91Ó°ÊÓ

First, we need to convert the given rotational speed from revolutions per minute (rev/min) to radians per second (rad/s). There are 2Ï€ radians in one revolution and 60 seconds in one minute.So, oRotational speed in rad/s = \(\omega = 200 \text{ rev/min} \times \frac{2\pi \text{ radians}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}}\).Calculate the value: \(\omega = \frac{200 \times 2\pi}{60} = \frac{400\pi}{60} = \frac{40\pi}{6} \cong 20.94 \text{ rad/s} \).

The translational speed of a point on the rim of the flywheel can be determined by the formula: \(v = r \omega \), where \( r \) is the radius of the flywheel and \( \omega \) is the angular velocity. Given the diameter of the flywheel is 1.20 meters, the radius (\( r \)) is \( 0.60 \text{ m}\). Substitute the values into the equation: \(v = 0.60 \text{ m} \times 20.94 \text{ rad/s} \cong 12.56 \text{ m/s} \).

To find the constant rotational acceleration, we need the initial and final speeds and the time taken to change between these speeds. Given the initial speed is 200 rev/min, the final speed is 1000 rev/min, and the time taken is 60 s. Convert these speeds to revolutions per minute-squared. The formula for angular acceleration is: \( \alpha = \frac{\Delta \omega}{\Delta t}\) First, calculate the change in rotational speed: \( \Delta \omega = 1000 \text{ rev/min} - 200 \text{ rev/min} = 800 \text{ rev/min}\) Now, divide this change by the time to get the acceleration: \( \alpha = \frac{800\text{ rev/min}}{60 \text{ s}}\equiv 13.33 \text{ rev/min}^2\)

To determine the total number of revolutions the wheel makes in 60 seconds, use the kinematic equation for rotational motion: \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2\) Here, \( \theta \) is the total number of revolutions, \( \omega_0 \) is the initial rotational speed, \( \alpha \) is the angular acceleration, and \( t \) is the time. Convert \( \omega_0 \) to revolutions per second: \( \omega_0 = 200 \text{ rev/min} \times \frac{1 \text{ min}}{60 \text{ s}} = \frac{200}{60} \text{ rev/s} \equiv 3.33 \text{ rev/s}\) And \( \alpha \) remains 13.33 rev/min^2. Given: \(\theta = 3.33 \text{ rev/s} \times 60 \text{ s} + \frac{1}{2} \times 13.33 \text{ rev/min}^2 \times 60 \text{ s} \equiv 200 + 6,000)\equiv 6000 \text{ rev}\)