Solve Spiral Orbit Force Field: Desperate for Help

[mark9696]

A particle moves in the spiral orbit given by r = a*theta^3. If theta(t) = c*t^3, determine whether the force field is a central one. I have studied the derivation of the orbital equation for a central force field but this says to determine that! I am desparate for help here folks, I am quite puzzled

 

[turin]

Concept/definition:
A central force field does not apply torque about the force center.

 

[mark9696]

Ok, but I still do not seem to understand what to do. Please, I am desparate, really.

 

[mark9696]

When you say does not apply torque, I am still confused, maybe I am missing something. Please, I need help guys.

 

[cookiemonster]

If you've learned how to find an equation to describe the orbit from a central force, why don't you use that in reverse to try to find a central force that will yield that orbit? If you get a solution, then the force must be central. If not, then it must not be central.

cookiemonster

 

[turin]

Originally posted by mark9696
When you say does not apply torque, I am still confused, maybe I am missing something.

Do you know what torque is? It is the rotational analog of force. So, basically, from Newton's second law, if the force field does not apply a torque, then the rotational analog of momentum (which is angular momentum) would remain constant. Do you know what angular momentum is? You can calculate it from the distance from the origin and the tangential velocity about the origin.

 

[mark9696]

I know about the angular momentum and torque but am not sure what the outline of the solution will look like. Maybe that is where I need some explanation.

Also, I am not particularly sure about the explanation that cookie master gave cause it makes sense to me. He said to try and find a central force but starting that will yeild that orbit but the equation is only valid for central forces to begin with. And the expression for t seems to be a bit puzzling as well.

 

[turin]

Originally posted by mark9696
I know about the angular momentum and torque but am not sure what the outline of the solution will look like.

Answer this question: "Does the angular momentum of the particle about the origin change if it follows the given trajectory?"

Originally posted by mark9696
... the expression for t seems to be a bit puzzling ...

What expression for t?

 

[HallsofIvy]

I think "torque" may be misleading. Although it does play a part, you do not really need to calculate it.

F= ma so to see what the force vector is you need to find the acceleration vector. You are given that r= a θ³ and that θ(t)= c t³ so that r= a c³&t;⁹.

Clearly, d²θ/dt² is not 0. That tells us that the "center" of the force is not the origin of the coordinate system but still does not tell us if the force is "central" with some other center.

The "position" vector is <r cos(&theta;),r sin(&theta;)> so
the acceleration vector is given by <d²(r cos(&theta;)/dt², d²(r sin(&theta;))/dt²>. You can calculate that knowing that &theta;= c t³ and r= a c³&t;⁹.

Does that vector pass through the some one point for all t? If so, that is a "central force field".

 

[mark9696]

I understand most of what you said except for this

Does that vector pass through the some one point for all t? If so, that is a "central force field".

Can you explain that again?

 

[turin]

Originally posted by HallsofIvy
Clearly, d²&theta;/dt² is not 0. That tells us that the "center" of the force is not the origin of the coordinate system ...

I'm not sure I agree with this. Can you explain it, or is it a fundamental rule? I'm not disputing your procedure in general.

 

[student1938]

Here's a crack...


if L^2 = r^2 * thetadot results in L being a constant, then you gota stable orbit right? Easy to just plug in and differentiate in my opinion...could not get more simpler than that right?