Determine the CM of a machine part that is a uniform cone of height h and radius

Question

Determine the CM of a machine part that is a uniform cone of height h and radius R. [Hint: Divide the cone into an infinite number of disks of thickness dz, one of which is shown at the figure (Figure 1) .]Express your answers in terms of the variables

Determine the CM of a machine part that is a uniform cone of height h and radius R. [Hint: Divide the cone into an infinite number of disks of thickness dz, one of which is shown at the figure (Figure 1) .]Express your answers in terms of the variables h and R. Enter your answers numerically separated by commas.

Explanation / Answer

We know from the symmetry that the center of mass lies on the z-axis:

                        xCM = 0,  yCM = 0.

For a differential element we use a circle at a height z, thickness dz, and radius r = (R/h)z.

If r is the density of the cone, the mass of this element is dm = rprdz.

We integrate over the cone to find zCM.

                         zCM = (

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