The apparatus of this lab are three cylinders (steel, copper, aluminum), a scale, and a caliper as seen below. The cylinders are the objects we are trying to find the density of, the scale measures the cylinders mass, and the caliper measures the cylinders height and diameter.
First we measured the mass, height, and diameter of all three cylinders:
Copper: diameter=1.27 cm, height=5.14 cm, mass=58.5 g
Aluminum: diameter=1.44 cm, height=4.84 cm, mass=21 g
Steel: diameter=1.44 cm, height=5.03 cm, mass=62.1 g
Next we needed to find of the density of the cylinders.
The definition of density is as stated:
Density = mass / volume
The volume of a cylinder is defined as:
Volume = π(radius^2)height
Since we can only measure the diameter of the cylinders the expression is rewritten as:
V = π(height)(diameter/2)^2
V = π((diameter^2)/4)(height)
δ = 4m / (π(d^2)h)
Because of uncertainty in or recorded measurements we must find the derivative of each variable; mass, diameter, height, and multiply it by the expected uncertainty of 0.0001 thus the equation for the uncertainty of density becomes:
δρ=|∂ρ/∂m|dm+|∂ρ/∂d|dd+|∂ρ/∂h|dh
Where:
|∂ρ/∂m| = 4 / (π(d^2)h) dm=0.0001 kg
|∂ρ/∂d| = (-8m) / (π(d^3)h) dd=0.0001 m
|∂ρ/∂h| = (-4m) / (π(d^2)(h^2)) dh=0.0001 m
After substituting the values in and adding the density with the uncertainty we get:
δρ Cooper = 8984.5 ± 174 kg/m
δρ Aluminum = 2664.2 ± 55.3 kg/m
δρ Steel = 7580.7 ± 131.9 kg/m
By the results we can conclude that the is a significant amount of error to be considered when conducting experiments.
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