A student determines the density ρ of steel by taking measurements from a steel wire Mass- 6.2 +-0.1g
Length- 25.0 +-0.1m
Diameter- 2.00 +-0.01mm
He uses the equation ρ= 4m/πd^2l
What is the percentage uncertainty in his calculated value of density ?

The percentage uncertainty in his calculated value of density is .

Explanation:

We can estimate the absolute uncertainty by the definition of total differential. That is:

(1)

Where:

- Partial derivative of the density with respect to mass, measured in .

- Partial derivative of the density with respect to diameter, measured in grams per cubic milimeter.

- Partial derivative of the density with respect to length, measured in grams per cubic milimeter.

- Mass uncertainty, measured in grams.

- Diameter uncertainty, measured in milimeters.

- Length uncertainty, measured in milimeters.

- Density uncertainty, measured in grams per cubic milimeters.

Partial derivatives are, respectively:

(2)

(3)

(4)

And we expand (1) as follows:

(5)

If we know that , , , , and , then the absolute uncertainty is:

And the expected density is:

(6)

The percentage uncertainty in his calculated value of density is:

(7)

If we know that and , then the percentage uncertainty is:

The percentage uncertainty in his calculated value of density is .