Suppose that the density of the wire in Example 4 is = 1 + k |cos

Suppose that the density of the wire in Example 4 is δ = 1 + k |cos θ| (k constant). Find the center of mass.

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EXAMPLE 4 Find the center of mass (centroid) of a thin wire of constant density 8 shaped like a semicircle of radius a. We model the wire with the semicircle y = Va²-x² (Figure 6.54). The dis- tribution of mass is symmetric about the y-axis, so x = 0. To find y, we imagine the wire divided into short subarc segments. If (x, y) is the center of mass of a subarc and is the angle between the x-axis and the radial line joining the origin to (x, y), then ỹ = a sin 0 is a function of the angle measured in radians (see Figure 6.54a). The length ds of the subarc containing (x, y) subtends an angle of de radians, so ds = a de. Thus a typical subarc segment has these relevant data for calculating y: length: ds a do mass: dm = distance of c.m. to x-axis: 8 ds y = a sin 0. = δα αθ Mass per unit length times length