A spinning wheel on a fireworks display is initially rotating in a counterclockwise direction. The wheel has an angular acceleration of -3.80 rad/s2. Because of this acceleration, the angular velocity of the wheel changes from its initial value to a final value of -21.0 rad/s. While this change occurs, the angular displacement of the wheel is zero. (Note the similarity to that of a ball being thrown vertically upward, coming to a momentary halt, and then falling downward to its initial position.) Find the time required for the change in the angular velocity to occur.
since it ends up at the same spot, it goes one way for the same time and distance that it goes the other way. Therefore find half the time and double it.
0 = Vo + a t
so a t = -Vo
Vo = +21 m/s (by symmetry)
so
- 3.8 t = -21
t = 21/3.8
we want 2t = 42/3.8
To find the time required for the change in angular velocity to occur, we can use the formula:
angular acceleration = (change in angular velocity) / (change in time)
In this case, the angular acceleration is given as -3.80 rad/s^2, and the change in angular velocity is -21.0 rad/s. We need to find the change in time.
Rearranging the formula, we have:
(change in time) = (change in angular velocity) / (angular acceleration)
Plugging in the values, we get:
(change in time) = (-21.0 rad/s) / (-3.80 rad/s^2)
Now we can solve for the change in time:
(change in time) = 5.5263 s
Therefore, the time required for the change in the angular velocity to occur is approximately 5.53 seconds.
To find the time required for the change in angular velocity to occur, we can use the formula:
ωf = ωi + αt
where:
ωf = final angular velocity = -21.0 rad/s
ωi = initial angular velocity (unknown)
α = angular acceleration = -3.80 rad/s²
t = time (unknown)
Since the angular displacement is zero, we can use the formula:
ωf² = ωi² + 2αθ
where θ = 0 (angular displacement)
Substituting the given values into the above equation, we get:
(-21.0 rad/s)² = ωi² + 2(-3.80 rad/s²)(0)
441.0 rad²/s² = ωi²
Taking the square root of both sides, we get:
ωi = ±sqrt(441.0 rad²/s²)
ωi ≈ ±21.0 rad/s
Since we are given that the initial direction is counterclockwise, the initial angular velocity is ωi = 21.0 rad/s.
Now, substituting the values into the first formula, we get:
-21.0 rad/s = 21.0 rad/s + (-3.80 rad/s²)(t)
Simplifying the equation, we have:
-42.0 rad/s = -3.80 rad/s²(t)
Dividing both sides by -3.80 rad/s², we get:
t ≈ 11.05 s
Therefore, the time required for the change in the angular velocity to occur is approximately 11.05 seconds.