Physics, 12.11.2020 14:00 Jerrygordon212

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 ?

Answers: 1

Show answers

Answers

Answer from: itsmichaelhere1

The percentage uncertainty in his calculated value of density is $\pm 0.713\,\%$ .

Explanation:

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

$\Delta \rho \approx \frac{\partial \rho}{\partial m}\cdot \Delta m + \frac{\partial \rho}{\partial d}\cdot \Delta d + \frac{\partial \rho}{\partial l}\cdot \Delta l$ (1)

Where:

$\frac{\partial \rho}{\partial m}$ - Partial derivative of the density with respect to mass, measured in $\frac{1}{mm^{3}}$ .

$\frac{\partial \rho}{\partial d}$ - Partial derivative of the density with respect to diameter, measured in grams per cubic milimeter.

$\frac{\partial \rho}{\partial l}$ - Partial derivative of the density with respect to length, measured in grams per cubic milimeter.

$\Delta m$ - Mass uncertainty, measured in grams.

$\Delta d$ - Diameter uncertainty, measured in milimeters.

$\Delta l$ - Length uncertainty, measured in milimeters.

$\Delta \rho$ - Density uncertainty, measured in grams per cubic milimeters.

Partial derivatives are, respectively:

$\frac{\partial \rho}{\partial m} = \frac{4}{\pi\cdot d^{2}\cdot l}$ (2)

$\frac{\partial \rho}{\partial d} = -\frac{8\cdot m}{\pi\cdot d^{3}\cdot l}$ (3)

$\frac{\partial \rho}{\partial l} = - \frac{4\cdot m}{\pi\cdot d^{2}\cdot l^{2}}$ (4)

And we expand (1) as follows:

$\Delta \rho \approx \frac{4\cdot \Delta m}{\pi\cdot d^{2}\cdot l} - \frac{8\cdot m\cdot \Delta d}{\pi\cdot d^{3}\cdot l}-\frac{4\cdot m\cdot \Delta l}{\pi\cdot d^{2}\cdot l^{2}}$

$\Delta \rho \approx \left(\frac{4}{\pi\cdot d^{2}\cdot l}\right)\cdot \left(\Delta m -\frac{m\cdot \Delta d}{d}-\frac{m \cdot \Delta l}{l} \right)$ (5)

If we know that $d = 2\,mm$ , $l = 25\,mm$ , $m = 6.2\,g$ , $\Delta m = \pm 0.1\,g$ , $\Delta d = \pm 0.01\,mm$ and $\Delta l = \pm 0.1\,mm$ , then the absolute uncertainty is:

$\Delta \rho \approx \pm\left[\frac{4}{\pi\cdot (2\,mm)^{2}\cdot (25\,mm)} \right]\cdot \left[(0.1\,g)-\frac{(6.2\,g)\cdot (0.01\,mm)}{2\,mm} -\frac{(6.2\,g)\cdot (0.1\,mm)}{25\,mm} \right]$

$\Delta \rho \approx \pm 5.628\times 10^{-4}\,\frac{g}{mm^{3}}$

And the expected density is:

$\rho = \frac{4\cdot m}{\pi\cdot d^{2}\cdot l}$ (6)

$\rho = \frac{4\cdot (6.2\,g)}{\pi\cdot (2\,mm)^{2}\cdot (25\,mm)}$

$\rho \approx 78.941\times 10^{-3}\,\frac{g}{mm^{3}}$

The percentage uncertainty in his calculated value of density is:

$\%e = \frac{\Delta \rho}{\rho}\times 100\,\%$ (7)

If we know that $\Delta \rho \approx \pm 5.628\times 10^{-4}\,\frac{g}{mm^{3}}$ and $\rho \approx 78.941\times 10^{-3}\,\frac{g}{mm^{3}}$ , then the percentage uncertainty is:

$\%e = \frac{\pm 5.628\times 10^{-4}\,\frac{g}{mm^{3}} }{78.941\times 10^{-3}\,\frac{g}{mm^{3}} }\times 100\,\%$

$\%e = \pm 0.713\,\%$

The percentage uncertainty in his calculated value of density is $\pm 0.713\,\%$ .

Answer from: Quest

explanation:

as the height from which the ruler was dropped increased, the speed of the ruler at ryan's hand increased.

Answer from: Quest

An atom (ours, not carroll's) consists of a positively charged nucleus, surrounded by one or more negatively charged particles called electrons. the positive charges equal the negative charges, so the atom has no overall charge; it is electrically neutral. most of an atom's mass is in its nucleus; the mass of an electron is only 1/1836 the mass of the lightest nucleus, that of hydrogen. although the nucleus is heavy, it is quite small compared with the overall size of an atom. the radius of a typical atom is around 0.1 to 0.25 nanometres (nm), whereas the radius of a nucleus is about 10-6 nm. * if an atom were enlarged to the size of the earth, its nucleus would be only 200 feet in diameter and could easily rest inside a small football stadium.read more on -

Another question on Physics

Physics, 22.06.2019 06:30

In positive numbers less than 1, the zeros between the decimal point and a non-zero number are blank significant

Answers: 1

Answer

Physics, 22.06.2019 09:00

What occurs during a combustion reaction

Answers: 3

Answer

Physics, 22.06.2019 14:00

Una carga puntual de 3 x 10-6 c se coloca a 12 cm de una segunda carga puntual de - 1,5 x 10-6 c. calcular la magnitud fuerza eléctrica entre las cargas

Answers: 2

Answer

Physics, 23.06.2019 01:00

Use vsepr theory to predict the molecular geometry of the molecule hcn.

Answers: 3

Answer

You know the right answer?

A student determines the density ρ of steel by taking measurements from a steel wire Mass- 6.2 +-0....

Questions