Astar exerts a gravitational force of magnitude $\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{25}}\mathbf{\hspace{0.33em}}\mathit{N}$on a planet. (a) What is the magnitude of the gravitational force that the planet exerts on the star? (b) If the mass of the planet were twice as large, what would be the magnitude of the gravitational force on the planet? (c)If the distance between the star and planet (with their original masses) were three times larger, what would be the magnitude of this force?

The given data can be listed below,

The gravitational force exerted by the star on the planet is,$F_{S}P=4\times {10}^{25}\hspace{0.33em}N$ .

The conservation of momentum is due to the fact that when your frame of reference is switched, the rules of physics stay intact, as expressed by Newton's third law of motion.

(a)

By using Newton’s third law, force on two points can be equated as,

$F_{X}Y=-F_{Y}X$

Here, $F_{XY}$ is the force exerted by XonYand$-F_{YX}$ is the force exerted by Y on Xin the opposite direction.

From the above equation. The gravitational force on the star by the planet is given by,

$$\begin{gathered} F_{PS}=-F_{SP} \\ =-4\times {10}^{25}\mathrm{N} \end{gathered}$$

Thus, the gravitational force on the star by the planet is trice the original distance is$-4\times {10}^{25}\hspace{0.33em}N$.

(b)

The gravitational force on the planet by the star is given by,

$F_{SP}=\frac{G{m}_s{m}_p}{r^2}$

Here, G is a universal gravitational constant whose value is $6.673\times {10}^{-11}\hspace{0.33em}N\cdot m^2/k{g}^2$, $m_s$is the mass of the star, $m_p$ is mass of the planet, and r is the distance between the centers of the planets.

When the mass of the planet increases by two times, the magnitude of the force is given by,

$F_{SP}^{\text{'}}=\frac{G\left(2m_S\right)m_P}{r^2}$

Substitute all the values in the above,

role="math" localid="1658067409581" $$\begin{gathered} F_{SP}^{\text{'}}=2F_{SP} \\ =2\left(4\times {10}^{25}\mathrm{N}\right) \\ =8\times {10}^{25}\mathrm{N} \end{gathered}$$

Thus, the magnitude of the gravitational force on the planet when mass of the planet is twice of original mass is role="math" localid="1658067442767" $8\times {10}^{25}\mathrm{N}$.

(c)

The distance between the star and the planet is given by,

$r^{\text{'}}=3r$

Here, ris the actual distance between the star and the planet.

The new gravitational force on the planet by the star is given by,