Chapter 15, Problem 47Q

(a)

To determine

The mass of the nucleus of a comet with side length $10\text{km}$ and density $1000{\text{kg/m}}^3$.

Answer to Problem 47Q

Solution:

${10}^{15}\text{kg}$.

Explanation of Solution

Given data:

The side length of the comet’s nucleus is $10\text{km}$ and the density of the comet is $1000{\text{kg/m}}^{\text{3}}$.

Formula used:

Write the expression for density.

$\rho =\frac{m}{V}$

Here, $m$ and $V$ are, respectively, the mass and density of the comet’s nucleus.

Write the expression for the volume for a cube.

$V=a^3$

Here, $a^{}$ is the side length of the cube.

Explanation:

Recall the expression for the mass density of the comet’s nucleus.

$\rho =\frac{m}{V}$

Rearrange the above expression for $m$.

$m=\rho V$

Since the expression for the volume of the comet’s nucleus is $V=a^3$, the above formula becomes,

$m=\rho a^3$

Substitute $1000{\text{kg/m}}^{\text{3}}$ for $\rho$ and $10\text{km}$ for $a$ in the above expression.

$\begin{array}{c}m=1000{\text{kg/m}}^{\text{3}}{\left(10\text{ km}\left(\frac{{10}^3\text{ m}}{1\text{ km}}\right)\right)}^3\\ =1000\times {10}^{12}\text{ kg}\\ ={10}^{15}\text{kg}\end{array}$

Conclusion:

Hence, the mass of the nucleus of the comet is ${10}^{15}\text{kg}$.

(b)

To determine

The average density of the tail of the comet which has a length of ${10}^8\text{km}$ and a width of ${10}^6\text{km}$, when $1$% of the mass of the nucleus of the comet evaporates to form the tail. Also, take the air density as $1.2{\text{ kg/m}}^3$.

Answer to Problem 47Q

Solution:

${10}^{-11}{\text{kg/m}}^3$

Explanation of Solution

Given data:

The length of the tail is ${10}^8\text{km}$, width of the tail is ${10}^6\text{km}$, the mass of the tail is 1% of the mass of the comet’s nucleus and air density is $1.2{\text{ kg/m}}^3$.

Formula used:

Write the expression for density.

$\rho =\frac{m}{V}$

Here, $m$ and $V$ are, respectively, the mass and density of the comet’s nucleus.

Explanation:

The mass of the comet’s nucleus was found in the previous part to be ${10}^{15}\text{kg}$ and it is given that mass of the tail is 1% of the mass of the nucleus.

$\begin{array}{c}{m}_{tail}=1\%\text{of}m\\ =1\left(\frac{1}{100}\times {10}^{15}\text{ kg}\right)\\ ={10}^{13}\text{kg}\end{array}$

Now, volume of the tail can be calculated as:

$V_{tail}=l\times b\times h$

Here, $l\text{and}b$ are, respectively, the length and width of the tail and $h$ is the side length of the comet’s nucleus.

Substitute ${10}^8\text{km}$ for $l$, ${10}^6\text{km}$ for $b$ and $10\text{km}$ for $h$.

$\begin{array}{c}{V}_{tail}={10}^8\text{ km}\left({10}^6\text{ km}\right)\left(10\text{ km}\right)\\ ={10}^{15}{\text{km}}^{\text{3}}\left({\left(\frac{{10}^3\text{ m}}{1\text{ km}}\right)}^3\right)\\ ={10}^{24}{\text{m}}^{\text{3}}\end{array}$

Thus, recall the expression for the density of the tail.

${\rho }_{tail}=\frac{m_{tail}}{V_{tail}}$

Here, $m_{tail}$, ${\rho }_{tail}$ and $V_{tail}$ are, respectively, the mass, volume and density of the comet’s tail.

Substitute ${10}^{13}\text{kg}$ for $m_{tail}$ and ${10}^{24}{\text{m}}^{\text{3}}$ for $V_{tail}$.

$\begin{array}{c}{\rho }_{tail}=\frac{{10}^{13}\text{ km }}{{10}^{24}{\text{m}}^{\text{3}}}\\ ={10}^{-11}{\text{kg/m}}^{\text{3}}\end{array}$

Conclusion:

Hence, the average density of the tail of the comet is ${10}^{-11}{\text{kg/m}}^{\text{3}}$.

(c)

To determine

Whether the passing of the comet’s tail through Earth leads to harmful effects on the health of human beings.

Answer to Problem 47Q

Solution:

The tail of the comet does not affect the health of human beings on Earth.

Explanation of Solution

Introduction:

The air we breathe has a density of about $1.2{\text{kg/m}}^{\text{3}}$. Any gas mixed with air that does not change the density of air much will not harm the human body.

Explanation:

The density of the tail of the comet is ${10}^{-11}{\text{kg/m}}^{\text{3}}$, which is very low compared to the density of the air we breathe. When Earth passes through the tail, the air in Earth’s atmosphere also mixes with the tail, but the mixed air is not harmful for human beings to breathe.

Conclusion:

Hence, the passing of the Earth through the comet’s tail is not harmful for human life.