Orbital equation

DB · Jan14-05, 04:54 PM

Im having a little trouble finding the orbital period of Earth using:

$LaTeX Code: t_{years}=2\\pi\\sqrt{\\frac{a^3}{GM}}$

"M" being the mass of the central body, obviously the sun, at 2 x 10^30 kg
"a" being the semi-major distance in Au = 1
So,
$LaTeX Code: \\approx$

$LaTeX Code: t_{years}=6.28\\sqrt{\\frac{1}{6.673*2}}*\\sqrt{\\frac {1}{10^{-11}*10^{30}}$

$LaTeX Code: t_{years}=6.28\\sqrt{.0749}*\\sqrt{10^{-19}}$

$LaTeX Code: t_{years}=6.28*.2736*(3.16*10^{-10})$

$LaTeX Code: t_{years}=5.428*10^{-10}$

$LaTeX Code: 5.428*10^{-10}\\neq1year$

I know I'm doing something terribly wrong, any help appreciated

One other question, the gravitational constant is in Newton/seconds right?

**dextercioby** · Jan14-05, 05:00 PM

1-st question:how did u get the orbit period in HOURS?????????
2-nd question:are u familiar working with big numbers?

Daniel.

PS.Use the constant correctly.Together with their units.

DB · Jan14-05, 05:04 PM

Originally Posted by dextercioby

1-st question:how did u get the orbit period in HOURS?????????

Ya big oopps, I realize my sheet says years, I don't know why I remembered hours, ill edit that thank you.

Originally Posted by dextercioby

2-nd question:are u familiar working with big numbers?

Familiar enough.

**dextercioby** · Jan14-05, 05:07 PM

WHAT?????It should be in SECONDS,because every other constant (in the RHS) is in SI units...

Daniel.

PS.Post your work.Again,pay attention with your units.
The result should be roughly $LaTeX Code: \\pi\\cdot 10^{7} s$ .

**Integral** · Jan14-05, 05:22 PM

You need to get all of quantities in the same units. Make it a habit to write the physical units of all of your quantities. What are your units of length? Are they the same in both the numerator and denominator?

**kirovman** · Jan14-05, 05:24 PM

When working with problems like these, you should be sure to convert to SI units. ie 1 Year = 3600 x 24 x 365 seconds.
If you are methodical in doing this everytime, you should avoid further problems.

And if required in the end result, you can convert back to years again without messing with your working.

Also AU is NOT the unit you want to be using for length. Convert to metres.

I think you need to take some classes on dimensional analysis...has anyone got any good links for that?

DB · Jan14-05, 05:32 PM

Thanks. Im confused though. I've been wroking it out over and over and keep getting weird answers, I really can't see what to change. Should "a" be in metres because G is in m^3 kg^-1/s^2 ?

And take a look here:
http://en.wikipedia.org/wiki/Orbital_period
under Small body orbiting a central body it says t is in years and a is au's.

DB · Jan14-05, 05:34 PM

O i just saw your reply kirovman, thanks im gonna keep trying.

**dextercioby** · Jan14-05, 05:40 PM

Originally Posted by DB

Thanks. Im confused though. I've been wroking it out over and over and keep getting weird answers, I really can't see what to change. Should "a" be in metres because G is in m^3 kg^-1/s^2 ?

And take a look here:
http://en.wikipedia.org/wiki/Orbital_period
under Small body orbiting a central body it says t is in years and a is au's.

I believe the problem is asking for the orbital period in seconds.We all know that the orbital period of Earth is 1 year,but we want to find that using physics and not our senses.

That formula
$LaTeX Code: T(years)=\\sqrt{a^{3}}$ is not physically acceptable/correct.

Daniel.

PS.Physics is rigor.Dimensional analysis of mathematical expressions is essential.

DB · Jan14-05, 05:56 PM

Bingo. Thanks guys. I'm happy

$LaTeX Code: t_s=2\\pi\\sqrt{\\frac{(149597887000 m)^3}{6.67 m^3/kg/s^2 * 2 kg}}*\\sqrt{\\frac{1}{10^{-11}*10^{30}}$

$LaTeX Code: t_s=2\\pi\\sqrt{\\frac{3.34*10^{33} m}{13.34}}*3.16*10^{-10}$

$LaTeX Code: t_s=6.28*1.58*10^{16}*3.16*10^{-10}$

$LaTeX Code: t_s=\\frac{\\frac{31354784}{3600}}{24}=362$

I made some stupid mistakes because I find it difficult dealing with m^3/kg/s^2, but now that I found out "a" should be in metres in worked out.
I'm really just doing this for the fun of it, for the sake of learning.

Thanks all.

DB · Jan14-05, 05:58 PM

Originally Posted by dextercioby

That formula
$LaTeX Code: T(years)=\\sqrt{a^{3}}$ is not physically acceptable/correct.

ya it caused me alot of stress

**dextercioby** · Jan14-05, 06:40 PM

Originally Posted by DB

ya it caused me alot of stress

Advice for future physics problems involving numerical calculations:
Always use the sign/s of approximation:
$LaTeX Code: \\approx$ or $LaTeX Code: \\sim$
,when dealing with nonexact figures.

In your case,all "equal to" signs should have been "appox.equal to".

Daniel.

PS.Speed of light in vacuo is the only exception.