A rotating wheel requires 6.00 s to rotate 27.0 revolutions. Its angular velocity at the end of the 6.00-s interval is 97.0 rad/s. What is the constant angular acceleration (in rad/s) of the wheel?
Why did the wheel become a mathematician? Because it wanted to calculate its constant angular acceleration!
To find the constant angular acceleration, we can use the formula:
angular acceleration = (final angular velocity - initial angular velocity) / time
Here, the initial angular velocity is zero since the wheel starts from rest. So we have:
angular acceleration = (97.0 rad/s - 0 rad/s) / 6.00 s
That simplifies to:
angular acceleration = 97.0 rad/s / 6.00 s
So, the constant angular acceleration of the wheel is approximately 16.17 rad/s. And now we know why the wheel started spinning so quickly - it was always inclined towards acceleration!
To find the constant angular acceleration of the wheel, we can use the following equation:
ω = ω0 + αt
where:
ω is the final angular velocity (97.0 rad/s),
ω0 is the initial angular velocity (which is 0 because the wheel starts from rest),
α is the angular acceleration (what we're trying to find), and
t is the time interval (6.00 s).
First, let's find the initial angular velocity using the given information that the wheel rotates 27.0 revolutions (or 27.0 * 2π radians) in 6.00 seconds. We use the formula:
θ = ω0t + (1/2)αt^2
where:
θ is the total angle rotated (27.0 * 2π radians),
t is the time interval (6.00 s), and
α is the angular acceleration.
Rearranging the equation and substituting the given values:
(27.0 * 2π) = (0)t + (1/2)α(6.00)^2
(54π) = 18α
Now, solve for α:
α = (54π) / 18
α = 3π rad/s^2
Therefore, the constant angular acceleration of the wheel is 3π rad/s^2.