all right so welcome back we're starting a new unit on something called rotational dynamics so we've already looked at kinematics in a few different ways for example we looked at one-dimensional kinematics and we got used to the idea of displacements and velocities and accelerations and then we took that idea and we moved it into two dimensions where we looked at projectiles and looked at how perpendicular vectors are independent from one another and act independently if we want to properly describe how things move in circles or if an object is rotating how it behaves and we have to look at this idea of rotational dynamics so before we get started here I want to ask you a quick question suppose I've got four objects on a ramp and they're all in the same inclined ramp and let's say for the sake of argument they all have the exact same mass and there's only just enough friction so that these objects all roll they don't slide down the hill they're kind of roll but that the four objects are different there's a the green one is a spherical shell so like like a basketball where all the masses on the outside the red one is a solid sphere so maybe like a pool ball the ring the blue one is a ring so a cylindrical shell and then the the gold one there is a is a solid cylinder so it's like a can of tuna and again for the sake of purposes let's assume they all have the same radius now if I let this go and I ask you which one is gonna win the race I would expect many of you would predict well I think it's gonna be a tie well you might be surprised to learn though is that when we in fact run this despite the fact they have the same mass the same radius they finished the race in much different times so for example it looks like that spear wins pretty handily just ahead of the cylinder and the ring is just way behind and so the question should be well why is that what is it about these different shapes that cause them to to move down the ramp at different rates if we want to do that as a comparison if I add a block of ice here that's just gonna slide so a frictionless cubes gonna run down this ramp if I run those against that cube well the cube wins the whole race hands down no problem so something about those shapes and something about the fact that they're roll and rotating seems to be changing their behavior and it's really about the shape themselves not the mass not the radius but that where the mass is distributed so we're gonna look at a few different ideas here the first one I want to introduce to you is this idea of rotational angle okay so if we have an object like a CD which is a compact disc for all you children out there if a CD is spinning around then you could describe the motion of that CD as it's spinning by talking about the rotational angle which is basically just the amount of rotation and what we're gonna see over and over again when we compare rotational dynamics to linear dynamics is that there's gonna be parallels so for example how far an object rotates the angle that it rotates through is really gonna be analogous to the linear distance that an object travel if you think about one-dimensional kinematics like displacement and velocity and acceleration the angle that you rotate through is gonna be analogous to the linear distance so we define the rotational angle we just call it theta and it's just the ratio of the arc length to the radius of curvature so basically you may have learned this in and your math classes that if you have a if you have a circle radius and it travels to a through an arc length here so it's got an arc length the arc length being the distance that it travels on the outer edge of the circle so you could imagine a single point here on the edge of the CD and it traveling a certain distance that would be your arc length and so the changing your angle here is just going to be the change in your arc length divided by the radius where this is our rotational angle and this is measured in radians and more on that in a second arc length is s and r is our radius so it's worth pointing out that for a given circle if you were to travel all the way around the circle all the way around the circle you would have completed one revolution if you haven't seen this idea of radians radians is a way of basically of measuring angles but instead of using degrees we're gonna use radians and what we say is that if you imagine a circle if you imagine a circle that had a radius of one affectionately called the unit circle and you imagine traveling all the way around outside that circle doing one complete loop of that circle you might ask how far have you traveled well if the radius is one if the circumference is 2 PI R then you would have traveled 2 pi and so this is how we kind of define our radians we say that 2 pi radians or rad is equal to one revolution so you could say that that's also equal to 360 degrees so for example if you had a if you did a 180 degree rotation that would just be PI radians and so you're gonna have people to convert between pi between radians and and degrees so if the rotational angle it gives us an idea of the of how far the object has rotated the the displacement if you will then the angular velocity gives us a measure of how fast and so the symbol for angular velocity is omega which just looks like a fancy w and it's just the measure of the rate of change of the rotational angle so how quickly the rotational angle is changing now just for just for a conceptual look here let's take a look at something here so imagine I've got two I've got two circles and I've got these these objects at the end of the circle now imagine that the second circle is much bigger than the first okay but when I when I turn it on here both of these both of these circles move around with these points these red points move around the individual circles with the same frequency which is to say they take the same amount of time to complete a loop or a cycle now when you look at this it's pretty obvious that the red dot over here on the right this red dot is traveling much faster than that red dot because it travels a much bigger distance in the same amount of time now imagine that these weren't two separate objects imagine that they were actually the same like imagine that you had a merry-go-round or a yes even a CD spinning if you tried to describe the motion of how fast this CD is spinning you'd run into a big problem because the question would be well how fast is what part of it spinning if you look in the very outer edge the CDs fin spinning very quickly if you look down closer to the center the CD isn't moving nearly as fast so how fast something spinning it's not really useful to describe it in terms of regular velocity in terms of meters per second and so instead what we talk about is okay well how many radians per second is it going through notice how both the outer edge and the area it doesn't matter how close we get to the center of the circle they both have the same angular velocity which is to say they both travel through the same number of radians in the same amount of time and so when we when we talk about rotating objects we're gonna have to talk about rotational angle instead of displacement angular velocity instead of velocity and so we define our angular velocity as our change in rotational angle over time but remember that we just learned back here that the rotational angle is Delta s over R so we could say that this is equal to Delta s over our times T well if you think about it s over T which is distance over time or displacement over time that's really just velocity so since velocity is just displacement over time then I can substitute this in here and I get velocity over radius so long story short the angular velocity is just equal to the velocity divided by the outer radius or I should say the angular velocity is equal to the velocity of a point on the outer edge divided by that radius and so the units for angular velocity are radians per second okay let's do a quick example so mm-hmm calculate the velocity of a the angular velocity pardon me of a point three meter radius car tire when the car travels at about 54 kilometers an hour so first thing we're gonna do is I have no use for a kilometers an hour divided by three point six and that'll be right around 15 meters per second I know that my angular velocity is equal to the velocity of a point on the outer edge of the tire mmm divided by the radius if it's traveling at fifteen meters per second and then the radius is zero point three zero zero meters that's gonna give me 50 and then rads per second now it's worth pointing out if you look closely here radians are kind of this funny unit they sort of they're a bit of a phantom unit they sort of disappear when we don't want them there and then they show up when we need them so if you look at the unit's meters per second divided by meters the meters cancel out you might look at that and say well that should just be in per seconds or even Hertz but as a matter of fact since it is a velocity an angular velocity it's gonna be radians per second we didn't include radians in here and so I should point out that what we're talking about here is we're taking the linear velocity or linear speed of the car how fast the car is traveling in the forward direction and we're translating that to how quickly the tires must be rotating of course the distance travelled by the car in one rotate is equal to the circumference of the car tire so if the car tire rolls one complete rotation that would equal the the linear distance traveled mm-hmm so what would be what would the angular velocity be for a car traveling at the same speed but the head tires which were four times larger so how quickly would the tires have to rotate if the tires were four times bigger so just think about that for a second when the tires have to spend faster or slower would they have to complete more or fewer revolutions per second and so since we know that Omega the angular velocity is velocity divided by R then we know that Omega is proportional to one over R if we have the same speed I'm not really worried about that V and so you could say something like well if Omega one is V 1 over R 1 and that equals 50 radians per second then the new Omega Omega 2 would just be V 1 over R 1/4 and so that would be 12.5 radians per second because of the proportionality okay well we've looked at angular rotational angle which is like displacement we've looked at angular velocity which is like linear velocity and so now of course we're gonna look at angular acceleration so in previous unit we looked at uniform circular motion so keep in mind with uniform circular motion the whole time we assumed the motion was uniform which is that to say that it wasn't accelerated well I should say it wasn't speeding up or slowing down the speed was staying constant and so with angular acceleration we're talking about a situation where not only is something rotating or moving in a circle but it's also speeding up so you could imagine like a merry-go-round ride that's not moving yet and then it slowly starts to speed up faster and faster and faster so the symbol for angular acceleration is alpha and we define it as the rate of change of the angular velocity and so alpha is just the change in omega over time and you can imagine here the units would just be radians per second squared okay so we'll do a couple of examples suppose a Rockler student puts their bicycle on a stand and starts the rear wheel spinning from rest to a final angular velocity of two or fifty rpms or revolutions per minute in five seconds what is the angular acceleration in radians per second squared okay well this really is just more of a problem of converting our units so we know that we've got 250 revolutions per minute okay that's what rpm stands for well I know that there are two pi radians for every one revolution and I know that there is one minute for every 60 seconds and so if I look closely here I can see that revolutions is going to cancel out minutes is going to cancel out and 250 revolutions per minute works out to be 20 6.18 if you want some sig figs rads per second once I know that then it's pretty straightforward to say okay well my angular acceleration is my change in angular velocity divided by time change in is final minus initial so 26.1 8-0 divided by five point zero seconds and this is gonna work out to be right around five point two radians per second squared so suppose they now slam on the brakes and causing an angular acceleration of negative eighty seven point three rads per second squared how long does it take the bike to stop well again angular acceleration is just gonna be a change in angular velocity divided by time and so time will just be my change in angular velocity divided by acceleration now I'm going from a final to a final angular velocity of zero I started at twenty six point one eight and I'm dividing that by a negative acceleration because it's slowing down and this ends up being 0.30 seconds so it's worth just pausing here for a second and just considering the relationship between circular motion and linear motion so like I kind of hinted at before we've already talked about things moving in circles we've talked about satellites in orbit we've talked about you know merry-go-rounds going around and people driving over in rollercoasters going over hills and things like that but what we sort of have like I said before what we sort of neglected at this point is we just assumed that everything was moving at a constant speed and so what happens if you have something that's yes moving in a circle but also speeding up and things get a little bit complicated so if an object experiences an angular acceleration then the velocity of any point on the object must be increasing or decreasing so the linear acceleration the linear acceleration is tangent to the path of the rotation and so then we'll use a slightly different symbol a with a little T for tangent so if you look at this diagram over here to kind of explain what's happening so it just ignore just ignore this arrow for a second suppose we've got an object that's spinning in a circle well we all know the story here that there's gonna be a force directed towards the center of the circle and so that's gonna be my centripetal acceleration is going to be inwards towards the center of the circle we know that at that point you've got a little point here on the outer edge of our circle and that the velocity at that point is tangent to the circle none of that is new what we're saying now though is that if this object is also speeding up so think about a merry-go-round that's not moving and it's slowly speeding up getting faster and faster and faster as it speeds up it must also be accelerating in this direction here and we call this a T so the acceleration which keeps it in a circle is still centripetal acceleration it's still this right here it's still equal to V squared over R nothing's changed there but on top of that we could have we could be causing this object to move faster and faster in a circle and then it would have an acceleration which you can see now is in the same direction as the velocity keep in mind that perpendicular vectors are independent so this acceleration the centripetal acceleration could never change the speed of the object because they're perpendicular but the linear acceleration here can so the linear acceleration we're gonna define as a change in velocity over time but you remember that we talked about linear velocity and we said that angular velocity is linear velocity divided by the radius or put another way the linear velocity is the angular velocity times the radius so I can substitute this into this expression here and I get I get Omega are over t what we just defined Omega R we just defined our angular velocity oh sorry we just defined our angular acceleration as being the change in omega over t and so you can see right here that's exactly what I have and so I can substitute this in here and I end up with alpha R so the linear acceleration of a point at the end of the circle is equal to the angular acceleration times the radius or just to reverse this all around the angular acceleration is equal to the linear acceleration divided by the radius okay so note again it says your note that the angular acceleration the angular acceleration is tangent to the circle whereas the centripetal acceleration points towards the center of the circle okay so with all of this having been said we've learned about angular we've learned about angular a rotational angle and how that's related to displacement we've learned about angular velocity how it's related to velocity we've learned about angular acceleration how it's relate to acceleration so now you're probably wondering hey we used to have some really fun formulas that we used to use for linear kinematics I wonder if we can use those again and the answer is you are in such luck that absolutely all the same formulas that we had in translational or one-dimensional kinematics so like your good friend V equals V naught plus 80 or D equals V naught tables when FA T squared or V equals V naught T plus 2 ad all of those have an analogy in the rotational world okay so for example V equals V naught plus 80 well we could just say that Omega equals Omega naught plus alpha T and so just keep in mind the the kind of the analogy we keep using so displacement we think of displacement the analogy to that is rotational angle and velocity the analogy there is angular velocity and acceleration the analogy there is angular acceleration so if you look at each of these cases you can see that we just substituted in different versions sorry the the analogous version of each of each variable okay so let's put these to work so a deep-sea fishermen hooks a big fish that swims away from the boat pulling the fish line from his fishing reel the whole system is initially at rest and the fishing line unwinds from the reel which has radius of 4.5 centimeters from its axis of rotation the reel is given an angular acceleration of 110 rads per second squared for 2 seconds what is the final angular velocity of the real well just like we would do with linear kinematics let's set up our variables we've got Omega Omega naught alpha theta and T so all the new players and so we know for example that it's initially at rest so the initial angular velocity is 0 being their acceleration is 110 rads per second the time is 2 seconds and so if I go back and look at my options here of all these options the one that's gonna give me the final angular velocity of the reel is this one Omega equals Omega naught plus a t subbing in the values we see the 0 plus 110 times 2 and so this is going to be 220 radians per second alright so at what speed is a fishing line leaving the reel after the two seconds collapse elapses so how fast and think about that to that the fishing line is on the outer edge of the reel so if you think about the reel spinning you think about a point at the edge of that reel spinning around the fishing line is traveling at that same linear velocity and so we could compare the way we sort of jump between angular and linear is through these formulas here so angular velocity is linear velocity over radius and so that's that the speed at a point on the outer edge so the velocity would equal the angular velocity times the radius which is 220 times 0.045 remember that was in centimeters and so nine point nine meters per second is how fast the line is actually coming off the reel okay so how many revolutions doesn't make the number of revolutions is going to depend on on theta and so if I look at my formulas here which one could I use to solve for theta knowing what I know I've actually got two options but I'll choose this one theta equals Omega naught T plus 1/2 alpha T squared this is again zero so this simplifies to 1/2 times 1 10 times 2 squared and so this is 220 radians now just be careful there that it didn't actually ask us for radians it's actually asking us for 4 how many revolutions and remember that 1 revolution is equal to 2 pi radians so I can multiply this by a conversion factor here of one revolution for every 2 pi radians and I get approximately 35 revolutions okay and then last but not least how many meters of fishing line came off the reel in this time again the way we jump between linear so this is a question how many meters is asking a question of linear distance how far did that fish swim away and so the way we jump between those two is with these formulas that the angular the rotational angle is equal to the arc length or the distance traveled divided by the radius and so the arc length which is equal to the linear distance is equal to theta times R so that's 220 times 0.045 which works out to be right around nine point nine meters and that's just a flew competitive with the eat earlier velocity don't look too much into that okay that's it for the first part of rotational dynamics