To find the center of gravity (also known as the center of mass) in the $z$-coordinate for a given body or system, you generally follow a systematic approach that depends on the distribution of mass. The general formula for the $z$-coordinate of the center of gravity is:

$z_{cg} = \frac{\int z \, dm}{\int dm}$

where:

• $z_{cg}$ is the $z$-coordinate of the center of gravity.
• $dm$ is an infinitesimal mass element.
• $z$ is the $z$-coordinate of the mass element.
• $\int$ denotes the integral over the entire mass distribution.

Steps to calculate the center of gravity in the $z$-coordinate:

1. Determine the mass distribution: If you have a continuous distribution, you'll integrate over the volume, surface, or length of the object depending on the geometry.

2. Use the density function: The mass element $dm$ can often be written as $\rho \, dV$ for volume, $\rho \, dA$ for surface area, or $\rho \, dl$ for length, where $\rho$ is the mass density.

   • For a volume distribution: $dm = \rho \, dV$
   • For a surface distribution: $dm = \sigma \, dA$
   • For a line distribution: $dm = \lambda \, dl$

3. Set up the integral: You would integrate $z \, dm$ over the appropriate limits. For a volume distribution, this could be:

$z_{cg} = \frac{\int_V z \rho \, dV}{\int_V \rho \, dV}$

where $V$ is the volume of the object.

4. Simplify: In many cases, if symmetry is present, this process can be simplified. For instance, for symmetric shapes, the center of gravity may lie along a central axis, and you only need to compute the integral along one coordinate.

Do you have a specific example or a particular object in mind for which you need to find the center of gravity in the $z$-coordinate? I can walk you through the calculation.

Follow-up questions:

1. What type of object or system are you dealing with? (e.g., uniform rod, disk, 3D shape)
2. Do you know the density or mass distribution of the object?
3. Are you considering a continuous distribution or a discrete set of masses?
4. Would you like to know how symmetry affects the center of gravity?
5. Are there any specific limits or boundaries to the problem you'd like to define?

Tip: If you're dealing with simple geometries (like a sphere, cylinder, or rectangular prism), symmetry can often help you directly infer the center of gravity without needing complex integrations.