Calculating RPM of 2D Object with Force Applied

[Stabback]

Hi there, I have been searching the web for a couple hours now and cannot find an answer to what I thought was a simple question. Say I had an object (to make it simple, let's say this is 2D with no external forces like gravity, friction, or air resistance) and wanted to calculate how fast it would spin when a force is applied at an angle to it's center of gravity.

I know the mass of the object, the force coming in, the angle at which the force is applied, and the distance from the center of gravity of the object to where the force is applied. I wish to know how many revolutions per minute the object would turn at after the force is applied.

Thanks for the help.

 

[berkeman]

The key is understanding the concept of the Moment of Inertia. Then you use a rotational acceleration equation that is analogous to the linear F=ma equation.

http://en.wikipedia.org/wiki/Moment_of_inertia

 

[Stabback]

I read through the Wikipedia article and am having troubles relating the information to what I need done. I'm sorry if I come off as a little thickheaded but my current schooling (completed as much physics as high school has allowed at the highest level) has barely touched on rotation. I am familiar with Classical physics, and have learned about rotating an object around an outside point, however I cannot seem to find out the rpm when an object is spun.

Once again thanks for your help, I would appreciate it if you could stick with me while I work out a way to do this.

 

[berkeman]

I may move this thread to a homework section later, but since it sounds more like a general interest question, I'll leave it here for now.

I'll highlight a couple things that hopefully will get you going. The moment of inertia is the way that we express the resistance to rotation, much the same way that mass represents the amount of resistance to linear acceleration due to a force F=ma.

For rotation, the equation is $$\tau = I \alpha$$ which says that the torque produces an acceleration alpha, which is inversely porportional to the moment of inertia I. Just as in F=ma we are saying that the force accelerates the mass at an acceleration that is in inverse proportion to its mass. Same force, twice the mass, half the acceleration. Same torque, twice the I, half the rotational acceleration alpha.

Then, just as we have kinematic equations of linear motion (relating acceleration a, velocity v, and position x), we have analogous equations for the rotational acceleration $$\alpha$$ the rotational velocity $$\omega$$ and the rotational angle $$\theta$$

List of typical moments of inertia http://en.wikipedia.org/wiki/List_of_moments_of_inertia

And see the "Algebraic Equations" partway down this page for linear and rotational equations:
http://en.wikipedia.org/wiki/Kinematics

So for your problem, you need to estimate the I of your object (or calculate it if you need to), then figure out how much torque the force applies (torque is force multiplied by the lever arm, if you are applying the force at a 90 degree angle), and then using the rotational kinematic equations, you will be able to figure out what the angular acceleration $$\alpha$$ is, and from that how the object spins up to some rotational speed $$\omega$$

Keep track of your units carefully as you do this work. Use units like kg, meters, seconds, radians (there are $$2\pi$$ radians in 360 degrees), etc. Does that help?

 

[berkeman]

And here's a PF Tutorial by Marlon that may help as well:

https://www.physicsforums.com/showthread.php?t=72040&highlight=moment

 

[Stabback]

Yep, this helps a lot. Also, this is not homework related in any way. I am mucking about in computer programming and trying to make a physics simulator. So yes it is general interest, although I could see it being moved to the homework section to help others search for this kind of information later on.

I will be calculating the exact value of I as all numbers must be precise in this simulation. There is a good example for a flat polygon on the first Wikipedia page which will help out a lot with deciding the moment of inertia, and I believe I should be fine looking through the kinematics page. Thanks for the quick help, and if you were interested I could post the final product when the simulation is complete.

 

[berkeman]

Glad to help. And yes, when you get some screenshots of your work, by all means post some here.

 

[Stabback]

No problem, but don't expect it soon, it could be a couple weeks before it's done :P. Thanks again.