Calculating radius based on escape velocity and density? PLEASE HELP

[pinkybear]

Homework Statement

A spherical asteroid has a density of 2600 kg/m^3. I throw a ball at the speed on 20 m/s. If the ball is to travel in a circular orbit, what is the largest radius of the asteroid possible to accomplish this?

Homework Equations

these are the equations I used..
escape velocity:
V(e)=sqrt(2GMm/r)
and
Density:
mass=Volume*Density
volume= 4/3(pi)r^3 (since the asteroid is spherical)
GM= 6.67*10^-11 (constant)

The Attempt at a Solution

m=4/3 pi r^3*2600
20^2=[2(6.7*10^-11)(4/3 pi r^3*2600)]/r
or simplified:
400=(1.453*10^-6)*r^2
or
r= 16593 or 17000 (2 s.f)

My answer is wrong.. I have 4 tries left. I'm thinking that I messed up on the GM(gravitational force) part, but I don't know how or why. Is it correct to use the constant?

 

[berkeman]

Homework Statement

A spherical asteroid has a density of 2600 kg/m^3. I throw a ball at the speed on 20 m/s. If the ball is to travel in a circular orbit, what is the largest radius of the asteroid possible to accomplish this?

Homework Equations

these are the equations I used..
escape velocity:
V(e)=sqrt(2GMm/r)
and
Density:
mass=Volume*Density
volume= 4/3(pi)r^3 (since the asteroid is spherical)
GM= 6.67*10^-11 (constant)

The Attempt at a Solution

m=4/3 pi r^3*2600
20^2=[2(6.7*10^-11)(4/3 pi r^3*2600)]/r
or simplified:
400=(1.453*10^-6)*r^2
or
r= 16593 or 17000 (2 s.f)

My answer is wrong.. I have 4 tries left. I'm thinking that I messed up on the GM(gravitational force) part, but I don't know how or why. Is it correct to use the constant?

I have to think about this some more, but to use the Escape Velocity equation, it would seem that you need the mass of the ball (which seems to have gotten dropped in your calculation later), and you are not given that anyway.

I think you need to use the circular motion equations and also the gravitational attraction equation that you list, but I'm having trouble seeing why there is a max radius of the sphere that works for 20m/s radial speed...

 

[berkeman]

Hey, I think I got it.

Use the two equations I mention, and use r for the radius of the orbit of the ball, and R for the radius of the sphere. Equate the force due to gravity to the force needed to keep the 20m/s ball moving in a uniform circular orbit, and do the expansions (mass = density * volume, etc.), to come up with an equation that relates R and r. Do you see a useful characteristic of this equation?...

 

[pinkybear]

Hey, I think I got it.

Use the two equations I mention, and use r for the radius of the orbit of the ball, and R for the radius of the sphere. Equate the force due to gravity to the force needed to keep the 20m/s ball moving in a uniform circular orbit, and do the expansions (mass = density * volume, etc.), to come up with an equation that relates R and r. Do you see a useful characteristic of this equation?...

Sorry, I don't understand what you mean. Why do I need the radius of the orbit of the ball? =(

 

[berkeman]

Sorry, I don't understand what you mean. Why do I need the radius of the orbit of the ball? =(

The problem says that the ball travels in a circular orbit. It also says that you are to find the biggest radius R of the sphere that is consistent with a circular orbit for the ball. There is a relationship between R and r that you will find if you work with the equations I mentioned. That relationship let's you solve the problem.

Show us your work with Newton's Law of Gravitation equation and the equation for the centripital force for Uniform Circular Motion...

 

[pinkybear]

using
F=m*(v^2/r)
and
F=GMm/r^2

i got
r=GM/v^2
M=4/3 pi r^3
r=sqrt((3*v^2)/(4*pi*G))

got r= 1.2*10^6

it's not the correct answer..

 

[berkeman]

using
F=m*(v^2/r)
and
F=GMm/r^2

i got
r=GM/v^2
M=4/3 pi r^3
r=sqrt((3*v^2)/(4*pi*G))

got r= 1.2*10^6

it's not the correct answer..
did I understand the method correctly though?

Did you drop the density term? You wrote: "M=4/3 pi r^3"

And can you explain why you set r=R in your work above (I'm not saying it's wrong...)?

 

[pinkybear]

Did you drop the density term? You wrote: "M=4/3 pi r^3"

And can you explain why you set r=R in your work above (I'm not saying it's wrong...)?

edited: oh wait, i think I get what you mean...
so I got sqrt((3*r*v^2)/(4*pi*G))=R <radius of asteroid
but what do I use for r?
r=R+a small number? which would equal R?EDIT!mggggggg I am so stupid ok.. let me do it again!

 

[berkeman]

edited: oh wait, i think I get what you mean...
so I got sqrt((3*r*v^2)/(4*pi*G))=R <radius of asteroid
but what do I use for r?
r=R+a small number? which would equal R?

Sorry, could you show each of your steps again? I get a different equation than you, and my equation (assuming it's right) implies what to do with R and r to solve the problem.

Start with:

F = m \frac{v^2}{r} = \frac{GMm}{r^2}

and express M as a funtion of R and the density, and simplify to a form like this:

R = f(r) Where f(r) has terms in it for velocity, density, and some constants...

 

[pinkybear]

Oh my god...I added the density and I got 2.3*10^4, which is correct... but I ran out of time. UGH! I hate my life.

But, thank you for your help.