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
m = ρ•V = ρ•π•D²•d/4 =
=7850•3.14•0.09•0.025/4 = 3.12 kg.
R(gyr) =sqrt(I/A) = sqrt(4•I/ π•D²) = =sqrt(4•0.156/3.14•0.09) =1.5 m.
ω= ε•t,
ε = ω/t = 12/60 = 0.2 rad/s².
2•π•N = ε•t²/2,
N = ε•t²/4• π =0.2•3600/4•3.14 =
=57.3 rev.
M - M(fr) =I•ε.
M(fr)= M - I•ε =
=10 – 0.156•0.2 = 9.969 N•m.
a) To calculate the mass of the flywheel, we can use the formula:
Mass = Volume x Density
The volume of a cylinder can be calculated using the formula:
Volume = π x r^2 x h
Where r is the radius and h is the height (thickness) of the flywheel.
Given:
Diameter (d) = 30 cm
Radius (r) = d/2 = 30/2 = 15 cm = 0.15 m
Thickness (h) = 2.5 cm = 0.025 m
Density (ρ) = 7850 kg/m^3
Now, we can calculate the volume:
Volume = π x (0.15)^2 x 0.025
Finally, we can calculate the mass:
Mass = Volume x Density
Substitute the values and calculate the mass.
b) The radius of gyration (k) is given by the formula:
k = √(I / M)
Where I is the moment of inertia and M is the mass of the flywheel.
We have already calculated the mass in part a). Substitute the values into the formula and calculate the radius of gyration.
c) The angular acceleration (α) can be calculated using the formula:
α = (ω_f - ω_i) / t
Where ω_f is the final angular velocity, ω_i is the initial angular velocity (which is 0 in this case since it starts from rest), and t is the time taken.
Substitute the values and calculate the angular acceleration.
To calculate the number of revolutions, we need to convert the angular velocity to revolutions per minute (rpm). Since there are 2π radians in a revolution and 60 seconds in a minute, we can use the following conversion:
Number of revolutions = ω_f / (2π) * 60
Substitute the value of the final angular velocity and calculate the number of revolutions.
The friction moment applied to the flywheel can be calculated using the formula:
Friction moment (τ) = I * α
Substitute the values of I and α, and calculate the friction moment.