course Phy 121 ’”‚ãÖ„©‘Ó©¼ó³vœ÷íÈ“‰assignment #029
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11:44:35 `q001. Note that this assignment contains 15 questions. If an object moves a distance along the arc of a circle equal to the radius of the circle, it is said to move through one radian of angle. If a circle has a radius of 40 meters, then how far would you have to walk along the arc of the circle to move through one radian of angle? How far would you have to walk to move through 3 radians?
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RESPONSE --> To walk through 1 radian you would have to walk 40 meters and to walk through 3, you would have to walk through 120 meters. confidence assessment: 3
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11:45:28 Since 1 radian of angle corresponds to the distance along the arc which is equal to the radius, if the radius of the circle is 40 meters then a 1 radian angle would correspond to a distance of 40 meters along the arc. An angle of 3 radians would correspond to a distance of 3 * 40 meters = 120 meters along the arc. Each radian corresponds to a distance of 40 meters along the arc.
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RESPONSE --> ok self critique assessment: 3
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11:48:13 `q002. On a circle of radius 40 meters, how far would you have to walk to go all the way around the circle, and through how many radians of angle would you therefore travel? Through how many radians would you travel if you walked halfway around the circle? Through how many radians would you travel if you walked a quarter of the way around the circle?
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RESPONSE --> You would have to walk around 251.33 meters around the circle. This corresponds to 251.33 m/ 40 meters = .28 radians. If you were to walk halfway around, it would be 3.142 radians. If you walked 1/4 around you would travel 1.57 radians. confidence assessment: 3
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11:48:55 The circumference of a circle is the product of `pi and its diameter, or in terms of the radius r, which is half the diameter, C = 2 `pi r. The circumference of this circle is therefore 2 `pi * 40 meters = 80 `pi meters. This distance can be left in this form, which is exact, or if appropriate this distance can be approximated as 80 * 3.14 meters = 251 meters (approx). The exact distance 2 `pi * 40 meters is 2 `pi times the radius of the circle, so it corresponds to 2 `pi radians of arc. Half the arc of the circle would correspond to a distance of half the circumference, or to 1/2 ( 80 `pi meters) = 40 `pi meters. This is `pi times the radius so corresponds to `pi radians of angle. A quarter of an arc would correspond to half the preceding angle, or `pi/2 radians.
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RESPONSE --> This is a different way of presenting it, but I think I have all the same corresponding answers. self critique assessment: 2
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11:49:54 `q003. On a circle of radius 6 meters, what distance along the arc would correspond to 3 radians? What distance would correspond to `pi / 6 radians? What distance would correspond to 4 `pi / 3 radians?
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RESPONSE --> 3 radians would be 18 meters. I am not sure what the other questions are asking. confidence assessment: 2
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11:51:29 3 radians along the arc would correspond to an arc distance of 3 times the radius, or 3 * 6 meters, or 18 meters. `pi / 6 radians would correspond to `pi / 6 times the radius, or `pi / 6 * 6 meters = `pi meters. 4 `pi / 3 radians would correspond to 4 `pi / 3 * 6 meters = 8 `pi meters.
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RESPONSE --> Oh, I get the next two questions. You are trying to solve the problem using pi meters instead of the actual numbers. self critique assessment: 2
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11:53:03 07-18-2007 11:53:03 `q004. If you were traveling around a circle of radius 50 meters, and if you traveled through 4 radians in 8 seconds, then how fast would you have to be moving?
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NOTES -------> 4 radians corresponds to 200 meters. For 8 seconds it would be 200/8 = 25 m/s.
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11:53:05 `q004. If you were traveling around a circle of radius 50 meters, and if you traveled through 4 radians in 8 seconds, then how fast would you have to be moving?
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RESPONSE --> confidence assessment: 3
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11:53:16 If you travel 4 radians along the arc you half traveled an arc distance of 4 times the radius, or 4 * 50 meters = 200 meters. If you traveled this distance in 8 seconds your average speed would be 200 meters / (8 seconds) = 25 m/s.
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RESPONSE --> ok. self critique assessment: 3
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11:54:31 `q005. Traveling at 3 radians / second around a circle of radius 20 meters, how fast would you have to be moving?
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RESPONSE --> 3 radians corresponds to 60 meters. Therefore, it would be 60 meters/ second. confidence assessment: 3
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11:54:42 3 radians along the arc is a distance of 3 times the radius, or 3 * 20 meters = 60 meters. Moving at 3 radians/second, then, the speed along the arc must be 3 * 20 meters / sec = 60 meters /sec.
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RESPONSE --> ok. self critique assessment: 3
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11:55:37 `q006. If you know how many radians an object travels along the arc of a circle, and if you know the radius of the circle, how do you find the distance traveled along the arc? Explain the entire reasoning process.
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RESPONSE --> The radians equal the radius of the circle. You can calculate the circumference of the circle and then divide by the radians, or actuall the radius. confidence assessment: 3
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11:55:48 The distance traveled along the arc of circle is 1 radius for every radian. Therefore we multiply the number of radians by the radius of the circle to get the arc distance.
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RESPONSE --> ok. self critique assessment: 3
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11:56:42 `q007. If you know the distance an object travels along the arc of a circle, and if you know the radius of the circle, how do you find the corresponding number of radians?
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RESPONSE --> You would divide the total distance by the radius since this is the number also corresponds to the radian. confidence assessment: 3
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11:56:49 An arc distance which is equal to the radius corresponds to a radian. Therefore if we divide the arc distance by the radius we obtain the number of radians.
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RESPONSE --> ok. self critique assessment: 3
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11:58:31 `q008. If you know the time required for an object to travel a given number of radians along the arc of a circle of known radius, then how do you find the average speed of the object?
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RESPONSE --> You would get the circumference of the circle using 2pir. Then you would divide by the radius in meters. confidence assessment: 3
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11:59:21 If you know the number of radians you can multiply the number of radians by the radius to get the distance traveled along the arc. Dividing this distance traveled along the arc by the time required gives the average speed of the object traveling along the arc.
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RESPONSE --> Oh, you don't have to get the circumference since we know the number of radians as well as the radius. self critique assessment: 2
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12:00:23 `q009. If you know the speed of an object along the arc of a circle and you know the radius of the circle, how do you find the angular speed of the object in radians/second?
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RESPONSE --> You would multiply the radius by the speed to get the number of seconds per radian rotation. confidence assessment: 3
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12:01:20 The speed of the object is the distance it travels along the arc per unit of time. The angular velocity is the number of radians through which the object travels per unit of time. The distance traveled and the number radians are related by the fact that the distance is equal to the number of radians multiplied by the radius. So if the distance traveled in a unit of time is divided by the radius, we get the number of radians in a unit of time. So the angular speed is found by dividing the speed along the arc by the radius.
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RESPONSE --> Ah, now that I make up a problem, I see that you would divide and not multiply by the radius. self critique assessment: 2
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12:03:34 `q010. We usually let `d`theta stand for the anglular displacement in radians between two points on the arc of the circle. We usually let `omega stand for the angular velocity in radians / second. We let `ds stand for the distance traveled along the arc of a circle, and we let r stand for the radius of the circle. If we know the radius r and the arc distance `ds, what is the anglular displacement `d`theta, in radians?
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RESPONSE --> We would divide `ds/r to get the radians. confidence assessment: 3
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12:03:50 Since an angular displacement of 1 radian corresponds to an arc distance equal to the radius, the anglular displacement `theta in radians is equal to the number of radii in the arc distance `ds. This quantity is easily found by dividing the arc distance by the radius. Thus `d`theta = `ds / r.
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RESPONSE --> ok. self critique assessment: 3
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12:04:55 `q011. If we know the radius r of a circle and the angular velocity `omega, how do we find the velocity v of the object as it moves around the arc of the circle?
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RESPONSE --> I an not sure about the angular velocity and velocity of the object. confidence assessment: 0
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12:06:20 The angular velocity is the number of radians per second. The velocity is the distance traveled per second along the arc. Since an angular displacement of 1 radian corresponds to an arc distance equal to the radius, if we multiply the number of radians per second by the radius we get the distance traveled per second. Thus v = `omega * r.
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RESPONSE --> Alright, now I see how each is different. If you multiply the radius, which is the radian, by the distance per second along the arc then you would get the velocity. self critique assessment: 2
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12:08:20 `q012. We can change an angle in degrees to radians, or vice versa, by recalling that a complete circle consists of 360 degrees or 2 `pi radians. A half-circle is 180 degrees or `pi radians, so 180 degrees = `pi radians. How many radians does it take to make 30 degrees, how many to make 45 degrees, and how many to make 60 degrees?
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RESPONSE --> I am not sure how to solve this without the actual radian number. confidence assessment: 0
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12:10:19 30 degrees is 1/6 of 180 degrees and therefore corresponds to 1/6 * `pi radians, usually written as `pi/6 radians. 45 degrees is 1/4 of 180 degrees and therefore corresponds to 1/4 * `pi radians, or `pi/4 radians. 60 degrees is 1/3 of 180 degrees and therefore corresponds to 1/3 * `pi radians, or `pi/3 radians.
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RESPONSE --> I was trying to compare it with 360 degrees, but now I see how to solve it. self critique assessment: 2
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12:13:58 `q013. Since 180 deg = `pi rad, we can convert an angle from degrees to radians or vice versa if we multiply the angle by either `pi rad / (180 deg) or by 180 deg / (`pi rad). Use this idea to formally convert 30 deg, 45 deg and 60 deg to radians.
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RESPONSE --> 30 degrees would be 30 * pirad/180 = pi rad/ 6. 45 degrees = 40 * pirad/180 = pi rad/ 4 60 degrees = 60 * pirad/180 = pi rad / 3 confidence assessment: 3
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12:15:28 To convert 30 degrees to radians, we multiply by the rad / deg conversion factor, obtaining 30 deg * ( `pi rad / 180 deg) = (30 deg / (180 deg) ) * `pi rad = 1/6 * `pi rad = pi/6 rad. To convert 45 degrees to radians we use the same strategy: {}45 deg * (`pi rad / 180 deg) = ( 45 deg / ( 180 deg) ) * `pi rad = 1/4 * `pi rad = `pi/4 rad. To convert 60 degrees: 60 deg * (`pi rad / 180 deg) = ( 60 deg / ( 180 deg) ) * `pi rad = 1/3 * `pi rad = `pi/3 rad.
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RESPONSE --> Well I thought I did this right, but we got different answers in the end. self critique assessment: 2
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12:17:12 `q014. Convert 50 deg and 78 deg to radians.
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RESPONSE --> 50 * pirad/180 = .277 pirad. 78 * pirad/180 = .433 pirad confidence assessment: 3
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12:18:50 50 deg * (`pi rad / 180 deg) = ( 50 deg / ( 180 deg) ) * `pi rad = 5/18 * `pi rad = (5 `pi/ 18) rad. 78 deg * (`pi rad / 180 deg) = ( 78 deg / ( 180 deg) ) * `pi rad = 78/180 * `pi rad = (13 `pi/ 30) rad.
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RESPONSE --> I see how we kind of have to group to get pi and the radians separately in the end. self critique assessment: 2
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12:20:00 `q015. Convert (14 `pi / 9) rad to degrees.
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RESPONSE --> I would just multiply that by 180, but that gives a really large number. confidence assessment: 0
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12:21:17 Since the angle is given radians, we need to multiply by deg / rad to get the angle in degrees. (14 `pi / 9) rad * ( 180 deg / (`pi rad)) = ( 14 `pi / 9 ) * (180 / `pi ) deg = ( 14 * 180 / 9) * (`pi / `pi) deg = 14 * 20 deg = 280 deg.
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RESPONSE --> Oh, I see now that you have to multiply by 180/pirad which cancels out the pi rad and leaves degrees. self critique assessment: 2
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