An object moves at constant angular velocity around a circle of radius 7.5 meters, making a revolution every 7 seconds. Starting at t = 0, when its angular position is 0 radians, what are the x and y coordinates of its position after 3.9 seconds, and after 12.4 seconds (positions measured relative to the center of the circle)?
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/( 7 seconds) = .8976 radians/second.
After 3.9 seconds, starting the clock at 0 radians when t = 0, the angular position will be `theta1 = `omega * 3.9 = ( .8976 radians/second)( 3.9 seconds) = 3.50064 radians.
On a circle of radius 7.5 meters, the x and y coordinates will therefore be
x1 = 7.5 meters * cos( 3.50064 radians) = -7.022 meters
and
y1 = 7.5 meters * sin( 3.50064 radians) = -2.636 meters.
After 12.4 seconds, the angular position will be
`theta2 = `omega * 12.4 = .8976 radians/second( 12.4 seconds) = 11.13024 radians.
On a circle of radius 7.5 meters, the x and y coordinates will therefore be
x2 = 7.5 meters * cos( 11.13024 radians) = 1.006 meters
and
y2 = 7.5 meters * sin( 11.13024 radians) = -7.433 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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