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Angular speed @ Ifr = 30 cm W =radian per sec and t =4 secFind the angle generated by P in time t_ b) Find the distance traveled by P along the circle in time Find...

Question

Angular speed @ Ifr = 30 cm W =radian per sec and t =4 secFind the angle generated by P in time t_ b) Find the distance traveled by P along the circle in time Find the linear speed of P

angular speed @ Ifr = 30 cm W = radian per sec and t =4 sec Find the angle generated by P in time t_ b) Find the distance traveled by P along the circle in time Find the linear speed of P



Answers

Suppose that point $P$ is on a circle with radius $r,$ and ray $O P$ is rotating with angular speed $\omega .$ For the given values of $r, \omega,$ and $t,$ find each of the following. (a) the angle generated by $P$ in time $t$ (b) the distance traveled by $P$ along the circle in time $t$ (c) the linear speed of $P$ $$r=30 \mathrm{cm}, \omega=\frac{\pi}{10} \mathrm{radian} \text { per } \sec , t=4 \mathrm{sec}$$

Okay, So here, given that R is equal to 30 centimeters, our omega is equal to high over 10 radiance per second. Radiance her second, and our tea is equal to four seconds. Okay, so then, for part A Well, our angular speed, a mega is equal to Fada over t. So therefore, we have Hi. Over 10 is equal to Fada over four. Um, so that would imply that data is equal to well, four pi over 10 and four pi over 10. Well, that's two pi over five. So here we have. Um, yeah, it's equal to high over five radiance. All right. And then, uh, be the distance traveled by P. Um, along the circle is going to be given, by essence, equal to our types data. So our types data is going to be 30 times two pi over five. Um, which is equal to, well, six times to pie, Right? 5% 5153266 times. You just have six times two pi, which is 12 time. So therefore, here we have that, as is equal to 12, 57 years, all right, and then for C uh, B is equal to s over tea. Okay, so it's gonna be equal to 12 pi over four, which is equal to three pack. So here we is, equal to three pie centimeters per second. So we have that deep is equal to three high centimeters for a second. Yeah.

Okay, so we're given here that R is equal to 27 years, and omega is equal to, well, pie over 12, um, radiance per second, and our T is equal to six seconds. Okay, so for parquet, Um, well, given the T is equal to six seconds, Um, you know, mega is equal to fade over t. So therefore, we have hi over 12 is equal to data over six. So that would then imply that data is equal to six pi over 12. Right? Which is equal to, well, pi over two. So therefore we have that data is equal to high over to radiance. Okay, on Ben for heartbeat. The distance traveled by P along the circle is well, is, um s is equal to our times data where we have our equal to 20 centimeters and data equal to equal to pi over two radiance. So therefore, we have 20 times pi over two, which is equal to well, 10 pie centimeters. So here we have. That s is equal to 10 Hi centimeters. All right. And then for part c. Well, um v here is equal to s over t. So therefore we just equal to 10 pi over six, which is equal to five pi over three. So here we have, um, so you go to five pi over three, uh, centimeters per seconds. Yeah. Uh huh.

In this problem. We're told that a point is going along. A circle with radius are in this case. Are is 20 centimeters. We're told the angular speed for this point is pi over 12 radiance per second. And we're told the total time it takes is six seconds, six seconds. So, in part A, the first thing we're being asked to do is find the angle that's generated by P in Time t. So we're looking forward the angle measure. Well, in order to do this, I'm going to use our formula for angular speed. So remember, the angular speed is equal to the angle. That's travel or angle that's generated divided by the time. And so now the reason why I'm using this formula is because we're told to angular speed is pi over 12. So I'm going to substitute that in and we're told t is six seconds. So if theta getting divided by six, so now we just need to solve for theta. Well, to do this, I just need to multiply both sides of our equation by six. Well, six times pi over 12 is six pi over 12, but we can reduce six by over 12 to pi over two. So what we found is that our angle will be pi over two radiance. That's the answer for party. Now let's go to part B in part B were asked to find the the total distance traveled by this point p on the circle Well, for trying to find a total distance traveled. We just need to find the arc length of this particular ah point and how far it's gone. Remember, the fine are calling, which is referred to as s. We just need to multiply the radius of the circle R times the angle that it goes that's generated well in this case, we were told our is 20 centimeters and the angle that we went we just found to be pi over two. So now when we multiply 20 by pi over two, that would give us 20 pi over two and 20 over to produces to 10. So what we found is that s will be 10 pi centimeters. Alright Now, lastly for part C, we're being asked to find the linear speed of this point. So we have a couple of formula so we can use I'm going to use the formula for when your speed is equal to the distance traveled as divided by the time because now we just have s which is 10 pie divided by our time, which we were told to be is six seconds. Well, we can reduce 10/6 to 5 thirds. So what we found is that are linear speed will be five pi over three Now for the units, It's our units for s, which is centimeters and our units for time are seconds. So our linear speed will be five pi over three centimeters per second.

So in the year in question we are told that there is a circle with a point P. On it. With the point P. With sender oh and radius 20 centimeters. It has given that array or P on the in this up good. The ray O. P. S. Moving at an angular speed omega which is given to us by by 12 gradients per second. And it is we are asked first to find the angle generated when the ray O. P. Most. But bye bye. 12 radiance per second in six seconds. Right? So in six seconds we can see that. We can use a formula that says. The angler state is given by teacher by T. Which is the angle generated by time. So we have all the values to substitute in this formula. So we can write by by 12 is equal to pita by six, from which we can write. Theta is equal to Bye bye 12 times six. Which is equal to bye bye to radiance. So this is the angry which is generated in six seconds. Now. Next we are asked in the question to find the distance traveled by the point P. Along the circle in this time. Right? So we can use a formula that says the length of arc has given us the radius times the angle generated tita. So if the angle generated inspired by two radiance and the radiant and the radius is already known as 20 centimeters, we can find the length of the ark through which the rate or P traveled. Right? So the GOP traveled like this for six seconds and reached over here. So this length is what we are going to find. So L. Is that distance? So the distance it travels in six seconds is 20 which is the radius times theta that is paid by two, which would be equal to 10 by centimeters. And lastly in the question, we are asked to find the linear speed of the point peak and the linear speed we is given by our times omega, where r is the radius and omega is the angular speed, the radius s 20 centimeters, and angular speed is by by 12, which gives us the linear speed as five bye bye, three centimetre per second. So this is the required answer. I hope you understood the method. Thank you.


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