Finding out the type of central force given the orbit

[saadhusayn]

Hi,

This is the statement of the problem of AP French's textbook "Newtonian Mechanics".

1. Homework Statement

The commander of a spaceship that has shut down its engines and is
coasting near a strange-appearing gas cloud notes that the ship is
following a circular path that will lead directly into the cloud (see the
figure). He also deduces from the ship's motion that its angular momentum
with respect to the cloud is not changing. What attractive (central)
force could account for such an orbit? (ANSWER : F ~ -1 / r^5)

Homework Equations

The Attempt at a Solution

From the information given, angular momentum = m * r^2 * (dtheta / dt) =
constant.
The acceleration of the ship is a combination of tangential and radial
acceleration, the resultant of which is along the line joining the
spacecraft and the cloud.
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I don't know where to proceed from here.

Thank you very much for your help.

 

[mfb]

From the information given, angular momentum = m * r^2 * (dtheta / dt) =
constant.

This allows you to find velocity as function of radius, which then leads to the potential.