Force on Planet Moving in Interstellar Dust

[BigD]

Homework Statement

A uniform, spherical planet of mass M and radius R moves SLOWLY with an essentially uniform speed v through a cloud of interstellar dust particles, whose density is ρ. The dust particles are attracted towards the planet, and some of them would eventually fall onto its surface.

Find the resulting retarding force on the planet due to the dust cloud.
Since the planet moves slowly, initial speed and final speed can be assumed to be the same.

Homework Equations

Angular momentum => Li = Lf
Momentum => Pi = Pf
Energy including
Potential energy = -GMm/R
Kinetic Energy = 1/2 (mv2)

The Attempt at a Solution

I assumed the planet would consume all the dust within a circular cross section. By using conservation of energy and angular momentum, I got the radius of this circle to be

R' = (R^2 + 2RGM/v^2)^1/2

Then I used

dm = ρAdx = ρ(πR'^2)dx

to get

F = dp/dt = (dm/dt)v = ρπR^2(v^2 + 2GM/R).

I was told this wasn't right. Can someone give me a hint as to what I did wrong?

 

[D H]

Why did you put the planet in orbit? Per the problem statement it is moving with an "essentially uniform velocity". No mention of an orbit.

 

[BigD]

Why did you put the planet in orbit? Per the problem statement it is moving with an "essentially uniform velocity". No mention of an orbit.

I didn't.

 

[BigD]

The circle I mentioned was not an orbit; it was the cross section swept out by the planet.

I could really use some help on this.

 

[D H]

A couple of questions.

1. How did you derive that R' = (R^2 + 2RGM/v^2)^1/2 ?

2. Are you sure that the statement 'initial speed and final speed can be assumed to be the same.' is correct? This doesn't make a bit of sense. It means that momentum is not conserved. Better would be to assume that the initial and final speeds are approximately the same. (In other words, you can ignore second-order effects.)

 

[BigD]

A couple of questions.

1. How did you derive that R' = (R^2 + 2RGM/v^2)^1/2 ?

2. Are you sure that the statement 'initial speed and final speed can be assumed to be the same.' is correct? This doesn't make a bit of sense. It means that momentum is not conserved. Better would be to assume that the initial and final speeds are approximately the same. (In other words, you can ignore second-order effects.)

1. I started by changing the frame of reference to the planets center of mass so that the dust moves at speed v. I assumed that at some perpendicular distance R' from the trajectory of the planet the dust particles would just barely miss the planet and that they would pass at the radius of the planet R with some velocity v' perpendicular to the radius vector. I used conservation of energy and angular momentum to solve for R' in

Rv' = R'v
(1/2)v^2 = (1/2)v'^2 - GM/R

I think this part of the problem is right because I got the same result from another method. I'm guessing I did something wrong in the rate at which the dust flows into the planet.

2. I'm sure that it's right. I assume it means that the force that we want to find is negligable compared with the planet's momentum.