After 10.0 s, a spinning roulette wheel at a casino has slowed down to an angular velocity of +1.56 rad/s. During this time, the wheel has an angular acceleration of -5.45 rad/s2. Determine the angular displacement of the wheel.
To determine the angular displacement of the wheel, we can use the kinematic equation for rotational motion:
ωf = ω0 + αt + 1/2 * α * t^2,
where ωf is the final angular velocity, ω0 is the initial angular velocity, α is the angular acceleration, and t is the time.
Given:
ω0 = 0 rad/s (initial angular velocity)
ωf = +1.56 rad/s (final angular velocity)
α = -5.45 rad/s^2 (angular acceleration)
t = 10.0 s (time)
Plugging in the values into the equation, we have:
1.56 = 0 + (-5.45) * 10.0 + 1/2 * (-5.45) * (10.0)^2.
Simplifying the equation, we find:
1.56 = -54.5 + (-2.725) * 100.
1.56 = -54.5 - 272.5.
1.56 = -327.
Since the equation does not hold true, there appears to be an error. It is not possible for the final angular velocity to be smaller than the initial angular velocity when the angular acceleration is negative.
Please recheck the given values or provide more information so that I can assist you further.