A merry-go-round is rotating at the constant angular speed of 3 RPM counterclockwise. The platform of this ride is a circular disk of radius 24
feet.
(a) If θ = 34 degrees, what are your xy coordinates after 4 minutes?
(b) If θ = 20 degrees, what are your xy coordinates after 45 minutes?
(c) If θ = −14 degrees, what are your xy coordinates after 6 seconds?
(d) If θ = -2.1 rad, what are your xy coordinates after 2 hours and 7 seconds?
since a merry-go-round is circulur theta is the arc of the circle (there is an image but I can't upload it onto here). i thought that maybe you would have to use the parametric equation of a circle to find the xy coordinate but i am not 100% positive
well, I will say that if θ is the central angle, then the arc length (distance traveled at the rim) is s = rθ
Now, if you can make θ a function of time, s(t) = r θ(t). But since you have already said what the rotation speed is, the time will tell you what θ is.
after t minutes, θ = 360*3t degrees
how would that give you the xy coordinate?
To answer these questions, we need to understand how the xy coordinates change in relation to the angular position θ, the constant angular speed of the merry-go-round, and the time passed. Here's how to approach each question:
(a) To find the xy coordinates after 4 minutes when θ = 34 degrees:
1. Convert the angular speed from revolutions per minute (RPM) to degrees per minute. Since 1 RPM equals 360 degrees, the angular speed is 3 * 360 = 1080 degrees per minute.
2. Multiply the angular speed by the time in minutes (4 minutes) to get the change in angle: 1080 * 4 = 4320 degrees.
3. Add the change in angle to the given θ value: 34 + 4320 = 4354 degrees.
4. Convert the degrees back to radians by dividing by 180 and multiplying by π: 4354 * π / 180 = 75.8709 radians.
5. Use the polar coordinate system conversion formulas to find the xy coordinates:
- x = r * cos(θ)
- y = r * sin(θ)
Here, r is the radius of the platform (24 feet).
Plug in the values: x = 24 * cos(75.8709) and y = 24 * sin(75.8709) to get the xy coordinates.
(b) To find the xy coordinates after 45 minutes when θ = 20 degrees:
1. Follow the same steps as in part (a) to calculate the change in angle: 1080 * 45 = 48600 degrees.
2. Add the change in angle to the given θ value: 20 + 48600 = 48620 degrees.
3. Convert the degrees to radians: 48620 * π / 180 = 848.2292 radians.
4. Use the polar coordinate system conversion formulas (x = r * cos(θ) and y = r * sin(θ)) with r = 24 to find the xy coordinates.
(c) To find the xy coordinates after 6 seconds when θ = -14 degrees:
1. Convert the angular speed of 3 RPM to degrees per second: 3 * 360 / 60 = 18 degrees per second.
2. Multiply the angular speed by the time in seconds (6 seconds) to get the change in angle: 18 * 6 = 108 degrees.
3. Add the change in angle to the given θ value: -14 + 108 = 94 degrees.
4. Convert the degrees to radians: 94 * π / 180 = 1.6406 radians.
5. Use the polar coordinate system conversion formulas (x = r * cos(θ) and y = r * sin(θ)) with r = 24 to find the xy coordinates.
(d) To find the xy coordinates after 2 hours and 7 seconds when θ = -2.1 radians:
1. Convert the angular speed of 3 RPM to radians per second: 3 * 2π / 60 = 0.3142 radians per second.
2. Multiply the angular speed by the time in seconds (2 hours and 7 seconds) to get the change in angle: 0.3142 * (2 * 60 * 60 + 7) = 2264.3142 radians.
3. Add the change in angle to the given θ value: -2.1 + 2264.3142 = 2262.2142 radians.
4. Use the polar coordinate system conversion formulas (x = r * cos(θ) and y = r * sin(θ)) with r = 24 to find the xy coordinates.
By following these steps, you'll be able to calculate the xy coordinates for each scenario.
what does θ have to do with anything?
For (a), after 4 minutes, the wheel has turned 3 complete revolutions.