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
To solve these problems, we will use the formulas and equations of rotational motion. Let's go through each part:
a) To calculate the mass of the flywheel, we can use the formula:
Mass (m) = Density (ρ) x Volume (V)
The volume of the flywheel can be calculated using the formula for the volume of a cylinder:
Volume (V) = π x (radius)^2 x height
For the flywheel, the radius (r) is given as half of the diameter (d), so r = 30cm / 2 = 15cm = 0.15m. The height (h) is given as 2.5cm = 0.025m.
Substituting these values into the volume formula, we get:
V = π x (0.15m)^2 x 0.025m
Now we can substitute the given density (7850 kg/m^3) and calculate the mass:
m = 7850 kg/m^3 x V
Calculate the volume using the formula, then multiply it by the density to find the mass.
b) The radius of gyration (k) is a measure of how the mass of an object is distributed around its axis of rotation. It can be calculated using the formula:
k = √(I / m)
Where the moment of inertia (I) is given as 0.156 kg m^2, and the mass (m) can be calculated in part (a).
Substitute these values and calculate the radius of gyration using the formula.
c) To calculate the flywheel's angular acceleration (α), we can use the formula:
α = (ω - ω0) / t
where ω is the final angular velocity, ω0 is the initial angular velocity (which is zero as it starts from rest), and t is the time taken.
Substitute the given values for ω, ω0, and t into the formula and calculate the angular acceleration.
Next, let's calculate the number of revolutions the flywheel makes. Since the flywheel starts from rest and reaches a final angular velocity of 12.0 rad/s after exactly 1 minute, we can use the formula:
θ = ω0t + 0.5αt^2
where θ is the total angle rotated, ω0 is the initial angular velocity, α is the angular acceleration, and t is the time taken.
Since the initial angular velocity (ω0) is zero, the formula simplifies to:
θ = 0.5αt^2
Substitute the calculated value of α and the given time (1 minute = 60 seconds) into the formula and calculate the total angle rotated.
Finally, to calculate the friction moment applied to the flywheel, we can use the formula:
Mf = Iα
where Mf is the friction moment and I is the moment of inertia.
Substitute the calculated value of α and the given moment of inertia into the formula and calculate the friction moment.