Final answer:
To stop a flywheel spinning at 361 rpm in 2.75 minutes, one must calculate the angular deceleration and apply Newton's second law for rotation to find the torque, which is approximately -115.8 N·m.
Step-by-step explanation:
To calculate the constant torque required to stop a spinning flywheel, we first need to determine the angular deceleration. The flywheel's initial angular velocity (ω0) can be found by converting the given revolutions per minute (rpm) to radians per second (rad/s). Since 361 rpm is approximately equal to 37.7 rad/s (using ω = 2π × rpm / 60), and the final angular velocity (ω) is 0 rad/s because we want to stop it, we can utilize the angular kinematic equation ω = ω0 + αt, where α is the angular deceleration and t is time. The time, given in minutes, must be converted to seconds, which gives us 165 seconds (2.75 min × 60 s/min).
The angular deceleration is then α = (ω - ω0) / t, which when plugged in gives us about -0.229 rad/s2. Next, torque (τ) is calculated using τ = I × α, where I is the moment of inertia of a uniform disk, which equals (1/2)mr2, with m being the mass and r the radius.
Substituting the known values, we get τ = (1/2) × 62.6 kg × (1.43 m)2 × -0.229 rad/s2, which results in a torque of approximately -115.8 N·m. The negative sign indicates that the torque is in the opposite direction of rotation, which is required to stop the flywheel.