Answer :
To calculate the change in angular momentum of the flywheel, we first need to understand the relationship between angular velocity and angular momentum.
Step 1: Convert Angular Velocities to Radians per Second
We start by converting the initial and final angular velocities from revolutions per minute (rev/min) to radians per second (rad/s). The conversion factor is:
[tex]1 \text{ rev} = 2\pi \text{ rad}[/tex]
and
[tex]1 \text{ minute} = 60 \text{ seconds}[/tex]
This means:
[tex]\text{Angular Velocity (rad/s)} = \text{Angular Velocity (rev/min)} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}}[/tex]
Calculating:
For the initial angular velocity ([tex]\omega_i[/tex]):
[tex]\omega_i = 600 \text{ rev/min} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} = 62.83 \text{ rad/s}[/tex]For the final angular velocity ([tex]\omega_f[/tex]):
[tex]\omega_f = 900 \text{ rev/min} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} = 94.25 \text{ rad/s}[/tex]
Step 2: Calculate the Change in Angular Momentum
The change in angular momentum ([tex]\Delta L[/tex]) can be calculated using the formula:
[tex]\Delta L = I \Delta \omega[/tex]
Where [tex]I[/tex] is the moment of inertia. However, we do not have it directly but we can derive it based on the torque ([tex]\tau[/tex]) using the relation:
[tex]\tau = I \alpha[/tex]
Where [tex]\alpha[/tex] is the angular acceleration, which can be calculated as:
[tex]\alpha = \frac{\Delta \omega}{\Delta t}[/tex]
The time interval [tex]\Delta t[/tex] is given as 10 seconds, so:
[tex]\alpha = \frac{\omega_f - \omega_i}{10 \text{ s}} = \frac{94.25 \text{ rad/s} - 62.83 \text{ rad/s}}{10 \text{ s}} = 3.144 \text{ rad/s}^2[/tex]
Now we substitute back to find the moment of inertia ([tex]I[/tex]):
[tex]350\pi = I \cdot 3.144\text{ (using torque)} \implies I = \frac{350\pi}{3.144} \approx 111.5 \text{ kg m}^2[/tex]
Step 3: Use the Change in Angular Velocity to Find Change in Angular Momentum
Now, we can calculate the change in angular momentum:
[tex]\Delta L = I \Delta \omega \Rightarrow \Delta L = 111.5 \text{ kg m}^2 \cdot (94.25 - 62.83)\text{ rad/s}[/tex]
Calculating:
[tex]\Delta L = 111.5 \cdot 31.42 \approx 3500 \text{ kg m}^2/s[/tex]
Conclusion
The change in angular momentum of the flywheel after 10 seconds is approximately 3500 kg·m²/s.
