At t = 0 seconds the angular position of an object is 0 radians, and it moves at constant angular velocity around a circle of radius 4.8 meters, making a revolution every 8.6 seconds. What are the x and y coordinates of its position at t = 3.9 seconds and at t = 12.66 seconds?
Moving through a revolution, which corresponds to angular displacement `dTheta = 2 `pi radians, in 8.900 seconds, the object will have an angular velocity of
`omega = `dTheta / `dt = 2 `pi radians/( 8.6 seconds) = .7306 radians/second.
After 3.9 seconds, starting the clock at 0 radians when t = 0, the angular position will be `theta1 = `omega * 3.9 = ( .7306 radians/second)( 3.9 seconds) = 2.84934 radians.
On a circle of radius 4.8 meters, the x and y coordinates will therefore be
x1 = 4.8 meters * cos( 2.84934 radians) = -4.597 meters
and
y1 = 4.8 meters * sin( 2.84934 radians) = 1.382 meters.
After 12.66 seconds, the angular position will be
`theta2 = `omega * 12.66 = .7306 radians/second( 12.66 seconds) = 9.249396 radians.
On a circle of radius 4.8 meters, the x and y coordinates will therefore be
x2 = 4.8 meters * cos( 9.249396 radians) = -4.727 meters
and
y2 = 4.8 meters * sin( 9.249396 radians) = .8375 meters.
Generalized Response: If an object moves through angle `dTheta in time `dt at constant angular velocity, then its angular velocity is
angular velocity = `omega = `dTheta / `dt.
If the object starts from the positive x axis at clock time t = 0, then by clock time t1 it will have moved through angular displacement
`theta1 = `omega * t1.
If the circle has radius r, then by the circular definitions of the sine and cosine functions the x and y coordinates relative to the center of the circle will be
x1 = r * cos(`theta1)
and
y1 = r * sin(`theta1).
At time t2 the angular positioni will be
`theta2 = `omega * t2
and the coordinates will be
x2 = r * cos(`theta2)
and
y2 = r * sin(`theta2).
The figure below shows a circle of radius r, with the standard starting position indicated. The angular velocity `omega is calculated from the known angular displacement and required time, and is indicated by the moving red radial line. The angular positions `theta1 and `theta2 are indicated, as are the corresponding x and y coordinates.
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