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so in the previous couple videos we defined all these new rotational motion variables and we define them exactly the same way we defined all these linear motion variables so for instance this angular displacement was defined the exact same way we defined regular displacement it's just this is the angular position as opposed to the position the regular position similarly this angular velocity was the angular displacement per time just like velocity was the regular displacement over time and the angular acceleration was the change in the angular velocity per time just like regular acceleration was the change in regular velocity per time and so because these definitions are exactly the same except for the fact that the linear motion variable is replaced with its angular counterpart all the equations results in principles we found and derived for the linear motion variables will also hold true for the rotational motion variables as long as you replace the linear motion variable in that equation with its rotational motion variable counterpart and it even works with graphs so let's say you had a velocity versus time graph and it looked like this since we already know from 1d motion that the slope of this velocity versus time graph is equal to the acceleration that means on an angular velocity versus time graph the slope is going to represent the angular acceleration because the relationship between Omega and alpha is the same as the relationship between V and a similarly the area underneath the curve on a velocity versus time graph represented the displacement so that means that the area under the curve on an Omega versus time graph an angular velocity versus time graph is going to represent the angular displacement and so if you remember from 1d motion the way we derived a lot of the 1d kinematic formulas that related the linear motion variables was by looking for areas under a velocity graph we could do the same thing for the rotational motion variables we could find this area related to Omega and alpha and we'd get the rotational kinematics formulas but we already know since these are all defined the same way the linear motion variables are defined we're going to get the exact same equations just with the linear motion variable replaced with its rotational motion variable so let's write those down first we'll write down the linee motion kinematic formulas if you remember they looked like this so there they are these are the four kinematic formulas that relate the linear motion variables but remember this only works these equations only work if the acceleration is constant but if the acceleration is constant these four kinematic formulas are a convenient way to relate all these kinematic linear motion variables now if you wanted rotational kinematics formulas you could go through the trouble that we went through with these to derive them using areas under curves but since we know the relationship between all these rotational motion variables is the same as the relationship between the linear motion variables I can make rotational motion kinematic formulas simply by replacing all of these linear variables with the rotational motion variable counterpart so let's do that so in other words instead of V the velocity the final velocity I would have Omega the final angular velocity instead of the initial the initial velocity I'd have the initial angular velocity instead of the acceleration I'd have the angular acceleration and time is just time so there's no such thing as angular time or linear time as far as we know there's only one time and that's T and that works in either equation so you could probably guess when are these rotational motion kinematic formulas going to be true it's going to be one the Alpha the angular acceleration is constant and so you can keep going through wherever you had an X that was the regular position you'd replace it with theta the angular position so I'll replace all these X's with Thetas we replace all of our accelerations with angular accelerations and then I'll finish cleaning up these the initial and V finals and then we've got them these are the rotational kinematics formulas they're only true if the angular acceleration is constant but if it is constant these are a convenient way to relate all these rotational motion variables and you can solve a ton of problems using these rotational kinematics formulas and in fact use these the exact same way you use these regular kinematic formulas you identify the variables that you know you identify the variable that you want to find and you use one of the formulas that lets you solve for that unknown variable so let me show you some examples do a couple examples using these formulas because it takes a while before you get the swing of them so let me copy these we're about to use these now let's tackle a couple examples of rotational kinematics formula problems so let me get rid of all this and let's tackle this problem let's say you had a four meter long bar that's why I've had this bar here the whole time to show that it can rotate it starts from rest and it rotates through five revolutions with a constant angular acceleration of thirty radians per second squared and the question is how long did it take for this bar to make the five revolutions so what do we do how do we tackle these problems you first identify all the variables that you know so it said that it revolved five revolutions that's the amount of angle that it's gone through but it's in weird units this is in units of revolutions so we know what the Delta Theta is five revolutions but we want our Delta Theta always to be in radians because look at our acceleration was given in radians per second squared you've got to make sure you compare apples to apples I can't have revolutions for Delta Theta and radians for acceleration you've got to pick one unit to go with and the unit we typically go with is radians so how many radians would five revolutions be one revolution is 2pi radians because one time around the entire circle is 2 pi radians that means five revolutions would be five times two pi radians which gives us ten pi radians so we've got our angular displacement what else do we know it tells us this 30 radians per second squared that is the angular acceleration so we know that alpha is 30 radians per second squared you can write the Radian you can leave it off sometimes people write the radians sometimes they leave it blank so you could write one over second squared if you wanted to that's why I left this blank over here but we could write radians if we wanted to and should this alpha be positive or negative well since this object is speeding up it started from rest that means it's sped up so our direction of the angular displacement has to be the same direction as this angular acceleration in other words if something's speeding up you have to make sure that your angular acceleration has the same sign as your angular velocity and your angular velocity will have the same sign as your angular displacement assistance we call this positive 10 pi radians and the object sped up we're going to call this positive 30 radians per second squared if this bar would have slowed down we'd have had to make sure that this alpha has the opposite sign as our angular velocity but that's only two rotational kinematic variables you always need three in order to solve for a fourth so what's our third rotational kinematic variables it's this that it says the object started from rest so this is code this is code word for Omega initial is zero initial angular velocity is zero because it starts from rest so that's we can say down here that's our third known variable and now we can solve we've got three we can solve for a fourth which one do we want to solve for it says how long so that's the time we want to know the time that it took all right so these are the variables that are involved we want to know the time we know the top three the way I figure out what kinematic formula to use is that I just asked which variables left out of all of these I've got my three knowns and my one unknown that I want to find which variable isn't involved and it's omega final so Omega final is not involved here at all so I'm going to use the rotational kinematics formula that does not involve Omega final I'll put these over here so I'll look through them first one's got Omega final I don't wanna use that one because I wouldn't know what to plug in here and I don't want to solve for it anyway I don't want the second one this third one has no Omega final so I'm going to use that one so let's just take this we'll put it over here so we know Delta Theta Delta Theta was ten PI radians and we know Omega initial was zero so this whole term is 0 0 times T is still zero so that's all zero and we have one half the angular acceleration was thirty and the time is what we want to know and you can't forget that that's squared so now we just solve this algebraically for time we multiply both sides by two that would give us 20 pi then we divide by 30 and that'll end up giving us 20 pi and technically that's 20 PI radians divided by 30 radians per second squared and then you have to take a square root because it's T squared and if you solve all this for T I get that the time ended up taking about 1.45 seconds and our units all canceled out the way they should here radians cancelled radians you ended up with second Squared's on the top you took a square root that gives you seconds to end with now the second part Part B says what was the angular velocity after rotating for five revolutions now there's a couple ways we could solve this because we solved for the time we know every variable except for the final angular velocity so I could use any of these now to me this first one is the simplest there's no squares involved there's not even a ratio or anything so let's use this we could say that Omega final is going to equal Omega initial that was just zero plus the angular acceleration was 30 and now that we know the time we could say that this time was one point four five seconds and that gives me a final angular velocity of forty three point five radians per second that's how fast this thing was revolving in a circle the moment it hit five revolutions so that was one example let's do another one let's carry our kinematic formulas with us we could use those so we get rid of all that let's check this one out says this four meter long bar is going to start this time it doesn't start from rest this time it starts with an angular velocity oh we're not going to rotate that whoa that'd be a more difficult problem we're going to rotate this this four meter long bar starts with an angular velocity of 40 radians per second but it decelerates to a stop after it rotates 20 revolutions and the first question is how fast is the edge of the bar moving initially in meters per second so in other words this point on the bar right here is going to have some velocity this way we want to know what is that velocity initially in meters per second well this isn't too hard we've got a formula that relates the speed to the angular speed you just take the distance from the axis to the point where you want to determine the speed and then you multiply by the angular velocity and that gives you what the speed of that point is so this R let's be careful this is always from the axis and in this case this is the axis right there the distance from the axis to the point we want to find is in fact the entire length of this bar so this will be four meters so they'll find the speed we could just say that equal to four meters since you want to know the speed of a point out here that's four meters from the axis and we multiply by the angular velocity which initially was 40 radians per second and we get the speed of this point on the rod four meters away from the axes is a hundred and sixty meters per second that's really fast and that's the fastest point on this rod if you're going to ask what the speed of the rod would be halfway that would be half as much because this would only be this R right here would only be two meters from the axis to that point only two meters and the closer in you go the smaller the R will be the smaller the speed will be so these are going to travel these points on the rod down here don't travel very fast at all because there are so small all these points have the same angular velocity they're all rotating with the same number of radians per second but the actual distance of the circle they're traveling through is different which makes all of their speeds different so that answers part a we've got how fast in meters per second it was going 160 meters per second and the next part asks what was the angular acceleration of the bar all right this one we're going to have to actually use a kinematic formula for we'll bring these back put them over here again the way you use these you identify what you know we know the initial angular velocity was 40 so this time we know Omega initial 40 radians per second said it revolved 20 revolutions that's Delta Theta but again we can't just write 20 we've got to write this in terms of radians if we're going to use these radians per second they have to all be in the same units so it's going to be 20 revolutions times 2 PI radians per revolution so that's 40 PI radians what's our third known you always need a third known to use a kinematic formula it's this it says it decelerates to a stop which means it stops that means Omega final the final angular velocity is zero and we want the angular acceleration that's alpha so this is what we want to know we want to know alpha we know the rest of these variables again to figure out which equation to use I figure out which one got left out and that's the time I was neither given the time nor was I asked to find the time since this was left out I'm gonna look for the that doesn't use time at all and that's not the first one that's not the second or the third it's actually the fourth so I'm gonna use this fourth equation so what do we know we know Omega final was zero so I'm going to put zero squared but zero squared is still zero equals Omega initial squared that's 40 radians per second squared and then is going to be plus two times alpha we don't know alpha but that's what we want to find so I'm going to leave that as a variable and then Delta theta we know Delta Theta was 40 PI radians since it was 20 revolutions and if you solve this algebraically for alpha you move the 40 over to the other side so you'll subtract it you get a negative 40 radians per second squared and then you got to divide by this 2 as well as the 40 PI radians which gives me negative six point three seven radians per second squared why is it negative well this thing slowed down to a stop so this angular acceleration has got to have the opposite sign to the initial angular velocity we call this positive 40 that means our alpha is going to be negative so recapping these are the rotational kinematics formulas that relate the rotational kinematic variables their only true if the angular acceleration is constant but when it's constant you can identify the three known variables and the one unknown that you're trying to find and then use the variable that got left out of the mix to identify which kinematic formula to use since you would use the formula that does not involve the variable that was neither given nor asked for

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