At a county fair there is a betting game that involves a spi

At a county fair there is a betting game that involves a spinning wheel. As the drawing shows, the wheel is set into rotational motion with the beginning of the angular section labeled \"I\" at the marker at the top of the wheel. The wheel then decelerates and eventually comes to a halt on one of the numbered sections. The wheel in the drawing is divided into twelve sections. Each of which is an angle of 30.0 degree. Determine the numbered section on which the wheel comes to a halt when the deceleration of the wheel has a magnitude of 0.200rad/s^2 and the initial angular velocity is (a) +1.20 rev/s and (b) +1.47 rev/s

Solution

(a)

initial angualar speed w0 = 1.2 rev/s = 1.2*2*pi rad/s = 7.54 rad/s

final angular speed wf = 0

angular acceleration alpha = -0.2 rad/s^2

from equation of motion

wf^2 - w0^2 = 2*alpha*theta

0 - 7.54^2 = -2*0.2*theta

theta = 142.129 rad = 142.129/(2pi) = 22.6 revolutions

final position = 0.6*360 = 216 degrees

numbered section = 216/30 = 7.2 = 8

(b)

initial angualar speed w0 = 1.47 rev/s = 1.47*2*pi rad/s = 9.24 rad/s

final angular speed wf = 0

angular acceleration alpha = -0.2 rad/s^2

from equation of motion

wf^2 - w0^2 = 2*alpha*theta

0 - 9.24^2 = -2*0.2*theta

theta = 213.44 rad = 213.44/(2pi) = 33.97 revolutions

final position = 0.97*360 = 324 degrees

numbered section = 349.2/30 = 11.64 = 12