OBJECTIVE: I will be able to calculate the energy of a rolling object during today's class,

WORDS FOR TODAY:

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

Torque: Force through a displacement (NOT WORK!)

FORMULAE OBJECTUS:

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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: 

The VERY last thing we're going to cover in this #*@!% unit is total kinetic energy of a rolling object.

Picture a uniform sphere rolling across a beautifully manicured green lawn on a gorgeous summer day with a very light breeze blowing.... it's the day AFTER you've just aced the APC exam...

But I digress..

Back to the sphere...

The sphere is in motion, so it clearly has KE (1/2mv²) as we would expect. However, it also has ROTATIONAL kinetic energy (1/2Iω²).

Soooooooooooo... that means that the TOTAL energy of the rolling sphere is... wait for it...

wait for it....

1/2mv² +1/2Iω²

Yay!!!!

We're done with formulae!

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Object Rotates About Its Center (But doesn't go anywhere)	Round Object Slides (Like My Globes)	Object rolls because of friction that grips that sphere where it makes contact with the surface

For more infor on rolling, slipping & sliding... Kahn Acadamy and You Tube

sketch my globe sliding across one of our tables without friction and show all forces and components of those forces.

now sketch a softball as it rolls across one our tables... what MUST be occuring there?

Now we'll work through AP C FR problem #3 from 2012 that has been publically released. At first it seems pretty rough, but once we get used to dealing with the lingo, it gets (a wee bit?) easier....

Let us commence... Oh before we go, did you know that the motto on the seal of the great state of Hawaii reads:

Ua Mau Ke Ea O Ka Aina I Ka Pono?

(The Life of the Land is Perpetuated in Righteousness)

But I digress...

The problem starts out with an image that essentially shows us all 3 forms of a rolling, sliding, both (although it is NOT described that way... hmph):

Please, please, please, annotate your work.... this is tough stuff! This is a LEARNING exercise, not a doing excercise!!!

The problem describes those three situations and then encourages us to express all our answers in terms of M, R, vo, mu and fundamental constants.

Then it gets a wee bit intimidating:

Let's back up a bit.... let's say I asked you to start with Newton's 2nd Law and derive a differential equation to then find the impulse of an object of mass m moving at a velocity v and stopping in a time of t seconds

So.. let's start with Newton's 2nd:

F=ma

We know that acceleration is the first derivative of velocity

F=m(dv/dt)

TA DA! We have a differential equation, although we usually take that a step further to have differential on each side, so let's do that now.

Fdt = mdv

But if we are talking about the macro world and not the infintessimal differential world that becomes:

F∆t = m∆v

Or more succinctly

F∆t = ∆p

Soooooooooooooooo

Let's get started:

Please, please, please, annotate your work.... this is tough stuff! This is a LEARNING exercise, not a doing excercise!!!

Write down Newton's 2nd in linear or rotational flavor (converse with your groupies and do that now).

 

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F = ma

F = mrω²

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Analyze what we know about a spherical object that is rotating AND rolling(converse with your groupies and do that now)

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Please, please, please, annotate your work.... this is tough stuff! This is a LEARNING exercise, not a doing excercise!!!

• A rotating sphere that is also rolling has kinetic energy and rotational kinetic energy
• A sphere will roll because of the friction between the rolling surface and the ball. If there is no rolling friction, the ball will slide, not roll.
• This is a force problem so let's write a sum-of-the-forces equation (ALWAYS!)

∑F_y=ma_y (Looks boring)

∑F_x=ma_x (Looks interesting so list forces!)

WAIT a sec, there's only one force in x, friction so...

F_x= f

but we know f = μmg so:

F_x= μmg

so... we're done with this part, right?

 

ugh, no...

F_x= -μmg

Now take one of the parts of that equation and represent it as the derivative of two other quantities(converse with your groupies and do that now)

 

 

F_x= -μmg

ma= -μmg

HEY! The m's cancel!!!

m(dv/dt) = -μmg

We DID IT! YAYYYYYYY!

The problem only wants us to go this far ("Derrive a differential equation that CAN BE used to find...", they aren't actually asking us to do that)

But notice that if we wanted to, we could go even further:

Now rearrange:

dv = -μgdt

Integrate both side:

∫dv = ∫-μgdt

Recognize that μg are constant and solve:

v = -μgt

Y*I*K*E*S but also very, very elegant. Diff eqs ROCK!

Now please work with your groupies on the next part

Please, please, please, annotate your work.... this is tough stuff! This is a LEARNING exercise, not a doing excercise!!!

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Now let's finish up the next coupla parts:

Coursework: #59

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