To solve the problem of determining the time required to stop the flywheel, we can follow these steps:
### Step 1: Convert the initial angular velocity from rpm to rad/s
The initial angular velocity (ω_initial) is given as 420 rpm. We need to convert this to radians per second.
\[
\omega_{\text{initial}} = 420 \, \text{rpm} \times \frac{2\pi \, \text{radians}}{1 \, \text{rotation}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}}
\]
Calculating this gives:
\[
\omega_{\text{initial}} = 420 \times \frac{2\pi}{60} = 14\pi \, \text{rad/s}
\]
### Step 2: Identify the final angular velocity and angular acceleration
The flywheel comes to a stop, so the final angular velocity (ω_final) is:
\[
\omega_{\text{final}} = 0 \, \text{rad/s}
\]
The angular acceleration (α) is given as a constant rate of slowing down:
\[
\alpha = -2 \, \text{rad/s}^2
\]
### Step 3: Use the angular motion equation
We can use the first equation of angular motion, which relates initial angular velocity, final angular velocity, angular acceleration, and time:
\[
\omega_{\text{final}} = \omega_{\text{initial}} + \alpha t
\]
Substituting the known values:
\[
0 = 14\pi + (-2)t
\]
### Step 4: Solve for time (t)
Rearranging the equation to solve for t:
\[
2t = 14\pi
\]
\[
t = \frac{14\pi}{2} = 7\pi \, \text{seconds}
\]
### Step 5: Calculate the numerical value of time
To get the numerical value, we can use the approximation of π:
\[
t \approx 7 \times 3.14 \approx 22 \, \text{seconds}
\]
### Conclusion
Therefore, the time required to stop the flywheel is:
\[
\text{Time} = 22 \, \text{seconds}
\]
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