To find the mass of an aluminum rod without a scale, one can calculate its volume using its dimensions and then apply the density of aluminum. The discussion highlights that the rod's shape, likely a circular cylinder, is crucial for volume calculation. Some participants suggest alternative methods, such as applying a constant force and measuring acceleration, though this may still require a scale for force measurement. The conversation also touches on using Coulomb's law or Hooke's Law for different acceleration methods. Ultimately, understanding the rod's dimensions and density is key to determining its mass.