Chapter 2, Problem 107A

Interpretation Introduction

Interpretation:

The density of the black hole needs to be determined.

Concept introduction:

Density of a sample of matter or an object is equal to the mass of the sample divided by the volume of the sample.

$\text{Density}=\frac{\text{Mass}}{\text{Volume}}$

Answer to Problem 107A

Density of the black hole is $4.5273\times {10}^{10}{\text{g/cm}}^{\text{3}}$.

Explanation of Solution

Given information:

The mass of black hole is about $500$ times the mass of Sun. The volume is same as moon.

Also,

$\begin{array}{l}\text{Mass}\text{of}\text{the}\text{Sun}=\text{1}{\text{.9891×10}}^{\text{30}}\text{kg}\\ \text{Volume}\text{of}\text{the}\text{Moon}=\text{2}{\text{.1968×10}}^{\text{10}}{\text{km}}^{\text{3}}\end{array}$

Calculation: Mass of the black hole can be determined by multiplying the mass of the Sun by $500$.

$\begin{array}{c}\text{Mass}\text{of}\text{black}\text{hole}=1.9891\times {10}^{30}\text{kg}\times 500\\ =9.9455\times {10}^{32}\text{kg}\end{array}$

As the volume of the black hole is equal to the volume of the moon, the unit should be converted to cm³.

$\begin{array}{c}\text{Volume}\text{of}\text{black}\text{hole}=\text{Volume}\text{of}\text{the}\text{moon}\\ =2.1968\times {10}^{10}{\text{km}}^{\text{3}}\end{array}$

Now,

$\begin{array}{c}\text{Density}=\frac{\text{Mass}}{\text{Volume}}\\ =\frac{9.9455\times {10}^{32}\text{kg}}{2.1968\times {10}^{10}{\text{km}}^{\text{3}}}\times \frac{\text{1000}\text{g}}{\text{1}\text{kg}}\times {\left(\frac{\text{1}\text{km}}{\text{1000}\text{m}}\right)}^3\times {\left(\frac{\text{1}\text{m}}{\text{100}\text{cm}}\right)}^3\\ =4.5273\times {10}^{10}{\text{g/cm}}^{\text{3}}\end{array}$

Hence, the density of the black hole is $4.5273\times {10}^{10}{\text{g/cm}}^{\text{3}}$.

Conclusion

Density of the black hole is $4.5273\times {10}^{10}{\text{g/cm}}^{\text{3}}$.