An exercise bicycle's flywheel is 30cm in diameter and 2.5 cm thick and is constructed from steel (density=7850 kg m3) its moment of inertia is 0.156 kg m2 about its axis of rotation
a) calculate the mass of the flywheel
b) calculate the flywheel's radius of gyration
c) whilst exercising a man applies a constant moment of 10.0Nm to the flywheel. Starting from rest the flywheel reaches an angular velocity of 12.0 rad s-1 after exactly 1 minute. Calculate the flywheel's angular acceleration, the number of revolutions the flywheel makes and the friction moment applied to the flywheel
a) To calculate the mass of the flywheel, we can use the formula:
mass = (density) * (volume)
The volume of the flywheel can be calculated using the formula for the volume of a cylinder:
volume = π * (r^2) * h
where r is the radius of the flywheel (which is half of its diameter) and h is the thickness.
Given:
diameter = 30 cm = 0.3 m
thickness = 2.5 cm = 0.025 m
density = 7850 kg/m^3
First, let's calculate the radius of the flywheel:
radius = diameter / 2 = 0.3 m / 2 = 0.15 m
Now, we can calculate the volume:
volume = π * (0.15^2) * 0.025 = 0.01767 m^3
Finally, we can calculate the mass:
mass = density * volume = 7850 kg/m^3 * 0.01767 m^3 ≈ 138.42 kg
Therefore, the mass of the flywheel is approximately 138.42 kg.
b) The radius of gyration can be calculated using the formula:
radius of gyration = √(moment of inertia / mass)
Given:
moment of inertia = 0.156 kg m^2
mass (calculated in part a) = 138.42 kg
Let's calculate the radius of gyration:
radius of gyration = √(0.156 kg m^2 / 138.42 kg) ≈ 0.1735 m
Therefore, the radius of gyration is approximately 0.1735 m.
c) To solve this part, we need to use the equation for angular acceleration:
angular acceleration (α) = (final angular velocity - initial angular velocity) / time
Given:
initial angular velocity (ω initial) = 0 (starting from rest)
final angular velocity (ω final) = 12.0 rad/s
time (t) = 1 minute = 60 seconds
Let's calculate the angular acceleration:
angular acceleration = (12.0 rad/s - 0 rad/s) / 60 s = 0.2 rad/s^2
The flywheel's angular acceleration is 0.2 rad/s^2.
To calculate the number of revolutions, we can use the formula:
number of revolutions = (final angular velocity / 2π) * time
number of revolutions = (12.0 rad/s / (2π)) * 60 s ≈ 11.47 revolutions
Therefore, the flywheel makes approximately 11.47 revolutions.
Finally, to calculate the friction moment applied to the flywheel, we can use the equation:
friction moment = (moment of inertia) * (angular acceleration)
friction moment = 0.156 kg m^2 * 0.2 rad/s^2 = 0.0312 Nm
Therefore, the friction moment applied to the flywheel is 0.0312 Nm.