Problem: An object moves at constant angular velocity around a circle of radius 4.6 meters, making a revolution every 8.900 seconds. Assume that its angular velocity is constant. Starting at t = 0, when its angular position is 0 radians, what are the x and y coordinates of its position after 7.9 seconds, and after 11.3 seconds (positions measured relative to the center of the circle)?
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Solution: 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/( 8.900 seconds) = .7059 radians/second.
After 7.9 seconds, starting the clock at 0 radians when t = 0, the angular position will be `theta1 = `omega * t1 = ( .7059 radians/second)( 7.9 seconds) = 5.576 radians.
On a circle of radius 4.6 meters, the x and y coordinates will therefore be
x1 = 4.6 meters * COS( 5.576 radians) = 3.496 meters
and
y1 = 4.6 meters * SIN( 5.576 radians) = -.9881 meters.
After 11.3 seconds, the angular position will be
theta2 = `omega * t2 = .7059 radians/second( 11.3 seconds) = 7.976 radians.
On a circle of radius 4.6 meters, the x and y coordinates will therefore be
x2 = 4.6 meters * COS( 7.976 radians) = -.5598 meters
and
y2 = 4.6 meters * SIN( 7.976 radians) = 4.565 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).
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Figure Description: 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.
