OPENING QUESTIONS:
What is the center of mass of an object?
How is rotational torque different from linear torque?
How is it possible to have multiple & different moments of inertia for the same object?
Contrast c.o.m. with m.o.i
Compare c.o.m. with m.o.i
Let's take about 15 minutes to finish up our tp activity. Also, when you have a derivation that you and your groupies are comfortable with, and it accurately (within 10-15%) predicates the appropriate drop heights, please make a pretty, annotated copy, give it to me, and I'll scan it and post it here. Also, please indicate any reasons why your calcs didn't *precisely* determine the correct drop heights as an addendum to your work.
OBJECTIVE: I will be able to calculate the moment of inertia (m.o.i.) of a solid object during today's class
WORDS FOR TODAY:
Moment of Inertia: The point about which a solid object is set to rotate
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:
Term
|
Symbol |
Formula
|
SI units
|
Description
|
Notes
|
radian
|
θ |
--- |
radians |
π (in radians) = 180o |
1 radian = 57.3 degrees |
period
|
T |
--- |
seconds |
the period = time for one full rotation |
|
angular speed
|
ω |
∆θ/dt |
radians/sec |
∆θ/dt is instantaneous speed
|
angles ALWAYS described in radians |
angular acceleration
|
α |
dω/dt |
rad/s/s | dω/dt is instantaneous acc | |
tangential velocity
|
vt |
vt = rω |
m/s |
linear velocity at any radial distance "r" |
|
tangential acceleration |
at |
at = rω2 rdω/dt |
m/s/s |
||
centripetal acceleration
|
ac |
ac = v2/r
or rω2 |
m/s2
|
acceleration of an object following a circular path |
Be careful -- radians (by definition) are unit less |
| Torque | τ |
τ=RFsinθ | Nm |
force exerted through displacement (NOT WORK!) | |
I |
depends on shape (see link here) | kgm2 |
The 'pivot point' about which an object rotates. Often associated with center of mass but it does NOT need to be the c.o.m | ||
Make SURE you know these by heart:
And we're going for ALL of these (soon)
|
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WORK O' THE DAY:
The moment of inertia of a single particle is easily determined:
I = mr2
Notice that we use the "r" vector here to indicate that we are interested in the distance from axis of rotation to the particle in question *regardless* of whether it is in x, y or z orientation.
Unfortunately, we pretty much never deal with one particle, so we have to "sum up" all the particles that make up a solid object:
I =∑mir2i
For our colleagues currently studying calc A, you'll soon come to find that whenever you see something like that (summing up a quantity off somethings), you KNOW that integration is lurking around the corner.
And so it is:
We can (and will) find the moment of inertia of an object by:
I =∫r2dm
Please take a few moments to discuss that integral with your group... what is the fly in that particular ointment?
So yes, getting r in terms of mass in order to integrate can be tricky. We'll come back to this on Monday with yet another of those "long-thin-rod" workthroughs.
For now though, let's go back to more sedate items
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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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Please work through example 10.4 on page 305. See especially the "what if" section... that relates *directly* to our toilet paper falling/unravelling activity.
Now please work through example 10.6 on page 306
Coursework:
#34, #35 & #36 on page 327
ANSWERS:
