Wyznaczanie gęstości materii

Introduction

The main part of abstract

Exercise 2

Calculate the volumevolumevolume of the figures presented below. Provide a mathematical notation of the formulas for the volume of a given figure.

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Na rysunku przedstawiony jest w punkcie a - prostopadłościan o wymiarach a równa się 6 cm, b równa się 4 cm, c równa się 12 cm; w punkcie b - walec o promieniu podstawy r równym 3 cm i wysokości 8 cm; w punkcie c - walec na prostopadłościanie oparty podstawą, walec o średnicy 2r równej 4 cm i wysokości h równej 8 cm, prostopadłościan o wymiarach a równa się 3 cm, b równa się 2 cm, c równa się 7 cm.

a)

Calculations:

$V=a\cdot b\cdot c=...$

b)

Calculations:

$V=\pi \cdot r^2\cdot h=...$

c)

Calculations:

$V=V_1+V_2=a\cdot b\cdot c+\pi \cdot r^2\cdot h=...$

Show solution

a) $V=a·b·c=6 cm·4 cm·12 cm=288 c{m}^3$

b) $V=\mathrm{\pi }·r^2·h=3,14·{\left(3 cm\right)}^2·8 cm=226,08 c{m}^3$

c) $V=V_1+V_2=a·b·c+\mathrm{\pi }·r^2·h=3 cm·2 cm·7 cm+3,14·{\left(\frac{4 cm}{2}\right)}^2·8 cm=\left(42+100,48\right) c{m}^3=142,48 c{m}^3$

Exercise 3

Look at figures.

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Na rysunku przedstawiony jest wzorcowy sześcian o masie równej 2,7 grama i objętości równej 1 centymetr sześcienny. W punkcie a jest jeden wzorcowy sześcian, w punkcie b są dwa wzorcowe sześciany, w punkcie c są trzy wzorcowe sześciany, w punkcie b z indeksem dolnym 1 jest figura złożona z dwóch wzorcowych sześcianów stycznych jednym bokiem, w punkcie c z indeksem dolnym 1 jest figura złożona z trzech wzorcowych sześcianów, ułożonych w prostopadłościan, stycznych jednym bokiem.

1. Check the changes in the massmassmass and volumevolumevolume of the cubes presented in figures (a), (b) and (c) which are made of aluminum.

2. Compare the results with the volumes and masses of bars made after melting two cubes (bIndeks dolny 1₁) and melting three cubes (cIndeks dolny 1₁).

3. Compare the mass‑volume ratio in every variant.

Show solution

(a) $V_a=1 c{m}^3,m_a=2,7 g, \frac{m_a}{V_a}=\frac{2,7 g}{1 c{m}^3}=2,7\frac{g}{c{m}^3}$

(b) $V_b=1 c{m}^3+1 c{m}^3=2 c{m}^3, m_b=2,7 g+2,7 g=5,4 g, \frac{m_b}{V_b}=\frac{5,4 g}{2 c{m}^3}=2,7\frac{g}{c{m}^3}$

(c) $V_c=1 c{m}^3+1 c{m}^3+1 c{m}^3=3 c{m}^3, m_c=2,7 g+2,7 g+2,7 g=8,1 g, \frac{m_c}{Vc}=\frac{8,1 g}{3 c{m}^3}=2,7\frac{g}{c{m}^3}$

(d) $V_{b1}=2·1 c{m}^3=2 c{m}^3, m_{b1}=2·2,7 g=5,4 g, \frac{m_{b1}}{V_{b1}}=\frac{5,4 g}{2 c{m}^3}=2,7\frac{g}{c{m}^3}$

(e) $V_{c1}=3·1 c{m}^3=3 c{m}^3, m_{c1}=3·2,7 g=8,1 g, \frac{m_{c1}}{V_{c1}}=\frac{8,1 g}{3 c{m}^3}=2,7\frac{g}{ c{m}^3}$

1. The weight to volumevolumevolume ratio in all cases was the same and was $2,7\frac{g}{c{m}^3}.$ This ratio does not depend on the way of examining individual cubes or combining them into one whole bar (e.g. by melting) as a rod. Thus, the ratio of body massmassmass to its volume is a physical quantity that characterizes the properties of the body.

2. DensitydensityDensity is a new physical quantity describing the properties of the body. Density of the body is determined on the basis of its mass and volume.

Density can be written mathematically in the form of the following formula:

Conclusion

The measurement result differs most often from the real value of the measured quantity. We say that the result of each physical measurement is burdened with measurement uncertaintymeasurement uncertaintymeasurement uncertainty.

In our case, we can write that the densitydensitydensity of the measured cuboidcuboidcuboid was:

This uncertainty results both from the measurement method and the accuracy of the used instrument. An inaccurate reading of the results from the scale of the instrument may also be the reason of uncertainty. The properties of the tested subject can also influence the uncertainty of measurement.

DensitydensityDensity is a physical quantity that is often used to describe the physical properties of the studied physical body. For bodies of regular shaped, the volumevolumevolume determination procedure can be simplified. For this purpose, the mathematical relations that associate the volume of the body with its linear dimensions should be used. To determine the density of the body, we need to measure its mass and volume and then use the following formula:

If we know the density and the volume of the body, we can calculate its mass. For this purpose, the following formula should be used:

The body volumevolumevolume of known massmassmass and density is calculated using the formula:

This uncertainty results both from the measurement method and the accuracy of the used instrument. An inaccurate reading of the results from the scale of the instrument may also be the reason of uncertainty. The properties of the tested subject can also influence the uncertainty of measurement.

Exercises

Glossary

Keywords

densitydensitydensity

massmassmass

measurement uncertaintymeasurement uncertaintymeasurement uncertainty

volumevolumevolume