shape of a space curve ( C ), with density function
`\( \rho(x, y, z) \)`, is given by
`\( \left((\int_C x \rho ds)/(\int_C \rho ds)`,
`(\int_C y \rho ds)/(\int_C \rho ds)`,
`(\int_C z \rho ds)/(\int_C \rho ds)\right) \)`.

(b) The center of mass for a helix
`~r(t) = < 2sin(t), 2cos(t), 3t >` with constant density ( K ) is
`\( \left(0, 0, (3\pi)/(2)\right) \)`.

Step-by-step explanation:

In the first step, determining the center of mass
`(\( \bar{x}, \bar{y}, \bar{z} \))` of a thin wire in the shape of a space curve involves integrating the product of each coordinate with the density function along the curve. The numerator of each coordinate's average involves the integral of the product of the coordinate and the density function, while the denominator is the integral of the density function alone.

This ratio gives the weighted average of each coordinate, providing the center of mass for the wire. For the second step, finding the center of mass of a helix represented by
`~r(t) = < 2sin(t), 2cos(t), 3t >` with constant density (K) involves integrating the coordinates along the helix and dividing by the total mass.

The resulting center of mass is
`(\( \bar{x}, \bar{y}, \bar{z} \)) = (0, 0, \( (3\pi)/(2) \))`. This implies that the center of mass is at the origin in the xy-plane, and the z-coordinate is
`\( (3\pi)/(2) \)`, indicating the distribution of mass along the helix.