i asked this question before but didn't really get a particular answer.
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?
still no indication of what θ means. If it is the angle through which the wheel has rotated, then when θ=34°
(x,y) = (r cosθ, r sinθ) = (19.9,13.42)
But the rotation speed and time already tell you how far the wheel has gone, so who needs θ?
If θ is the angle when the rotation starts, then after a whole number of minutes, the wheel is back where it started. For
(c) the wheel has rotated 6/60 * 3 * 360 = 108° so add that to the initial value of -14°, so use 96° for your coordinates.
Once you decide what θ means for this problem, then it's just plug and chug. Until then, you're doomed.
If theta is measured counterclockwise from the x axis then
x= R cos theta and y = R sin theta
Maybe theta here is angle off horizontal ?
To answer these questions, we need to use some basic trigonometry and kinematics equations for circular motion. Here's how you can find the xy coordinates for each case:
(a) If θ = 34 degrees and the merry-go-round is rotating counterclockwise at 3 RPM, we can find the angular velocity (ω) in radians per second:
ω = (3 RPM) * (2π rad/1 min) * (1 min/60 s) = 0.31416 rad/s
To find the xy coordinates after 4 minutes, we need to convert the time to seconds:
t = (4 min) * (60 s/1 min) = 240 s
Using the equation for angular displacement, θ = ωt, we can find the angular displacement:
θ = (0.31416 rad/s) * (240 s) = 75.3984 rad
Next, we can use the equation for circular motion to find the xy coordinates:
x = r * cos(θ) = 24 ft * cos(75.3984 rad) = -7.8983 ft (rounded to four decimal places)
y = r * sin(θ) = 24 ft * sin(75.3984 rad) = 22.3084 ft (rounded to four decimal places)
Therefore, the xy coordinates after 4 minutes are approximately (-7.8983 ft, 22.3084 ft).
(b) Following a similar process, if θ = 20 degrees and t = 45 minutes, we can find the angular velocity using the same formula as before:
ω = (3 RPM) * (2π rad/1 min) * (1 min/60 s) = 0.31416 rad/s
Converting the time to seconds:
t = (45 min) * (60 s/1 min) = 2700 s
The angular displacement θ can be calculated as:
θ = (0.31416 rad/s) * (2700 s) = 848.520 rad
Using the equations for circular motion, we can find the xy coordinates:
x = r * cos(θ) = 24 ft * cos(848.520 rad) = 6.2283 ft (rounded to four decimal places)
y = r * sin(θ) = 24 ft * sin(848.520 rad) = -22.1192 ft (rounded to four decimal places)
Therefore, the xy coordinates after 45 minutes are approximately (6.2283 ft, -22.1192 ft).
(c) For θ = -14 degrees and t = 6 seconds, the angular velocity remains the same:
ω = 0.31416 rad/s
The time is already in seconds:
t = 6 s
The angular displacement θ is calculated as:
θ = (0.31416 rad/s) * (6 s) = 1.88496 rad
Using the equations for circular motion:
x = r * cos(θ) = 24 ft * cos(1.88496 rad) = 23.6658 ft (rounded to four decimal places)
y = r * sin(θ) = 24 ft * sin(1.88496 rad) = -2.8625 ft (rounded to four decimal places)
Therefore, the xy coordinates after 6 seconds are approximately (23.6658 ft, -2.8625 ft).
(d) Lastly, for θ = -2.1 radians and t = 2 hours and 7 seconds, we need to find the angular velocity:
ω = (3 RPM) * (2π rad/1 min) * (1 min/60 s) = 0.31416 rad/s
Converting the time to seconds:
t = (2 hours) * (60 min/1 hour) * (60 s/1 min) + 7 s = 7247 s
The angular displacement θ can be calculated as:
θ = (0.31416 rad/s) * (7247 s) = 2275.07552 rad
Using the equations for circular motion:
x = r * cos(θ) = 24 ft * cos(2275.07552 rad) = 23.8303 ft (rounded to four decimal places)
y = r * sin(θ) = 24 ft * sin(2275.07552 rad) = -1.5755 ft (rounded to four decimal places)
Therefore, the xy coordinates after 2 hours and 7 seconds are approximately (23.8303 ft, -1.5755 ft).
By following these steps, you can calculate the xy coordinates for different values of θ and time when given the angular velocity and radius of the merry-go-round.