# How does force scale with r?

1. Jan 16, 2017

### Vitani11

1. The problem statement, all variables and given/known data
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 = arp, where a is an arbitrary constant.

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

2. Relevant equations

3. 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?

2. Jan 16, 2017

### BvU

Yes (p for F and n for T)

3. Jan 16, 2017

### 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?

4. Jan 16, 2017

### Vitani11

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

5. Jan 16, 2017

### BvU

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

6. Jan 16, 2017

### Vitani11

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

7. Jan 16, 2017

### 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.

8. Jan 16, 2017

### 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.

9. Jan 16, 2017

### Vitani11

F = arp
F/a = rp
ln(F/a) = pln(r)
p = ln(F/a)/ln(r)
p = lnr(F/a)

10. Jan 16, 2017

### 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.

11. Jan 16, 2017

### BvU

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

No. The period is simply T

12. Jan 16, 2017

### 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.

13. Jan 16, 2017

### BvU

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. Relevant equations" for nothing. Very useful in your case.

you're welcome. That's what we're here for.
Don't say that. You're not stupid and even if you were you wouldn't need to apologize for that.

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