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At for help in understanding the concepts that are important in this problem. Take two quarters and lay them on a table. Press down on one quarter so it cannot move...

Question

At for help in understanding the concepts that are important in this problem. Take two quarters and lay them on a table. Press down on one quarter so it cannot move. Then, starting at the $12: 00$ position, roll the other quarter along the edge of the stationary quarter, as the drawing suggests. How many revolutions does the rolling quarter make when it travels once around the circumference of the stationary quarter? Surprisingly, the answer is not one revolution. (Hint: Review the paragraph jus

At for help in understanding the concepts that are important in this problem. Take two quarters and lay them on a table. Press down on one quarter so it cannot move. Then, starting at the $12: 00$ position, roll the other quarter along the edge of the stationary quarter, as the drawing suggests. How many revolutions does the rolling quarter make when it travels once around the circumference of the stationary quarter? Surprisingly, the answer is not one revolution. (Hint: Review the paragraph just before Equation $8.12$ that discusses how the distance traveled by the axle of a wheel is related to the circular arc length along the outer edge of the wheel.)



Answers

(II) The axle of a wheel is mounted on supports that rest on a rotating turntable as shown in Fig. $46 .$ The wheel has angular velocity $\omega_{1}=44.0 \mathrm{rad} / \mathrm{s}$ about its axle, and the turntable has angular velocity $\omega_{2}=35.0 \mathrm{rad} / \mathrm{s}$ about a vertical axis. (Note arrows showing these motions in the figure.) $(a)$ What are the directions of $\overline{\omega}_{1}$ and $\vec{\omega}_{2}$ at the instant shown? $(b)$ What is the resultant angular velocity of the wheel, as seen by an outside observer, at the instant shown? Give the magnitude and direction. (c) What is the magnitude and divection of the angular acceleration of the wheel at the instant shown? Take the $z$ axis vertically upward and the direction of the axle at the moment shown to be the $x$ axis pointing to the right.

Hello, everyone. This is problem. 72 from chapter 10 says place two cores on table with their rims touching. Then, while holding 1/4 fixed, roll the other one around without slipping around the circumference of the fixed quarter. Tell us until it's complete. One round trip. How many revolutions is the rolling quarter made about its center? Okay, if we draw the situation out, we have just basically 2/4. That air touching just at a point. And, um, we're doing is we're rolling this quarter around without slipping around the outside of the inter quarter. Okay, so we know that this quarter is moving around a circle that has radius twice. What a, um, twice the radius of a normal quarters. This is like two are here. And so when this gives in the circle at ah, moving at the same, I guess the same relative angular velocity. We know that this one is going to have twice the linear speed and thus should cover twice the distance twice the yet twice the distance around the circumference. And so what that means is that, um, this quarter should make two full revolutions

So let's go with them. Examined all the answer twisters for this one. So hey says that it's going to be the speed at which the wheel is ruling and we're looking to represent this graph. Well, the speed if it's going the speed is going to be constant. So if we wanted to draw a graph with speed, it should be like something like this, because the speed of that gonna be slowing down and then stopping and going up again and then slowing down and stopping that would not make any sense. So the graph is just to the biologist ruling at a constant rate because the tolls tells us it's going to be at a constant rate, so automatic he can't be a B, says the distance. The wheel is from its starting point. Well, the real ISC So the wheel is going that way. So the distance from its starting point right here is just gonna go farther and farther and farther away. So if you want to make a graph out of that, that's just gonna be something like this. So it definitely can't be b forsee. It says the distance of the mark on the rim to the center of the wheel. Well, the mark. Even if it's going to go that way and is going to go like this, the distance is always going to be the radius of the circle. So the Radius is always going to be constant, that we weird. If her Radius was getting bigger and smaller and getting to zero and stuff like that, it's just going to constant. So again it's going to look like that. So C doesn't work. But for D, it says the distance of the mark on the rim from the ground. Well, that makes a lot of sense, sweetie, because we know that from the mark. As we go this way, it gets closer and closer to the ground and hits the ground at a certain point and then gets fired and farther away. And then it gets closer and closer to the ground. Isaac turns again and then hits the mark. The mark hits the ground again, so that's exactly what we're illustrating right here. So it has to be D

All right. Swimming a drawl. Rather intricate picture here. Here's a circle. Previous B. Here's the outer circle. Ready to say? Say the point is here where we started. You're right. Triangle right there. No open notice is that this ark here is the same length. Is this arc here? Okay, so here is a year's pay. This says very being. And so if we look at this angle now, what I wanted I want this angle right here. Call Alfa and noticed that recalled this ark s this large ankle here. Call it beta. Well, Dana, Times B is equal to s, but data times a day also equals. Asked by the Ark Ling form, such is expressing that these two arks have the same length. Some particular data story particular beta is equal to a over B times they. And now, if I draw another line here noticed that this upper angle here is data and that this forms of writing right there. So I have this relationship that Ada is. He called to his Vader plus Alfa plus pie over too. I'm not actually the complement of alphabets office here. This angle, the compliment about just going to be pie over to minus Alfa. Come. All right. So what I want to do is I want to solve her. Oh, so Alfa, don't move out. Over. Been having Alsa. It's going to be equal to high and then said Pirate to aspire to plus data and then minus beta beta is a hay over b tense there. And so what we have then is the Alfa is hi minus Hey my being B times. They're stealing a little bit of outre And this is our key relationship here because we notice that the exit mike ornaments of this point here's the point that's being traced. X and my co ordinates are going to be the X and Y coordinates of the center cultural find using this triangle. But then we'll have to subtract these other coordinates here, and that's what really used half of it. So the X coordinate is going to be well, it's the X coordinate is a minus B. So that's a minus B and then Cosa Dada and then minus because I offer that's this little distance here sets the adjacent side of Alfa and the high part nous is the radius being and why is going to be a minus? B plus a minus B signed data and then minus He signed off. And the key thing is that we have Alfa actually in terms of a bee and died because over your and beer Constance hand data is the parameter now we want So here is actually our parametric equations that we have. And now if we set be to be a before and do quite a good bit of algebra, we can actually get the result. But this is kind of the difficult part specializing how to get the initial Parametric equations and back.

Question 30 in question. 30 were given a graph and told that, assuming that the rate of change of this graph or in other words, the slope of this graph remains constant. What is the angular displacement after eight seconds When looking at the graph of a line, we know that the equation can be written as y equals M X plus be using the information that we're given in the picture. We know that the Y value is the angular velocity. So I can rewrite Why, as Omega here, I'm gonna leave my slope as M for now and looking at the X axis. My ex is represented by time and my Y intercept the BUE value is my velocity a time zero In other words, it is my initial velocity. To find the slope of this graph, I would need to look at change in vertical or change in angular velocity over change in horizontal or change in time. But we know that change and angular velocity over change in time is really our acceleration or angular acceleration. So the equation of this line really looks like this. Well, this is one of our cinematic equations. What a substitute the values here to find my acceleration and then plug it into my formula for the equation of the line. I begin by choosing two points on the line. My change and angular velocity will be six minus negative nine. And my change in time will be five minus zero. Simplifying this well, Give me that. The angular acceleration is three radiance per second squared. In other words, that is my slope to find the angular displacement. After eight seconds, I'm going to make a list of information that I know starting with time, which I'm told is eight seconds. Then I know that my acceleration is three radiance per second squared and from the chart, I know that my initial angular velocity is negative. Nine radiance per seconds. And I am after angular displacement. So again, I'm going to choose a king a Matic equation that has these four values by then substitute the values that I've just found and sulfur angular displacement. So after eight seconds, my angular displacement will be 24 radiance


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