Question

A spinning wheel on a fireworks display is initially rotating in a counterclockwise direction. The wheel has an angular acceleration of -4.20 rad/s2. Because of this acceleration, the angular velocity of the wheel changes from its initial value to a final value of -26.0 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.

Answer 1

The time it takes for the change in the angular velocity to occur is 6.5 seconds

Step-by-step explanation:

From the question, we have;

The angular acceleration of the wheel, α = -4.20 rad/s²

The final angular velocity of the wheel, ω = -26.0 rad/s

From the information, we have the initial direction of rotation of the wheel = Counterclockwise

The angular displacement while the change occurs = 0 rad/s

Therefore, the initial angular velocity of the wheel, ω₀ = 0 rad/s

We have;

`\alpha = (\omega -\omega _0 )/(t - t_0)`

Where;

t - t₀ = The time it takes for the change to occur

`\alpha = (\omega -0 )/(t - 0)`

When t₀ = 0, we have;

t - t₀ = t

`\alpha = (\omega )/(t)`

`\therefore t = (\omega)/(\alpha )`

`\therefore t = (-26 \, rad/s)/(-4 \, rad/s^2 ) \approx 6.5 \, seconds`

The time it takes for the change in the angular velocity to occur is t - t₀ = 6.5 seconds