Chapter 4, Problem 36Q

To determine

The gravitational force exerted by Sun on Saturn in comparison to the gravitational force exerted by Sun on Earth. Also compare the acceleration of Saturn and Earth when the mass of Saturn is 100 times that of Earth and the semi major axis of Saturn is 10 au.

Answer to Problem 36Q

Solution:

Gravitational forces are equal and acceleration of Saturn is 100 times less than that of Earth.

Explanation of Solution

Given data:

Mass of Saturn is 100 times the mass of Earth.

Distance of Saturn from the Sun is 10 au and distance of Earth from Sun is 1 au.

Formula used:

Newton’s law of universal gravitation is stated by an equation as,

$F=G\left(\text{}\frac{m_1{m}_2}{r^2}\right)$

Here, $m_1$ is the mass of first object, $m_2$ is the mass of second object, r is the distance between the objects, and G is the universal constant of gravitation with a value of $6.67\times {10}^{-11}\text{ N}\cdot {\text{m}}^{\text{2}}{\text{/kg}}^{\text{2}}$

Explanation:

From Newton’s law of universal gravitation, the gravitational force F is proportional to the mass and inversely proportional to the square of the distance r.

Use Newton’s equation and write the expression for the gravitational force of the Sun on the Earth,

$F_{Sun-earth}=G\left(\frac{Mm}{r^2}\right)$

Here, M is the mass of the Sun, m is the mass of the Earth, and r is the distance between Sun and Earth.

Substitute 1 au for r,

$F_{Sun-earth}=G\left(\frac{Mm}{{\left(1\text{ au}\right)}^2}\right)$ …… (1)

Similarly, write the equation for gravitational force of the Sun on Saturn.

$F_{Sun-Saturn}=G\left(\frac{M{m}_{saturn}}{{r^{\prime }}^2}\right)$

Here, $m_{saturn}$ is the mass of Saturn and $r^{\prime }$ is the distance of Saturn from the Sun.

Substitute 100m for $m_{saturn}$ (as mass of Saturn is 100 times the mass of Earth) and 10 au for $r^{\prime }$.

$\begin{array}{c}{F}_{Sun-Saturn}=G\left(\frac{M\left(100m\right)}{{\left(10\text{ au}\right)}^2}\right)\\ =G\left(\frac{Mm}{{\left(1\text{ au}\right)}^2}\right)\end{array}$ …… (2)

Observe from equations 1 and 2 that the ratio of the gravitational forces between Sun and Earth and Sun and Saturn is 1.

$\frac{F_{Sun-Saturn}}{F_{Sun-earth}}=1$

Newton’s law of universal gravitation is stated by an equation as,

$F=G\left(\text{}\frac{m_1{m}_2}{r^2}\right)$

Consider that mass $m_1$ is considerably greater in size. So, the gravitational force of $m_1$ will accelerate the mass $m_2$ with an acceleration $a_2$.

Using Newton’s second law which states that the external force is the product of mass of the object and the acceleration of the object, the above equation can be written as,

$\begin{array}{c}{m}_2{a}_2=G\left(\text{}\frac{m_1{m}_2}{r^2}\right)\\ a_2=G\left(\text{}\frac{m_1}{r^2}\right)\end{array}$

So, the above expression can be written as,

$a_2\propto \text{}\frac{m_1}{r^2}$ …… (3)

Use the relation derived in equation (3) to write the expression for the acceleration of the Earth due to gravitational pull on the Sun of mass M.

$a_{Earth}\propto \text{}\frac{M}{r^2}$

Use the relation derived in equation (3) to write the expression for the acceleration of Saturn due to gravitational pull on the Sun of mass M.

$a_{Saturn}\propto \text{}\frac{M}{{r^{\prime }}^2}$

Determine the ratio of $a_{Earth}$ and $a_{Saturn}$.

$\begin{array}{l}\frac{a_{Earth}}{a_{Saturn}}=\text{}\frac{{r^{\prime }}^{2}M}{M{r}^2}\\ \frac{a_{Earth}}{a_{Saturn}}={\left(\frac{r^{\prime }}{r}\right)}^2\end{array}$

Substitute 1 au for r and 10 au for $r^{\prime }$,

$\begin{array}{c}\frac{a_{Earth}}{a_{Saturn}}={\left(\frac{10\text{ au}}{1\text{}\text{ au}}\right)}^2\\ a_{Earth}=100a_{Saturn}\end{array}$

Conclusion:

From Newton’s law of gravity, the gravitational force is proportional to the mass and inversely proportional to the square of the distance between the planet and the Sun. So, for both Earth and Saturn, the gravitational force of the Sun on them are equal. However, since the acceleration of a planet does not depend on the planet’s mass, but only on the Sun’s mass and the distance between them, the acceleration is 100 times less for Saturn than it is for Earth.