A flywheel in the form of a uniformly thick disk of radius 1.48 m, has a mass of 43.6 kg and spins counterclockwise at 303 rpm. Calculate the constant torque required to stop it in 2.50 min.
To calculate the constant torque required to stop the flywheel, we need to first determine the initial angular velocity and the final angular velocity. Then we can use the equation for torque to find the torque required.
1. First, let's convert the initial angular velocity from rpm (revolutions per minute) to rad/s (radians per second):
Initial angular velocity (ω1) = 303 rpm × (2π rad/1 rev) × (1 min/60 s)
Simplifying, we get:
ω1 = 303 × 2π × (1/60) rad/s
2. Next, let's convert the time to seconds:
Time (t) = 2.50 min × 60 s/min
3. Now, let's determine the final angular velocity using the equation for angular acceleration:
ω2 = 0 rad/s (since we want to stop the flywheel)
4. We can use the equation for angular acceleration to find the angular acceleration (α) of the flywheel:
α = (ω2 - ω1) / t
5. Now that we have the angular acceleration, we can calculate the moment of inertia (I) of the flywheel using the equation:
I = (1/2) × mass × radius^2
6. Finally, we can calculate the torque (τ) required to stop the flywheel using the equation:
τ = I × α
Let's substitute the values into the equations to find the answer:
1. ω1 = 303 × 2π × (1/60) rad/s = 31.87 rad/s
2. t = 2.50 min × 60 s/min = 150 s
3. ω2 = 0 rad/s
4. α = (ω2 - ω1) / t = (0 - 31.87) / 150 rad/s^2 = -0.2125 rad/s^2
5. I = (1/2) × mass × radius^2 = (1/2) × 43.6 kg × (1.48 m)^2 = 46.79 kg·m^2
6. τ = I × α = 46.79 kg·m^2 × (-0.2125 rad/s^2) = -9.94 N·m
Therefore, the constant torque required to stop the flywheel in 2.50 min is approximately -9.94 N·m (negative sign indicating that the torque is in the opposite direction to the initial motion).