How Does Force Scale with Radius in Orbital Motion?

[Vitani11]

Homework Statement

a. A particle in circular orbit is experiencing an attractive force F towards its center of the orbit. It its period T scales as rⁿ, where r is the radius of the orbit, how does the force scale with radius r? That is, find p in the relation F = ar^p, where a is an arbitrary constant.

b. Find n and p corresponding to the circulation motion under gravitational force and verify keplers third law.

Homework Equations

The Attempt at a Solution

I solved part a. for P and I guess that's apparently not what the question it asking for according to my professor who created this problem. I know how to verify Keplers third law for b., don't really know what n and p are supposed to represent or anything. scaling factors? the exponent for r?

 

[BvU]

Yes (p for F and n for T)

 

[Vitani11]

Okay, but I solved for P. What am I supposed to be doing that is not solving for P even though it says find P?

 

[Vitani11]

I just get something natural log of F/a with base r equal to P.

 

[BvU]

What did you find and what do you know about the gravitational force ?

 

[Vitani11]

Found that the ln_r(F/a) = P and I know the force falls off as 1/r². I'm not sure what to do with that information.

 

[Vitani11]

I plugged in gravitational force for F.. my problem is that I always thought P was 2 because that's just in the equation for gravitational force.

 

[BvU]

The exercise for (a) starts with: ##T## scales as ##r^n## and asks what that yields for p. There should be some dependence on ##n## in that expression. How come you don't have that ? Can you show your steps in detail ?

Then, for (b) you fill in the ##p=2## that you know for gravity and deduct what the value of the corresponding ##n## is.

 

[Vitani11]

F = ar^p
F/a = r^p
ln(F/a) = pln(r)
p = ln(F/a)/ln(r)
p = ln_r(F/a)

 

[Vitani11]

I know that for Force the period is 1/T² for gravitational force. I mean that doesn't really mean anything but that is the relationship.

 

[BvU]

p = ln_r(F/a)

No. There is a dependence on ##n## that you are leaving out. You are not using the fact that these orbits are circular.

I know that for Force the period is 1/T² for gravitational force

No. The period is simply T

 

[Vitani11]

Still don't understand. There is no n in the equation, and I guess I have a massive gap in my knowledge. I guess I'll just not get this one. Thank you anyway and sorry for my stupidity.

 

[BvU]

A particle in circular orbit is experiencing an attractive force F towards its center of the orbit

What force is needed to maintain a circular orbit ? Gives you a relationship with ##v##. What provides that force ?

There is also a relationship between ##v## and ##T##. Given that ##T\propto r^n## that helps you to find ##p## in ##F_{\rm grav}\propto r^p##.

The template doesn't have a section "2. Homework Equations " for nothing. Very useful in your case.

Thank you anyway

you're welcome. That's what we're here for.

and sorry for my stupidity.

Don't say that. You're not stupid and even if you were you wouldn't need to apologize for that.