91影视

Given data:

• The spherical planet has mass$M$ .
• The spherical planet has volume $V$.
• The spherical planet has radius $R$.
• The spherical planet has diameter $D=2R$.

The acceleration due to gravity of a planet is the ratio of the product of the Universal Gravitational constant and mass of the planet to the square of their radius.

The equation of the density of the planet is expressed as-

$\begin{array}{c}\mathrm{蚁}=\frac{\mathrm{M}}{\mathrm{V}}\\ \mathrm{M}=\mathrm{蚁V}\end{array}$

Here, $蚁$ is the density of the planet,$M$ is the mass of the planet and$V$ is the volume of the planet.

The equation of the acceleration due to gravity is expressed as-

$g=\frac{GM}{R^2}$

Here, $g$is the acceleration due to gravity, $G$ is the gravitational constant and $R$is the radius of the spherical planet.

For $M=蚁V$and $D=2R$,

$\begin{array}{c}g=\frac{G蚁V}{{\left(\frac{D}{2}\right)}^2}\\ =\frac{4G蚁V}{D^2}鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�\left(1\right)\end{array}$

a)

From the expression of the acceleration due to gravity, the graph of $g$as a function of $D$has been described below-

This graph is applicable for all the eight major planets if the average density of the planets is constant. However, if the average density of the planet changes, then the graph will also change.

Thus, the graph of $g$versus$D$will change with the change in the average density.

b)

From the equation the equation of the average density of the planets is expressed as-

$\mathrm{蚁}=\frac{{\mathrm{gD}}^2}{4\mathrm{GV}}鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�\left(2\right)$

As the planets are spherical in nature, then the equation of the volume of the planet can be expressed as-

localid="1655794647879" $V=\frac{4}{3}蟺R^3$

Then equation becomes-

$\begin{array}{c}蚁=\frac{g{D}^2}{4G脳\left(\frac{4}{3}蟺R^3\right)}\\ =\frac{3g{D}^2}{16G蟺R^3}\end{array}$

For $D=2R$,

$\begin{array}{c}蚁=\frac{3g{D}^2}{16G蟺{\left(\frac{D}{2}\right)}^3}\\ =\frac{24g}{16G蟺D}鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�鈰�\left(3\right)\end{array}$

For the planet Mercury,

Substituting $\mathbf{D}\mathbf{=}\mathbf{4879}\mathbf{\text{鈥塳尘}}$ , $G=6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}$and $g=3.7\text{鈥尘}/{\text{s}}^{\text{2}}$in the equation 3)

$\begin{array}{c}\mathrm{蚁}=\frac{24脳3.7\text{鈥尘}/{\text{s}}^{\text{2}}}{16脳6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳3.14脳4879\text{鈥塳尘}}\\ =\frac{24脳3.7\text{鈥尘}/{\text{s}}^{\text{2}}}{3.35脳{10}^{鈭�09}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳4879脳{10}^3\text{鈥尘}}\\ =\frac{24脳3.7\text{鈥尘}/{\text{s}}^{\text{2}}}{0.0163\text{鈥塏}鈰�{\text{m}}^3/{\text{kg}}^{\text{2}}}\\ =5447.85{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\end{array}$

Hence, further as,

$\begin{array}{c}蚁=5447.85{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\\ =5447.85{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/{\text{m}}^3鈰�{\text{s}}^{\text{2}}脳\frac{1}{\text{kg}鈰�{\text{m}}^{\text{2}}/{\text{s}}^{\text{2}}}\\ =5447.85\text{鈥塳驳}/{\text{m}}^{\text{3}}\end{array}$

For the planet Venus,

Substituting $D=12104\text{鈥塳尘}$, $G=6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}$and $g=8.9\text{鈥尘}/{\text{s}}^{\text{2}}$in the equation $\left(3\right)$,

$\begin{array}{c}蚁=\frac{24脳8.9\text{鈥尘}/{\text{s}}^{\text{2}}}{16脳6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳3.14脳12014\text{鈥塳尘}}\\ =\frac{24脳8.9\text{鈥尘}/{\text{s}}^{\text{2}}}{3.35脳{10}^{鈭�09}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳12104脳{10}^3\text{鈥尘}}\\ =\frac{24脳3.7\text{鈥尘}/{\text{s}}^{\text{2}}}{0.0405\text{鈥塏}鈰�{\text{m}}^3/{\text{kg}}^{\text{2}}}\\ =2192.59{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\end{array}$

Hence, further as,

$\begin{array}{c}蚁=2192.59{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\\ =2192.59{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/{\text{m}}^3鈰�{\text{s}}^{\text{2}}脳\frac{1}{\text{kg}鈰�{\text{m}}^{\text{2}}/{\text{s}}^{\text{2}}}\\ =2192.59\text{鈥塳驳}/{\text{m}}^{\text{3}}\end{array}$

For the planet Earth,

Substituting $D=12756\text{鈥塳尘}$, $G=6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}$ and$g=9.8\text{鈥尘}/{\text{s}}^{\text{2}}$in the equation$\left(3\right)$

$\begin{array}{c}蚁=\frac{24脳9.8\text{鈥尘}/{\text{s}}^{\text{2}}}{16脳6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳3.14脳12756\text{鈥塳尘}}\\ =\frac{24脳9.8\text{鈥尘}/{\text{s}}^{\text{2}}}{3.35脳{10}^{鈭�09}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳12756脳{10}^3\text{鈥尘}}\\ =\frac{24脳9.8\text{鈥尘}/{\text{s}}^{\text{2}}}{0.0427\text{鈥塏}鈰�{\text{m}}^3/{\text{kg}}^{\text{2}}}\\ =2079.62{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\end{array}$

Hence, further as,

$\begin{array}{c}蚁=2079.62{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\\ =2079.62{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/{\text{m}}^3鈰�{\text{s}}^{\text{2}}脳\frac{1}{\text{kg}鈰�{\text{m}}^{\text{2}}/{\text{s}}^{\text{2}}}\\ =2079.62\text{鈥塳驳}/{\text{m}}^{\text{3}}\end{array}$

For the planet Mars,

Substituting $D=6792\text{鈥塳尘}$, $G=6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}$and $g=3.7\text{鈥尘}/{\text{s}}^{\text{2}}$in the equation $\left(3\right)$

$\begin{array}{c}蚁=\frac{24脳3.7\text{鈥尘}/{\text{s}}^{\text{2}}}{16脳6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳3.14脳6792\text{鈥塳尘}}\\ =\frac{24脳3.7\text{鈥尘}/{\text{s}}^{\text{2}}}{3.35脳{10}^{鈭�09}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳6792脳{10}^3\text{鈥尘}}\\ =\frac{24脳3.7\text{鈥尘}/{\text{s}}^{\text{2}}}{0.0227\text{鈥塏}鈰�{\text{m}}^3/{\text{kg}}^{\text{2}}}\\ =3911.89{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\end{array}$

Hence, further as,

$\begin{array}{c}蚁=3911.89{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\\ =3911.89{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/{\text{m}}^3鈰�{\text{s}}^{\text{2}}脳\frac{1}{\text{kg}鈰�{\text{m}}^{\text{2}}/{\text{s}}^{\text{2}}}\\ =3911.89\text{鈥塳驳}/{\text{m}}^{\text{3}}\end{array}$

For the planet Jupiter,

Substituting $D=142,984\text{鈥塳尘}$, $G=6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}$and $g=23.1\text{鈥尘}/{\text{s}}^{\text{2}}$in the equation 3),

$\begin{array}{c}蚁=\frac{24脳23.1\text{鈥尘}/{\text{s}}^{\text{2}}}{16脳6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳3.14脳142984\text{鈥塳尘}}\\ =\frac{24脳23.1\text{鈥尘}/{\text{s}}^{\text{2}}}{3.35脳{10}^{鈭�09}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳142984脳{10}^3\text{鈥尘}}\\ =\frac{24脳23.1\text{鈥尘}/{\text{s}}^{\text{2}}}{0.4789\text{鈥塏}鈰�{\text{m}}^3/{\text{kg}}^{\text{2}}}\\ =1157.65{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\end{array}$

Hence, further as,

$\begin{array}{c}蚁=1157.65{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\\ =1157.65{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/{\text{m}}^3鈰�{\text{s}}^{\text{2}}脳\frac{1}{\text{kg}鈰�{\text{m}}^{\text{2}}/{\text{s}}^{\text{2}}}\\ =1157.65\text{鈥塳驳}/{\text{m}}^{\text{3}}\end{array}$

For the planet Saturn,

Substituting $D=120536\text{鈥塳尘}$, $G=6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}$and $g=9.0\text{鈥尘}/{\text{s}}^{\text{2}}$in the equation 3),

$\begin{array}{c}蚁=\frac{24脳9.0\text{鈥尘}/{\text{s}}^{\text{2}}}{16脳6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳3.14脳120536\text{鈥塳尘}}\\ =\frac{24脳9.0\text{鈥尘}/{\text{s}}^{\text{2}}}{3.35脳{10}^{鈭�09}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳120536脳{10}^3\text{鈥尘}}\\ =\frac{24脳9.0\text{鈥尘}/{\text{s}}^{\text{2}}}{0.4037\text{鈥塏}鈰�{\text{m}}^3/{\text{kg}}^{\text{2}}}\\ =535.05{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\end{array}$

Hence, further as,

$\begin{array}{c}蚁=535.05{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\\ =535.05{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/{\text{m}}^3鈰�{\text{s}}^{\text{2}}脳\frac{1}{\text{kg}鈰�{\text{m}}^{\text{2}}/{\text{s}}^{\text{2}}}\\ =535.05\text{鈥塳驳}/{\text{m}}^{\text{3}}\end{array}$

For the planet Uranus,

Substituting$\mathbf{D}\mathbf{=}\mathbf{51118}\mathbf{\text{鈥塳尘}}$, $G=6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}$and $g=8.7\text{鈥尘}/{\text{s}}^{\text{2}}$in the equation 3),

$\begin{array}{c}蚁=\frac{24脳8.7\text{鈥尘}/{\text{s}}^{\text{2}}}{16脳6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳3.14脳51118\text{鈥塳尘}}\\ =\frac{24脳8.7\text{鈥尘}/{\text{s}}^{\text{2}}}{3.35脳{10}^{鈭�09}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳51118脳{10}^3\text{鈥尘}}\\ =\frac{24脳8.7\text{鈥尘}/{\text{s}}^{\text{2}}}{0.1712\text{鈥塏}鈰�{\text{m}}^3/{\text{kg}}^{\text{2}}}\\ =1219.62{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\end{array}$

Hence, further as,

$\begin{array}{c}蚁=1219.62{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\\ =1219.62{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/{\text{m}}^3鈰�{\text{s}}^{\text{2}}脳\frac{1}{\text{kg}鈰�{\text{m}}^{\text{2}}/{\text{s}}^{\text{2}}}\\ =1219.62\text{鈥塳驳}/{\text{m}}^{\text{3}}\end{array}$

For the planet Neptune,

Substituting $D=49528\text{鈥塳尘}$, $G=6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}$and $g=11.0\text{鈥尘}/{\text{s}}^{\text{2}}$in the equation 3),

$\begin{array}{c}蚁=\frac{24脳11.0\text{鈥尘}/{\text{s}}^{\text{2}}}{16脳6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳49525脳51118\text{鈥塳尘}}\\ =\frac{24脳11.0\text{鈥尘}/{\text{s}}^{\text{2}}}{3.35脳{10}^{鈭�09}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳49525脳{10}^3\text{鈥尘}}\\ =\frac{24脳11.0\text{鈥尘}/{\text{s}}^{\text{2}}}{0.1659\text{鈥塏}鈰�{\text{m}}^3/{\text{kg}}^{\text{2}}}\\ =1591.32{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\end{array}$

Hence, further as,

$\begin{array}{c}蚁=1591.32{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/\text{N}鈰�{\text{m}}^3鈰�{\text{s}}^{\text{2}}\\ =1591.32{\text{鈥塳驳}}^{\text{2}}鈰�\text{m}/{\text{m}}^3鈰�{\text{s}}^{\text{2}}脳\frac{1}{\text{kg}鈰�{\text{m}}^{\text{2}}/{\text{s}}^{\text{2}}}\\ =1591.32\text{鈥塳驳}/{\text{m}}^{\text{3}}\end{array}$

Thus, the average density of the planets in the decreasing order are of the planets Mercury, Mars, Venus, Earth, Neptune, Uranus, Jupiter and Saturn which are $5447.85\text{鈥塳驳}/{\text{m}}^{\text{3}}$,$3911.89\mathrm{kg}/{\text{m}}^{\text{3}}$, $2192.59\mathrm{kg}/{\text{m}}^{\text{3}}$,$2079.62\text{鈥塳驳}/{\text{m}}^{\text{3}}$, $1591.32\text{鈥塳驳}/{\text{m}}^{\text{3}}$,$1219.62\text{鈥塳驳}/{\text{m}}^{\text{3}}$, $1157.65\text{鈥塳驳}/{\text{m}}^{\text{3}}$,$535.05\text{鈥塳驳}/{\text{m}}^{\text{3}}$respectively.

c)

Along with the planet Earth, the other eight major planets are also not uniform as they spin around their axis and around the sun. However, the force of spin mainly acts against the gravity and it eventually causes the planets to more bulge out around their respective equator.

Thus, due to the spinning force, the eight major planets have also variable average densities.

d)

If the average density of Saturn is same as of the Earth, then from the equation 3), the equation of the acceleration due to gravity of Saturn can be expressed as,

$蚁=\frac{24g}{16G蟺D}$

For $蚁=2079.62\text{鈥塳驳}/{\text{m}}^{\text{3}}$, $D=120536\text{鈥塳尘}$and $G=6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}$,

$\begin{array}{c}2079.62\text{鈥塳驳}/{\text{m}}^{\text{3}}=\frac{24g}{16脳6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳3.14脳120536\text{鈥塳尘}}\\ g=\frac{16脳6.67脳{10}^{鈭�11}\text{鈥塏}鈰�{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}脳3.14脳120536脳{10}^3\text{鈥尘}脳2079.62\text{鈥塳驳}/{\text{m}}^{\text{3}}}{24}\\ =34.99\text{鈥塏}鈰�{\text{m}}^{\text{2}}鈰�\text{m}鈰�\text{kg}/{\text{kg}}^{\text{2}}鈰�{\text{m}}^{\text{3}}\\ =34.99\text{鈥塏}/\text{kg}\end{array}$

Hence, further as,

$\begin{array}{c}g=34.99\text{鈥塏}/\text{kg}\\ =34.99\text{鈥塳驳m}/{\text{kgs}}^{\text{2}}\\ =34.99\text{鈥尘}/{\text{s}}^{\text{2}}\end{array}$

Thus, the value of $g$the at the Saturn鈥檚 surface is $34.99\text{鈥尘}/{\text{s}}^{\text{2}}$.