A rotating wheel requires 5.00 s to rotate 30.0 revolutions. Its angular velocity at the end of the 5.00 s interval is 100.0 rad/s. What is the constant angular acceleration of the wheel?
average angular speed = Wav = 30 * 2pi/5 =37.7 rad/s
so
(w end + w begin)/2 = 37.7
(100 + w begin)/2 = 37.7
100 + w begin = 75.4
w begin = -24.6 rad/s
alpha = (w end - w begin)/t
= 124.6/5 = 25 rad/s^2
To find the constant angular acceleration of the wheel, we can use the formula:
ω² = ω₀² + 2αθ
Where:
ω = Final angular velocity = 100.0 rad/s
ω₀ = Initial angular velocity = 0 (since the problem does not mention it)
α = Angular acceleration (what we want to find)
θ = Number of revolutions = 30.0 revolutions
First, we need to convert the number of revolutions to radians. Since 1 revolution is equal to 2π radians, we have:
θ = 30.0 revolutions * 2π radians/revolution
θ = 60π radians
Now we can substitute the values into the formula:
(100.0 rad/s)² = 0 + 2α(60π radians)
Simplifying:
10000 = 120πα
Now, isolate α by dividing both sides of the equation by 120π:
α = 10000 / (120π)
Using a calculator to evaluate this expression, we get:
α ≈ 26.53 rad/s²
Therefore, the constant angular acceleration of the wheel is approximately 26.53 rad/s².