Find mass of Jupiter using Jupiter's moon, Io

[totallyclone]

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
F_c=mv²/r
GmM/r_o²
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/r_o²=mv²/r_o
M=v²r_o/G
M=4∏²r_o³/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.

 

[staysys]

Your problem is not the time units, but the distance.

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

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

 

[rude man]

r₀ is given in km. What unit should you be using in your (correct) formula?

 

[totallyclone]

Your problem is not the time units, but the distance.

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

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

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

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

Fun=ma
GmM/r_o²=mv²/r_o
GM/r_o²=v²/r_o
M=4∏²r_o³/GT²

Would T=1.77days be in seconds then if I made the r_o to m?

 

[totallyclone]

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 r_o in terms of metres.

 

[totallyclone]

M=4∏²r_o³/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...

 

[staysys]

M=4∏²r_o³/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...

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?

 

[rude man]

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.

 

[Sunil Simha]

M=4∏²r_o³/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∏²r_o³/GT

 

[SteamKing]

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.