A set of density rods is designed to illustrate the concept of density. The idea is to create cylinders of equal diameters and masses, but varying lengths, to show which have the largest and smallest densities.
(a) Derive a formula that will predict the ratio of the length of one rod to the length of another rod. Assume the radius and mass of each cylinder is the same and express your answer in terms of the specific gravity of each rod. (Use the following as necessary: SG1, SG2.)
(b) If you want to design a set of five density rods (made of aluminum, iron, copper, brass, and lead), determine the ratios of the lengths of each rod to the length of the lead rod, the densest material in the group. The specific gravities of the elements are approximately SGalum = 2.7, SGiron = 7.8, SGcopper = 8.9, SGbrass = 8.5, SGlead = 11.3.
Volume = pi R^2 L
so
mass = rho L pi R^2
rho = density = 1000 kg/m^3 * SG
because density of water is 1000 kg/m^3
so
mass = 1000 SG L pi R^2
if mass and R are the same then
L * SG= maas /(1000 pi R^2) = constant
so
Ln * SGn = L1 * SG1
Damon, thanks but is that for A or B and what is the answer, please star it or underline it or something that it shows the final answer for it.
Hey, I showed you how to do it. Mow use what I showed you.
To derive the formula for the ratio of the length of one rod to another, we can start with the definition of density:
Density (ρ) = Mass (m) / Volume (V)
Since the radius and mass of each cylinder are the same, we can write the volume of a cylinder in terms of its length (h) and radius (r):
Volume (V) = π * r^2 * h
Substituting this into the density formula, we get:
ρ = m / (π * r^2 * h)
We also know that specific gravity (SG) is defined as the ratio of the density of a substance to the density of water:
SG = ρ / ρ_water
Since we are assuming all the rods have the same radius (r), their densities will be proportional to their masses. Therefore, we can write:
SG = m / m_water
Rearranging this equation, we can solve for the mass of the rod:
m = SG * m_water
Substituting this into the density formula, we get:
ρ = (SG * m_water) / (π * r^2 * h)
Now, we can compare the ratio of the lengths of two rods by taking the ratio of their densities:
(length1 / length2) = sqrt(ρ2 / ρ1)
Substituting the expressions for density, we have:
(length1 / length2) = sqrt((SG2 * m_water) / (SG1 * m_water))
= sqrt(SG2 / SG1)
Therefore, the formula to predict the ratio of the length of one rod to another rod is:
(length1 / length2) = sqrt(SG2 / SG1)
For part (b), we can use the specific gravity values to determine the ratios of the lengths of each rod to the length of the lead rod:
Aluminum length ratio = sqrt(SGlead / SGalum)
Iron length ratio = sqrt(SGlead / SGiron)
Copper length ratio = sqrt(SGlead / SGcopper)
Brass length ratio = sqrt(SGlead / SGbrass)