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:
Density = Mass / Volume
Rearranging the formula for mass, we get:
Mass = Density x Volume
The volume of the flywheel can be calculated using the formula for the volume of a cylinder:
Volume = π x (radius)^2 x height
Given:
Diameter = 30 cm
Radius = Diameter / 2 = 15 cm = 0.15 m
Height = 2.5 cm = 0.025 m
Density = 7850 kg/m^3
Plugging in the values, we can calculate the volume:
Volume = π x (0.15 m)^2 x 0.025 m
Now we can calculate the mass:
Mass = Density x Volume
b) The radius of gyration is calculated using the formula:
Radius of Gyration = √(Moment of Inertia / Mass)
Given:
Moment of Inertia = 0.156 kg m^2 (as given in the problem statement)
Mass (calculated in part a)
c) To calculate the angular acceleration, we can use the formula:
Angular acceleration = (Final angular velocity - Initial angular velocity) / Time
Given:
Initial angular velocity (ω1) = 0 rad/s (starting from rest)
Final angular velocity (ω2) = 12.0 rad/s
Time (t) = 1 minute = 60 seconds
Using the values:
Angular acceleration = (12.0 rad/s - 0 rad/s) / 60 s
To calculate the number of revolutions, we need to find the total angle rotated and then convert it into revolutions. The formula to calculate the angle is:
Angle (θ) = Angular velocity x Time
Given:
Angular velocity = 12.0 rad/s
Time = 1 minute = 60 seconds
Using the values:
Angle (θ) = 12.0 rad/s x 60 s
Finally, to calculate the friction moment applied to the flywheel, we can use the formula:
Friction moment = Applied moment - (Moment of inertia x Angular acceleration)
Given:
Applied moment = 10.0 Nm (as given in the problem statement)
Moment of Inertia = 0.156 kg m^2 (as given in the problem statement)
Angular acceleration (calculated in part c)
To solve this problem, we'll use the following formulas:
a) Mass of the flywheel = density x volume
b) Radius of gyration = sqrt(moment of inertia/mass)
c) Angular acceleration = (final angular velocity - initial angular velocity) / time
Number of revolutions = (final angular velocity - initial angular velocity) / (2*pi)
Friction moment = Total moment - Applied moment
Now, let's calculate each part step by step:
a) Mass of the flywheel = density x volume
The volume of the flywheel can be calculated using the formula for the volume of a cylinder: V = πr^2h
Here, r is the radius and h is the thickness.
r = diameter/2 = 30/2 = 15 cm = 0.15 m
h = 2.5 cm = 0.025 m
V = π(0.15)^2(0.025) = 0.0177 m^3
Mass = density x volume = 7850 kg/m^3 x 0.0177 m^3 = 138.645 kg
Therefore, the mass of the flywheel is 138.645 kg.
b) Radius of gyration = sqrt(moment of inertia/mass)
Given moment of inertia = 0.156 kg m^2
Radius of gyration = sqrt(0.156 kg m^2 / 138.645 kg)
= sqrt(0.001125)
= 0.0336 m
Therefore, the radius of gyration is 0.0336 m.
c) Angular acceleration = (final angular velocity - initial angular velocity) / time
Given: Initial angular velocity (ω1) = 0 rad/s
Final angular velocity (ω2) = 12.0 rad/s
Time (t) = 1 minute = 60 seconds
Angular acceleration = (12.0 rad/s - 0 rad/s) / 60 s
= 0.2 rad/s^2
Number of revolutions = (final angular velocity - initial angular velocity) / (2*pi)
= (12.0 rad/s - 0 rad/s) / (2*pi)
= 1.91 revolutions
Friction moment = Total moment - Applied moment
Total moment is given as 10.0 Nm
Friction moment = Total moment - 10.0 Nm
Therefore, the angular acceleration is 0.2 rad/s^2, the number of revolutions is 1.91 revolutions, and the friction moment applied to the flywheel is Total moment - 10.0 Nm.