Question

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?

Short answer

(a) The gravitational force on the star by the planet is $-4\times {10}^{25}\hspace{0.33em}N.$

(b) The magnitude of the gravitational force on the planet when mass of the planet is twice of original mass is

(c) The magnitude of force when the distance between the star and planet is trice the actual distance is localid="1658065983046" $4.45\times {10}^4\hspace{0.33em}N.$

Step-by-step solution

Identification of given data

The given data can be listed below,

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

Concept of Newton’s third law

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.

Determination of is the magnitude of the gravitational force that the planet exerts on the star

(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$-4\times {10}^{25}\hspace{0.33em}N$.

Determination of magnitude of the gravitational force on the planet when the mass of the planet is twice of the original mass

(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" $8\times {10}^{25}\mathrm{N}$.

Determination of the magnitude of this force when distance between the star and planet is trice the original distance

(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,