Determine the CM of a machine part that is a uniform cone of height h and radius R, Figure.?

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  • 9 months ago
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    Since the cone is radially symmetrical, we know the center of mass is somewhere on the axis.

    If we divide the cone into disks, each with volume dV, the center of mass is at the average:

    CM = ∫ z dV / ∫ dV

    CM = ∫ z (π r² dz) / ∫ (π r² dz)

    Using similar triangles:

    r / R = z / h

    r = R/h z

    Substituting:

    CM = ∫₀ʰ z (π (R/h z)² dz) / ∫₀ʰ (π (R/h z)² dz)

    CM = ∫₀ʰ (π R²/h² z³ dz) / ∫₀ʰ (π R²/h² z² dz)

    CM = ∫₀ʰ z³ dz / ∫₀ʰ z² dz

    CM = (¼ z⁴) |₀ʰ / (⅓ z³) |₀ʰ

    CM = (¼ h⁴) / (⅓ h³)

    CM = ¾ h

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