What Is the Correct Calculation for Satellite Speed in a Stable Orbit?

[BitterSuites]

[SOLVED] Centripetal Acceleration

Homework Statement

In order for a satellite to move in a stable circular orbit of radius 6689 km at a constant speed, its centripetal acceleration must be inversely proportional to the square of the radius of the orbit.

What is the speed of the satellite? The universal gravitational constant is 6.67259e-11 and the mass of the Earth is 5.98e24.

Answer in units of m/s.

Homework Equations

a = v^2/r
F = Gm1m2/r^2
F = ma

The Attempt at a Solution

Ok, a = v^2/r becomes v = sqrt of ar.

From the problem, a = r^-1/2 (or am I wrong?) so a = .012227

sqrt of ar = sqrt of 81.9863
v=9.0-4358

This however, is incorrect. Somehow G is important, but I'm not seeing it. Please help lead me to the right equations.

 

[bfr]

Centripetal acceleration=force of gravity

(m2)(v^2/r)=(G*m1*m2)/r^2

You can find acceleration from there.

 

[BitterSuites]

I already have acceleration from r=-1/2 as told from the problem itself. I need to solve for v. Can I use the above equation if I don't know v or m1?

 

[Saladsamurai]

There is only one force acting on the satellite, the force of gravity. You know that F_g=\frac{GMm}{r^2}.

By Newton's 2nd you also know that \sum F=ma .

Now ask yourself how you can use this along with what you were given to find v.

 

[tiny-tim]

From the problem, a = r^-1/2 (or am I wrong?) so a = .012227

Yes, you're right - you're wrong!

The question is telling you that a is proportional to r^-1/2. All it means is that a = F/m, and F is proportional to r^-1/2, so a must be also.

This is a questioner intending to be helpful, but actually being a little confusing.

Your "relevant equations" are correct, so just solve them (and ignore the help).

 

[mysqlpress]

mv^2/r =GMm/r^2
v^2=GM/r^2
v= sqrt GM/r^2

GM=gRe^2 as well

 

[BitterSuites]

Ok, so I did it this way and still ended up with the incorrect answer. Where am I still messing up?

F=Gm1m1/r^2
9.8 = 6.67259e-11*M*5.98e24/6689^2
4.38471e8=6.67259e-11*M*5.98e24
7.33242e-17=6.67259e-11*M
M=.000001

F=ma
9.8=.000001a
a = 8.91812e6

a=v^2/r
8.91812e6 = v^2/6689
v^2=5.96533e10
v=244240

So where did I go wrong again? :D

 

[mysqlpress]

Ok, so I did it this way and still ended up with the incorrect answer. Where am I still messing up?

F=Gm1m1/r^2
9.8 = 6.67259e-11*M*5.98e24/6689^2
4.38471e8=6.67259e-11*M*5.98e24
7.33242e-17=6.67259e-11*M
M=.000001

F=ma
9.8=.000001a
a = 8.91812e6

a=v^2/r
8.91812e6 = v^2/6689
v^2=5.96533e10
v=244240

So where did I go wrong again? :D

it's 6689km... not 6689m

 

[BitterSuites]

I tried v = sqrt GM/r^2 and it is also incorrect.

 

[BitterSuites]

it's 6689km... not 6689m

Wow. I'm special. Let me throw that in and see what happens :D

 

[mysqlpress]

I tried v = sqrt GM/r^2 and it is also incorrect.

it's my typo
should be v=sqrt GM/R

you see, from my deviation :)

 

[BitterSuites]

Phew. On the last try I got it correct. My calculations above worked once I converted km to m. *judges himself harshly*

v = sqrt GM/r^2 didn't work even with the km converted to m.

 

[BitterSuites]

Ah :) Thanks for everyone's help! I'm marking it solved now.