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
Answer this question
a) To calculate the mass of the flywheel, we can use the formula:
Mass = Volume x Density
The volume of the flywheel can be calculated using the formula for the volume of a cylinder:
Volume = πr²h
where r is the radius and h is the thickness.
Given:
Diameter = 30cm
Radius = 15cm = 0.15m
Thickness = 2.5cm = 0.025m
Density = 7850 kg/m³
Volume = π(0.15m)²(0.025m) = 0.01767 m³
Mass = Volume x Density = 0.01767 m³ x 7850 kg/m³ = 138.4695 kg
Therefore, the mass of the flywheel is approximately 138.47 kg.
b) The radius of gyration (k) can be calculated using the formula:
k = √(I/m)
where I is the moment of inertia and m is the mass.
Given:
Moment of inertia (I) = 0.156 kg m²
Mass (m) = 138.4695 kg
k = √(0.156 kg m² / 138.4695 kg) ≈ 0.0564 m
Therefore, the radius of gyration of the flywheel is approximately 0.0564 m.
c) The angular acceleration (α) can be calculated using the formula:
α = (ω - ω₀) / t
where ω is the final angular velocity, ω₀ is the initial angular velocity (which is zero since the flywheel starts from rest), and t is the time.
Given:
Final angular velocity (ω) = 12.0 rad/s
Time (t) = 1 minute = 60 s
α = (12.0 rad/s - 0 rad/s) / 60 s = 0.2 rad/s²
The flywheel makes one revolution when it rotates by an angle of 2π radians. Since the angular velocity is given in rad/s, we can calculate the number of revolutions as follows:
Number of revolutions = (ω / 2π)
Number of revolutions = 12.0 rad/s / (2π rad/rev) ≈ 1.91 rev
The friction moment applied to the flywheel can be calculated using the formula:
Friction moment (M) = α x I
Given:
α = 0.2 rad/s²
I = 0.156 kg m²
M = 0.2 rad/s² * 0.156 kg m² ≈ 0.0312 Nm
Therefore, the flywheel's angular acceleration is 0.2 rad/s², it makes approximately 1.91 revolutions, and the friction moment applied to the flywheel is approximately 0.0312 Nm.
a) To calculate the mass of the flywheel, we can use the formula:
Mass = Density * Volume
The volume of a cylindrical flywheel can be calculated using the formula for the volume of a cylinder:
Volume = π * (radius)^2 * height
Given the diameter of the flywheel is 30 cm, the radius (r) would be half of that, which is 15 cm (or 0.15 m). The height (h) of the flywheel is given as 2.5 cm (or 0.025 m).
Plugging these values into the formula, we have:
Volume = π * (0.15 m)^2 * 0.025 m
Volume = π * 0.0225 m^3 * 0.025 m
Volume = π * 0.0005625 m^3
Now we can calculate the mass using the given density of steel (7850 kg/m^3):
Mass = Density * Volume
Mass = 7850 kg/m^3 * 0.0005625 m^3
Mass = 4.409 kg
b) The radius of gyration (k) is a measure of how the mass of an object is distributed around its axis of rotation. It is calculated using the formula:
k = √(I / mass)
Given the moment of inertia (I) is 0.156 kg m^2 and the mass is 4.409 kg (calculated in part a), we can now calculate the radius of gyration:
k = √(0.156 kg m^2 / 4.409 kg)
k = √0.035374058 m
k ≈ 0.188 m
c) The angular acceleration (α) can be calculated using the formula:
α = Δω / Δt
Where Δω is the change in angular velocity and Δt is the change in time.
Given that the flywheel reaches an angular velocity of 12.0 rad/s after 1 minute, we need to convert 1 minute to seconds:
1 minute = 60 seconds
Now we can calculate the angular acceleration:
α = (12.0 rad/s - 0 rad/s) / (60 s - 0 s)
α = 12.0 rad/s / 60 s
α = 0.2 rad/s^2
To calculate the number of revolutions the flywheel makes, we can use the formula:
Number of revolutions = (angular distance) / (2π)
The angular distance can be calculated using the formula:
Angular distance = (angular velocity) * (time)
Given the angular velocity is 12.0 rad/s and the time is 1 minute (or 60 seconds), we can calculate the angular distance:
Angular distance = (12.0 rad/s) * (60 s)
Angular distance = 720 rad
Now we can calculate the number of revolutions:
Number of revolutions = 720 rad / (2π rad/revolution)
Number of revolutions ≈ 114.59 revolutions
Finally, to calculate the friction moment applied to the flywheel, we can use the formula:
Friction moment = (moment of inertia) * (angular acceleration)
Given the moment of inertia is 0.156 kg m^2 and the angular acceleration is 0.2 rad/s^2, we can calculate the friction moment:
Friction moment = (0.156 kg m^2) * (0.2 rad/s^2)
Friction moment = 0.0312 Nm
a) To calculate the mass of the flywheel, we can use the formula:
Mass = Volume x Density
Given that the flywheel's diameter is 30 cm and thickness is 2.5 cm, we can find its volume using the formula for the volume of a cylinder:
Volume = π x (radius^2) x thickness
First, we need to convert the diameter to radius by dividing it by 2:
Radius = 30 cm / 2 = 15 cm = 0.15 m
Substituting the values:
Volume = π x (0.15^2) x 0.025 m^3
Now we can calculate the mass:
Mass = Volume x Density = π x (0.15^2) x 0.025 m^3 x 7850 kg/m^3
b) The radius of gyration is given by the formula:
Radius of Gyration = √(Moment of Inertia / Mass)
Given that the moment of inertia of the flywheel is 0.156 kg m^2 (as stated in the question), and we just calculated the mass in part (a), we can now calculate the radius of gyration.
Radius of Gyration = √(0.156 kg m^2 / Mass)
c) To calculate the flywheel's angular acceleration, we can use the formula:
Angular Acceleration = (Final Angular Velocity - Initial Angular Velocity) / Time
Given the final angular velocity is 12.0 rad/s, the initial angular velocity is 0 rad/s (starting from rest), and the time is 1 minute (which needs to be converted to seconds):
Angular Acceleration = (12.0 rad/s - 0 rad/s) / (1 min x 60 s/min)
To calculate the number of revolutions the flywheel makes, we can use the formula:
Number of Revolutions = Final Angular Velocity / (2π)
Given the final angular velocity is 12.0 rad/s, we can substitute the values:
Number of Revolutions = 12.0 rad/s / (2π)
Finally, to calculate the friction moment applied to the flywheel, we use the formula:
Friction Moment = Moment of Inertia x Angular Acceleration
Given the moment of inertia is 0.156 kg m^2 (as stated in the question) and we just calculated the angular acceleration in part (c), we can now calculate the friction moment.