What is the Gravitational Force and Acceleration of Jupiter on its Moon Io?

[RubenL]

Homework Statement

Io (pronounced “EYE oh”) is one of Jupiter’s moons discovered by Galileo. Io is slightly larger
than Earth’s Moon.

The mass of Io is 8.92*10²² kilograms
and the mass of Jupiter is 1.9*10²⁷
kilograms. The distance between the
centers of Io and Jupiter is r = 4.22*10 meters.


1. Calculate the magnitude of the gravitational force of
attraction that Jupiter exerts on Io.


2. Calculate the magnitude of the acceleration of Io due
to the gravitational force exerted by Jupiter.

Homework Equations

F_g = G m₁ m₂/r²

Io = m₁ = 8.92*10²²
Jupiter = m₂ = 1.9*10²⁷
r = 4.22*10
G = 6.67*10^-11

The Attempt at a Solution

1. F_g = (6.67*10^-11)(8.92*10²²)(1.9*10²⁷) / (4.22*10⁸)²

= 6.36*10²²N


2. How can i find the acceleration of Io?
Do i use F=ma => a=F/m => a = (6.35*10²²) / (8.93*10²²) = 7.12*10⁴³ ?? this does not seem right.

 

[JesseC]

Looks like you multiplied instead of dividing in the last line of your post.

 

[RubenL]

2. How can i find the acceleration of Io?
Do i use F=ma => a=F/m => a = (6.35*10²²) / (8.93*10²²) = 7.12*10⁴³ ?? this does not seem right.

Looks like you multiplied instead of dividing in the last line of your post.

Thanks Jesse, i kept on making that mistake. Until i realized it was the way i typed it into my calculator...

2. How can i find the acceleration of Io?
Do i use F=ma => a=F/m => a = (6.35*1022) / (8.93*1022) = 0.71m/s²?

However, i still would like to know if i have used to correct method and achieved the correct result?
Considering Earth's moon's gravitational attraction (roughly 1.6m/s²), compared to Io the larger moon, you would expect Io to have a slightly greater gravitational attraction due to its greater mass. Therefore i am assuming that my answer is still not correct.

 

[haruspex]

Considering Earth's moon's gravitational attraction (roughly 1.6m/s²), compared to Io the larger moon, you would expect Io to have a slightly greater gravitational attraction due to its greater mass. Therefore i am assuming that my answer is still not correct.

As you can see from the equations you have used, the acceleration of the satellite towards the parent body is independent of the mass of the satellite. It depends on the mass of the parent and distance from it. Although Jupiter is so much more massive than the Earth, Io's extra distance from it more than makes up for this.

 

[RubenL]

As you can see from the equations you have used, the acceleration of the satellite towards the parent body is independent of the mass of the satellite. It depends on the mass of the parent and distance from it. Although Jupiter is so much more massive than the Earth, Io's extra distance from it more than makes up for this.

I see, i was not taking the distance into consideration in my assumption. Io is farther away from Jupiter than is the Moon from Earth.Therefore Io's force of gravity will be less then i expected.
In addition I have only calculated the acceleration of the force of gravity towards Io's surface and not that of Io's acceleration towards the parent body Jupiter.

Therefore i haven't answered the question correctly. I can not seem to figure out the answer to this question, i feel like i am missing some key point of information... i understand i must calculate the Fg of Jupiter and then divide it by the distance "r" (assuming i also figured out how to implement the strength decrease of Fg per distance).

 

[haruspex]

I have only calculated the acceleration of the force of gravity towards Io's surface and not that of Io's acceleration towards the parent body Jupiter.

No, you did it correctly (apart from the numerical error). You computed the force of attraction between Jupiter and Io, then you divided by the mass of Io get the acceleration Io experiences as a result of that force.

 

[RubenL]

Thats right, i calculated the Fg between m₁ and m₂ and divided it by Io's mass to produce the acceleration of that force.
Not solely Io's Fg, because in order to do that i would have needed to calculate "Fg = Gm_Io / r²" (where r would be the radius of Io's center to its surface).

Thank you for clearing that up :)