Open Box. Consider a hollow box with the top missing. The sides have negligible thickness and each has length L and mass m. (a) Find the x-coordinate of the center of mass.

x_{cm} = L / 2

Explanation:

The center of mass is defined by

          = 1 / M ∑ m_{i}  x_{i}

where M is the total mass of the system

in this case the system is continuous so, for which we use the density

      ρ = dm / dx

      dm = ρ dx

substituting

        x_{cm} = 1 / M ∫ x ρ dx

        x_{cm} = ρ / M ∫ x dx

we integrate and evaluate from x = 0 to x = L

        x_{cm} = ρ / M (L² /2 -0)

we introduce the density which is constant

        ρ = M / L

        x_{cm} = 1 /M (M/L)  L² / 2

        x_{cm} = L / 2

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