Final answer:
To calculate the constant torque required to stop a flywheel, convert the angular velocity to rad/s, determine the angular acceleration for the given time to stop, and use the moment of inertia with angular acceleration to calculate the required torque.
Step-by-step explanation:
To calculate the constant torque required to stop a spinning flywheel, we can use the relationship between torque (\(\tau\)), angular acceleration (\(\alpha\)), and the time it takes to stop the flywheel (t). Given the initial angular velocity (\(\omega_0\)), the final angular velocity (\(\omega\) is zero as the flywheel stops), and the time to stop, we can find the angular acceleration using \(\alpha = \frac{\Delta \omega}{\Delta t}\).
First, convert angular velocity from revolutions per minute (rpm) to radians per second (rad/s) using \(\omega_0 = 297 \times \frac{2\pi}{60}\). Then, determine the angular acceleration needed to bring the flywheel to a stop in 1 minute (60 seconds) with \(\alpha = \frac{-\omega_0}{60}\). The negative sign indicates the flywheel is decelerating. Finally, using the formula \(\tau = I\alpha\), calculate the torque, where I is the moment of inertia of the disk, I = \frac{1}{2}mr^2. Knowing the mass (m) and the radius (r), we find the moment of inertia. Multiplying this by the calculated angular acceleration gives us the required torque.
Example Calculation:
Given: m = 82.1 kg, r = 1.48 m, \(\omega_0\) = 297 rpm
\(\omega_0\) in rad/s = \(297 \times \frac{2\pi}{60}\)\(\alpha = \frac{-\omega_0}{60}\)I = \frac{1}{2} \times 82.1 \times (1.48)^2\(\tau\) = I\alpha
This calculation will yield the torque needed to stop the flywheel in 1 minute.
Remember to observe the direction of rotation and the sign convention used for torque and angular acceleration. A positive direction for torque agrees with the counterclockwise rotation.