A flywheel has a constant angular deceleration of 1.3 rad/^2. (a) find the angle through which the flywheel turns as it comes to rest from an angular of speed of 230 rad/s.
We can use the kinematic equation for rotational motion:
ωf^2 = ωi^2 + 2αθ
Where:
ωf = final angular velocity (0 rad/s, because the flywheel comes to rest)
ωi = initial angular velocity (230 rad/s)
α = angular deceleration (-1.3 rad/s^2, because it is decelerating)
θ = angle through which the flywheel turns
Solving for θ:
0^2 = (230 rad/s)^2 + 2(-1.3 rad/s^2)θ
0 = 52900 rad^2/s^2 - 2.6 rad/s^2 θ
2.6 rad/s^2 θ = 52900 rad^2/s^2
θ = 52900 rad^2/s^2 / 2.6 rad/s^2
θ ≈ 20346.15 rad
Therefore, the angle through which the flywheel turns as it comes to rest from an angular speed of 230 rad/s is approximately 20346.15 radians.