ROTATIONAL MOTION & DYNAMICS 04 - Calculating MOI & Parallel Axis Theorem

OPENING QUESTIONS:

*Y*I*K*E*S*

  1. Let's say we balance a meter stick (horizontally) precisely on its center point. Why would we (correctly) say that that meter stick is not experiencing any torque?

  2. If we move the pivot point 10. cm to the right, the meter stick now *does* experience torque. Why? What is that torque?

  3. What can we say generally about linear torque based on those examples?

  4. Rotational torque depends on (which) distance?

  5. What the heck is m.o.i. anyhow? Is there a way we can quantify that somehow... ? (HINT: Think units!)

  6. Write down ALL the rotational formulae (AND their rotational analogs, if present) that we've learned so far in 153 seconds...

  7. While we're at it, how are elastic collisions and inelastic collisions similar? How are they different?

  8. What at inelastic collisions and perfectly inelastic collisions?

OBJECTIVE: I will be able to calculate moments of inertia for various objects during this unit.

WORDS FOR TODAY:

Moment of Inertia: A measure of how much an object will resist rotation about a specific axis of rotation

FORMULAE & TERMS:

Torque: Force through a displacement (NOT WORK!)

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There are a veritable FLOOD of new terms to get comfortable with in this unit... so let's start NOW:

Linear & Rotational Formulae

WORK O' THE DAY: 

Starting with:

I =∑mir2i

we can quickly determine that this:

I =∫r2dm

Can be a powerful tool to find the m.o.i. of a solid object.

Please notice once again that we have two totally different quantities to relate here, displacement (r) and a tiny amount of mass (dm).

The key to the puzzle is to find some way to get r in terms of m. Notice that once again we're referring to that "long, thin rod" scenario.

Take a few moments to digest that term, what are the weasel words and their implications to that integral?

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The good news is that the m.o.i of common shapes has long since been calculated (what is the importance of the term "Homogeneous" here? What about "Rigid"?

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If you are new to integration or need a refresher on long thin rod type (mass per unit length) problems, please click HERE. You absolutely need to know how to do these type of problems. They will almost certainly show up on the test.

Notice how similar the approach is to integrating an object of uniform volume (mass per unit volume) is found at the bottom of THAT link. These are NICE to know, although I've not seen a two dimensional let alone three dimensional problem on the test... although they ARE part of our learning goals.

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Fortunately the m.o.i. of various common shapes are shown above. Notice MOST of those involve rotation about the center of mass, although not ALL do.

If we already know the m.o.i. about an axis of rotation that PASSES THROUGH THE CENTER OF MASS, we can find the m.o.i. about a second axis as long as it is PARALLEL to the first. (not so) oddly enough, we call this the Parallel Axis Theorem. The AP seems to really like this, so make sure you are comfy with it too.

I = Icm + MD2

Where Icm = the m.o.i. about the c.o.m, M = mass and D2 = the distance to the axis of rotation through the c.o.m.

I DO NOT recommend going through the derivation in the book for this, unless you're a glutton for mathematical punishment.

 

I = Icm + MD2

 

Coursework:

Please review the examples for calculating the m.o.i. for a long thin rod AND for a cylinder. That methodolgy will show up frequently in BOTH mechanics and E & M. I have annotated notes for both on our study guide page.. the direct link is HERE

#39 on page 328.

Find the m.o.i. of a 10.0 meter long, thin rod 15 cm to the right of the center of the rod.

ANSWERS:

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STUDY GUIDE