12. Rotational Kinematics

Rotational Position & Displacement

# Displacement in Multiple Revolutions

Patrick Ford

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Hey, guys. So now we're gonna talk about rotational displacement. When you go around the circle multiple times, let's check it out. So if you make one full revolution, a revolution is a complete circle. A complete cycle, a complete rotation. Ah, complete spin. There's all these words. You make a full one around the circle, you've gone a total change in angle off three hundred sixty degrees. Everyone knows that a full app is three hundred sixty degrees or two Pi radiance. Okay, therefore Delta X, Remember, the Delta X equation is Delta X equals R Delta theta. That's how you link these two variables. Okay, it becomes two pi r. And what I've done here is I replaced Delta Theta with two pi because that's what Delta theta is. If you make a revolution, um, you might recognize this equation that Delta X equals two pi r. This is the circumference equation. The definition of circumference is the size, the length of the border of a circle, and that's linear distance. If you're driving your car around a circle, your odometer, which tells you how far you've traveled, would give you a quantity equal to two pi r Okay, so that's the linear distance as you go in a rotational motion around the circle. Now, that's if you spend once. If you spend once you get to be sixty, what happens if you spend twice? Then you get to be sixty times, too. Eso If you spend any end times, then your delta theta is three sixty times and or two pi times and obviously radiance. So instead of having a delta x of two pi r, you get a delta x of two pi r times n where n is the number of rotations. Okay, two more things you may need to know if you wanna know how many revolutions you go through, and we'll do. Example An example of this just now. All you gotta do is divide your number of angles by either three sixty degrees or by two pi. Okay. For example, if I tell you, I spun, um, seven hundred and twenty degrees. Andi, I wanna know how many how many revolutions that is you divided by three. Sixty and you get to That means I spun twice. Okay, Something with if it's in radiance. If it's in pies, you could just divided by two pints. And the last thing is, let's say you go around the circle many times and you end up over here, okay? If I want to know how far you end up, you don't gotta draw this. If I wanna know how far I end up, all you gotta do is you keep subtracting by three sixty until your angles less than three. Sixty. So, for example, if you, um, spun four hundred and ten degrees and I wanna know how far from zero you end up, four hundred ten is more than three. Sixty. So you made multiple revolutions. All I gotta do, subtract by three. Sixty, and you see that the answer is fifty. You keep doing this until your final answer is less than three sixty, which it is. So we're good to go. If it wasn't, you would subtract by three sixty again. Okay, Same thing with radiance. If it's in ratings, it just keeps attracting by two pi until the answer is less than two pi. Let's do a quick example here. All right, so starting from zero degrees of stroller ball circle starting from here, the two degrees is always the positive X axis, You make two point two revolutions. So we're using the letter N to represent the number of revolutions two point two around a circular path of radius. Um, twenty. You have a circular path of rate is twenty. That means that your radio distance from the middle little are is twenty you could use little are a big are interchangeably, little arch. Technically more correct because it's not big guards reserved for the radius of like a disc. Little R is the distance from the center. Okay, but the words were used kind of interchangeably. All right, so what is the rotational displacement in degrees? Hey, rotational displacement is delta theta, and we want that in degrees. I'm gonna put a little d e g here to indicate that we want to do this degrees and not in ratings. Well, if you spin once, you spend three hundred sixty degrees. But if you spend two point two times, you just multiply them, okay? And this is going to give you seven hundred and ninety two degrees. Seven hundred ninety two degrees is your How much is button Cool for? That's it for part B. How many degrees from zero are you again? We're just gonna subtract seven ninety two until we get to a number that's less than three hundred sixty. So I want to know how far from zero. Okay, so seven ninety two minus three. Sixty. That gives you four. Thirty two. We're gonna have to keep going because we're not below three. Sixty, minus three. Sixty. And then finally, that's the answer. Seventy two degrees. Okay, that's the final answer. And for part C, what is your linear displacement leader? Displacement. Remember, is Delta X. And if I wanna know Delta X, Delta X is our delta fatum R is the distance here. Twenty and Delta theta has to be in radiance. Delta Data has to be in radiance, so I cannot use seven ninety two. Okay, It's just a little gentle seventy two. I cannot use that. I'm gonna have to use in Radiance Radiance. This is going to be two pi times two point two right to piles of four rotation times two point two because we rotated two point two times. So if you multiply all of this, put in the calculator. Pious people in one four, one five. But you calculate has a button for that. Um, if you do all this, you get the distance is two hundred and seventy six meters. Okay, that's it for this. Hopefully makes sense. Let me know if you have any questions.

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