Conversion generally means changing the units, but not the quantity. So, you can’t convert between mass density and force per cubic meter. But if the only force acting on a mass is gravity, you can calculate the force per cubic meter from the density.

## Mass Density and Force Density

Mass density, ρ, is the mass per unit volume, ρ = m/V. You can measure this in kilograms per cubic meter. You can also talk about the density of other quantities such as force. The concept of force density arises in both continuum mechanics and electrostatics. The force density, or force per unit volume, f, is the net force (F) on a region of matter divided by the volume (V) that contains it: f = F/V. If the force is in Newtons (N) and the volume is in cubic meters, f has units N/m^3.

## Calculating the Force Density

If the only force on the mass is gravity, then the force density is equal to the mass density times the acceleration due to gravity, g = 9.81 m/s^2: f=ρg. This is analogous to calculating weight (W) in Newtons from mass (m) in kilograms: W=mg. It can be confusing, since in countries other than the U.S., the word kilogram also means the weight of one kilogram of matter. This is equal to 9.8 Newtons and approximately 2.2 pounds.

#### References

- Mechanics; Benjamin Crowell
- Physics; Douglass C. Giancoli
- University of Cambridge: Dynamics
- MIT: Macroscopic Force Densities

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