Question

density homework

1. consider two metal cubes, cube 1 and cube 2. (make sure you fully understand these before commencing.)

a. if the two cubes have the same mass, but cube 1 has half the volume as cube 2, which has a greater density?
b. the two cubes are both solid copper, but cube 1 has twice the volume as cube 2. which has a greater density?
c. cube 1 has a mass of 5 kg and cube 2 has a mass of 20 kg. if they occupy the same volume, which has a greater density?
d. cube 1 has a greater density than cube 2. which occupies a larger volume: a 2 - kg block of cube 1 or a 2 - kg block of cube 2?

1. a 12.0 cm³ bar of lead metal has a mass of 136 g. what is the density of lead?
2. liquid mercury has a density of 13.546 g/cm³. what is the mass of 5.0 cm³ of mercury?

Explanation:

Step1: Recall density formula

The density formula is $
ho=\frac{m}{V}$, where $
ho$ is density, $m$ is mass, and $V$ is volume.

a.

Step1: Analyze given conditions

Given $m_1 = m_2$ and $V_1=\frac{1}{2}V_2$. Using the density formula $
ho=\frac{m}{V}$, for cube 1, $
ho_1=\frac{m_1}{V_1}$, and for cube 2, $
ho_2=\frac{m_2}{V_2}$. Substituting $m_1 = m_2$ and $V_1=\frac{1}{2}V_2$ into the density - expressions, we get $
ho_1=\frac{m_1}{\frac{1}{2}V_2}=2\frac{m_1}{V_2}$ and $
ho_2=\frac{m_2}{V_2}$. Since $m_1 = m_2$, $
ho_1>
ho_2$. So cube 1 has a greater density.

b.

Step1: Recall property of same - material

The density of a homogeneous material is a characteristic property and does not depend on the volume. Since both cubes are solid copper (same material), their densities are the same.

c.

Step1: Analyze given conditions

Given $V_1 = V_2$, $m_1 = 5kg$ and $m_2 = 20kg$. Using the density formula $
ho=\frac{m}{V}$, for cube 1, $
ho_1=\frac{m_1}{V_1}$, and for cube 2, $
ho_2=\frac{m_2}{V_2}$. Substituting $V_1 = V_2$, $m_1 = 5kg$ and $m_2 = 20kg$ into the density - expressions, we know that $
ho_2=\frac{20}{V_2}$ and $
ho_1=\frac{5}{V_1}$. Since $V_1 = V_2$, $
ho_2>
ho_1$. So cube 2 has a greater density.

d.

Step1: Rearrange density formula for volume

From $
ho=\frac{m}{V}$, we can get $V=\frac{m}{
ho}$. Given $m_1 = m_2=2kg$ and $
ho_1>
ho_2$. Substituting into the volume formula, $V_1=\frac{m_1}{
ho_1}$ and $V_2=\frac{m_2}{
ho_2}$. Since $m_1 = m_2$ and $
ho_1>
ho_2$, $V_2>V_1$. So the 2 - kg block of cube 2 occupies a larger volume.

2.

Step1: Apply density formula

Given $m = 116g$ and $V = 12.0cm^3$. Using the density formula $
ho=\frac{m}{V}$, we substitute the values: $
ho=\frac{116g}{12.0cm^3}\approx9.67g/cm^3$.

3.

Step1: Rearrange density formula for mass

From $
ho=\frac{m}{V}$, we can get $m=
ho V$. Given $
ho = 13.546g/cm^3$ and $V = 5.0cm^3$. Substitute the values: $m=(13.546g/cm^3)\times5.0cm^3 = 67.73g$.

Answer:

a. Cube 1
b. They have the same density
c. Cube 2
d. The 2 - kg block of cube 2

1. $9.67g/cm^3$
2. $67.73g$