A Scenario I Thought Up Today not sure how to describe with math?

[Jamey]

Hello, this is my first post. I'm not a big fan of long introductions, so I'm going to spare you my life story, for now, and discuss something I thought about on my lunch break today.

Alright, suppose there is a rigid sphere of radius $\textit{r}$ traveling at a constant velocity, $\vec{v}$ in a perfect vacuum and, further, assume that it isn't in a gravitational field at all. Suppose further that it is rotating about an arbituary axis and that its rotational velocity, $\omega$, is changing with time due to some applied force that we'll just call $\vec{F}$. Derive a relation which accurately describes the rotational motion and the trajectory of this sphere.

Okay, so, let me show you my line of reasoning and please tell me if I am on the right track...
I first remembered from basic physics that velocity is distance divided by time. I remembered also that angular velocity is the angle over time. I then remembered that linear velocity is the radius times the angular velocity. Expressed mathematically, this is linear velocity, $\vec{v} = r\omega$, where $\omega = \frac{\theta}{t}$. So, I reasoned, the relation would have to involve a derivative with respect to time. I thought that maybe I could differentiate omega and get $\frac{d\omega}{dt} = \frac{d\theta}{dt}$ then put that into the linear velocity equation to get $\vec{v} = r\frac{d\theta}{dt}$.
At this point, I don't know what to do next and think that something is missing. I think I need to come up with more information to write a complete, mathematical description. Maybe, though, it is because I haven't formally taken physics for a while (I plan to next semester though). What I am asking is how to complete this formulation and whether or not I am on the right track? I don't really know, though, this is just a kind of thought experiment to get my mind ready. I'm studying some physics and math in addition to the classes I am taking at my local community college on the side with the help of schaum's. So, to reiterate, what do I do and am I thinking right or is what I am saying kind of silly?

 

[bahamagreen]

"Suppose further that it is rotating about an arbituary axis and that its rotational velocity, ω, is changing with time due to some applied force that we'll just call F⃗. Derive a relation which accurately describes the rotational motion and the trajectory of this sphere."

I'm thinking a couple of things, all very diffuse and hand-wavy. :)

The applied force is constant? So the change in rotational velocity is accelerating (not oscillating or anything else?)

The applied force needs more description; is this a single force that presents from one external direction toward the sphere, or a pair of opposed opposite-edge forces that may or may not "stay" on particular spots on the sphere's surface, making a couple, or a torque induced by an internal mass rotation relative to the sphere? "Center of effort" is not quite the right term, but these specifications will make a difference to the rotation and movement of the center of mass.

The arbitrary axis is going to need to be comprised of some orientation components with respect to v and to the nature of the force; a rotational axis parallel to the direction of v is going to change v (vector) differently than if the axis is in the plane perpendicular to v. There may be a trigonometric aspect that addresses the components of the axial rotation with respect to the parallel and perpendicular planer directions.

So, very hand-wavy...

Maybe "first order" might represent the geometric specification of the force and a way to characterize the axis as orientation components subject to that force with respect to the COM..

Maybe "second order" might be an additional treatment that captures any gyroscopic precession or other classical artifacts that might need to be rolled into the solution.

 

[guitarphysics]

Few comments:
The sphere can't travel at a constant velocity if a constant force is being applied to it. Also, that's not what you get when you differentiate omega (the time derivative of theta is already omega, so the time derivative of omega is the second time derivative of theta).

You need dynamical equations (Newton's second law) for the rotational and linear motion in order to describe the system, and you haven't done any of this, so essentially you haven't done much yet. Start by fixing the stuff I mentioned above, and then look a bit into Newton's second law and what it implies for rotational and linear movements. Then you'll be able to describe the system (you'd need to know the mass of the sphere, but whatever).

(Also, did you mean arbitrary axis? By the way- don't get too worried about vectors with the velocity and all that yet when discussing rotational motion, it'll just confuse you for now. Simply use scalars when talking about angular velocity and all that).

 

[guitarphysics]

bahamagreen, I think your points are valid, but Jamey has far more fundamental problems to deal with first (I mentioned them above in my post).

 

[Jamey]

God I'm stupid, you're right about the velocity and the force thing. Okay, and I need to use Newton's second law. Also, I've never taken calculus-based physics so I don't know much about how to put calculus into physics. I don't know exactly what it is I am trying to do other than learn this stuff...

Anyway, the flaws you have pointed out are disconcerting to me because it makes me feel like the whole thing was very silly. Let me try again with the additional tips you have given me and I'll post some of my ideas once I have refreshed my memory of these elementary concepts... please don't laugh at me I'm not trying to be hand-wavy or make stuff up, I just need to practice...

 

[guitarphysics]

Don't worry, we all look silly when we don't know- go through my older posts if you want, they'll make you feel better ;).
Physics takes a lot of effort, so expect to suck at most things at the beginning; but you'll see gradual improvement over time if you put in the work! It's a great process.

Let us know how you do on this problem, have fun.