Find mass of Jupiter using Jupiter's moon, Io

In summary, the mass of Jupiter can be found using the data for Io by using the equation M=4∏2ro3/GT2, where r0 is the mean distance from Jupiter (422x10^3 km) and T is the period (1.77 days). However, it is important to check for unit consistency in the equation, as the gravitational constant G has units of m^3 kg^-1 s^-2, and the orbital radius of Io should be converted to meters. Once these corrections are made, the calculated mass of Jupiter should be 1.90x10^27 kg, which agrees with the value found through a simple Google search.
  • #1
totallyclone
54
0

Homework Statement


Io - mass: 8.9x1022kg
period: 1.77 days
mean distance from Jupiter: 422x103km

Find the mass of Jupiter using the data for Io.

Homework Equations


ƩF=ma
Fc=mv2/r
GmM/ro2
v=2∏r/T

The Attempt at a Solution


I am just unsure what should I put the time in?? Seconds, hours, days, months, years?? I'm getting a different answer and as I google the mass of Jupiter, it is about 1.90x1027kg.

I should be getting the same or really close to it. This is how I tackled the question.

Let m=mass of Io, M=mass of Jupiter

ƩF=ma
GmM/ro2=mv2/ro
M=v2ro/G
M=4∏2ro3/GT2
M=4∏2(422x103)3/(6.67x10-11)(1.7x24x3600)2
M=1.90x1018

I'm getting the same answer BUT different exponents. I know I'm missing something out. :frown:
 
Last edited:
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  • #2
Your problem is not the time units, but the distance.

Try checking for unit consistency in your equation:
[itex]M=4\pi^{2}r_{0}^{3}/(GT^{2})[/itex]

what should the units of [itex]r_{0}[/itex] be in if you want everything to cancel on the right side except for mass?
 
  • #3
r0 is given in km. What unit should you be using in your (correct) formula?
 
  • #4
staysys said:
Your problem is not the time units, but the distance.

Try checking for unit consistency in your equation:
[itex]M=4\pi^{2}r_{0}^{3}/(GT^{2})[/itex]

what should the units of [itex]r_{0}[/itex] be in if you want everything to cancel on the right side except for mass?

If ro is given in km, shouldn't I also use it in km?

I double checked the given for Io and my givens are correct.
tumblr_mk8yuuItBI1qe908uo1_500.jpg


Fun=ma
GmM/ro2=mv2/ro
GM/ro2=v2/ro
M=4∏2ro3/GT2

Would T=1.77days be in seconds then if I made the ro to m?
 
  • #5
rude man said:
r0 is given in km. What unit should you be using in your (correct) formula?

I realized I made T in terms of seconds, so I think I should make ro in terms of metres.
 
  • #6
M=4∏2ro3/GT2
M=4∏2(4.22x108)3/(6.67x10-11)(1.77x24x60x60)2
M=2.91x1032

Aaaaah! Still not getting it right? Ah, I must be doing something wrong... :confused:
 
  • #7
totallyclone said:
M=4∏2ro3/GT2
M=4∏2(4.22x108)3/(6.67x10-11)(1.77x24x60x60)2
M=2.91x1032

Aaaaah! Still not getting it right? Ah, I must be doing something wrong... :confused:

But it looks right to me.
I just typed in the exact same equation you wrote, with your numbers and I got 1.90x10^27kg as it should be.
Can you double check that you didn't make a mistake in calculating the answer?
 
  • #8
staysys said:
But it looks right to me.
I just typed in the exact same equation you wrote, with your numbers and I got 1.90x10^27kg as it should be.
Can you double check that you didn't make a mistake in calculating the answer?

Me too.
 
  • #9
totallyclone said:
M=4∏2ro3/GT2
M=4∏2(4.22x108)3/(6.67x10-11)(1.77x24x60x60)2
M=2.91x1032

I think you forgot to square the time period. You get 2.91x1032 when you put M=4∏2ro3/GT
 
  • #10
1. By not checking the units of G, you made several mistakes in your initial calculation.
G = 6.67*10^-11 m^3 kg^-1 s^-2 This constant can be easily googled and it shows the proper units. Just because a known datum is given in a certain set of units does not ensure that it can be used without checking to see if those units are compatible with the other elements of an equation.
2. The orbital radius of Io is 422*10^3 km. How many meters is this?

If you correct these errors and redo your calculation, you should get the correct mass of Jupiter.
 

FAQ: Find mass of Jupiter using Jupiter's moon, Io

1. How does Io's orbit help us find the mass of Jupiter?

Io's orbit around Jupiter is directly influenced by the planet's mass. By studying Io's orbital motion and its distance from Jupiter, we can calculate the planet's mass using Newton's Law of Gravitation.

2. What data do we need to collect from Io to calculate Jupiter's mass?

To find the mass of Jupiter using Io, we need to measure Io's orbital period, distance from Jupiter, and its velocity. These data points can be obtained through observations using telescopes or spacecraft missions.

3. Can we use other moons of Jupiter to find its mass?

Yes, we can use other moons of Jupiter, such as Europa and Ganymede, to calculate the planet's mass. However, Io's orbit is more suitable for this calculation because it is closer to Jupiter and has a shorter orbital period, making the measurements more precise.

4. How accurate is the method of using Io's orbit to find Jupiter's mass?

This method is highly accurate and has been used by scientists for decades. With advanced technology and precise measurements, we can determine Jupiter's mass with an accuracy of up to 1%. However, the accuracy may vary depending on the quality of the data collected.

5. What can we learn from knowing the mass of Jupiter?

Knowing the mass of Jupiter is essential for understanding the formation and evolution of the solar system. It also helps us understand the dynamics of planetary motion and the interactions between celestial bodies. Additionally, it can provide insights into the composition and internal structure of the planet.

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