Density and error/uncertainty

[cod3blu3]

Requesting some help. In my physics lab this week we were required to measure a number of different spheres, find the mean of their diameters, find volume, and finally find density.

The whole purpose of the lab is to find the density and its errors using something about partial derivatives. Now bear with me here, I'm just starting calculus 2 this semester and as far as i know, we haven't learned anything about partial derivatives yet. My physics professor is one of those guys who uses all of these complicated words and formulas when he explains something and all i really want is a straight forward answer "for dummies"

So let's start with an example, and someone please walk me though how to solve it.

In lab we measured a ball that was 1cm in diameter.

the 6 separate measurements were 1.000cm,1.005cm,1.010cm,1.005cm,1.000cm,1.010cm

giving:
mean = 1.005
std dev = .0044721
mass = 1.8g +/- .1g

Can someone please 1) calculate volume with error. 2) calculate density with its error. 3) explain in an easy to understand manner?

All i really need to know is how to find the density and its error for this lab.


sorry if this is in the wrong place and for not using the format provided.

 

[jbsteele]

1) To calculate the volume with its error, we can use the formula for the volume of a sphere which is V = (4/3)*pi*r^3. Since the diameter of the sphere is 1cm, we can take the radius to be 0.5cm. Therefore, V = (4/3)*pi*(0.5cm)^3 = 0.523 cm^3. The error in this volume is given by the standard deviation of the diameter measurements. We can calculate the error in the volume as (ΔV/V)*100 = (0.0044721/1.005)*100 = 4.46%. Therefore, the volume of the sphere is 0.523 cm^3 with an error of 4.46%. 2) To calculate the density with its error, we can use the formula for density which is ρ = m/V. In this case, m is the mass of the sphere which is 1.8g +/- .1g, and V is the volume which we just calculated to be 0.523 cm^3 with an error of 4.46%. Therefore, the density of the sphere is ρ = 1.8g/0.523 cm^3 = 3.44 g/cm^3 with an error of 5.9%. 3) Density is a measure of how much mass a certain object has per unit of volume. To calculate the density of an object, we need to find its mass and volume. The mass can be measured directly by using a balance, while the volume can be calculated using the formula for the volume of a sphere (V = (4/3)*pi*r^3). The density of the object is then calculated using the formula ρ = m/V, with the error being calculated using partial derivatives.