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 need to use its volume and density. The volume of a cylinder (which the flywheel resembles) can be calculated using the formula V = πr^2h, where r is the radius and h is the thickness. In this case, the radius is half the diameter, so r = 30cm / 2 = 15cm = 0.15m. The thickness is given as 2.5cm = 0.025m.
Now, we can calculate the volume: V = π(0.15m)^2(0.025m) = 0.01768 m^3.
Given that the density is 7850 kg/m^3, we can now calculate the mass: mass = density * volume = 7850 kg/m^3 * 0.01768 m^3 = 138.308 kg.
Therefore, the mass of the flywheel is approximately 138.308 kg.
b) The radius of gyration, denoted as k, is calculated using the formula k = √(I / m), where I is the moment of inertia and m is the mass.
In this case, the moment of inertia is given as 0.156 kg m^2 and the mass is 138.308 kg.
Therefore, the radius of gyration is k = √(0.156 kg m^2 / 138.308 kg) ≈ 0.144 m.
c) To calculate the flywheel's angular acceleration, we can use the formula α = Δω / Δt, where α is the angular acceleration, Δω is the change in angular velocity, and Δt is the change in time.
The angular velocity change is given as 12.0 rad/s (final angular velocity) - 0 rad/s (initial angular velocity) = 12.0 rad/s.
The time change is given as 1 minute = 60 seconds.
Therefore, the angular acceleration is α = (12.0 rad/s) / (60 s) = 0.2 rad/s^2.
To calculate the number of revolutions the flywheel makes, we need to convert the angular velocity to revolutions per minute (rpm). There are 2π radians in one revolution and 60 seconds in one minute. Therefore, the formula to convert from rad/s to rpm is:
rpm = (ω * 60) / (2π)
Using the final angular velocity, this gives us: rpm = (12.0 rad/s * 60) / (2π) ≈ 114.59 rpm.
Finally, to calculate the friction moment applied to the flywheel, we can use the formula M = I * α, where M is the moment and α is the angular acceleration.
Given that I is 0.156 kg m^2 and α is 0.2 rad/s^2, we have M = 0.156 kg m^2 * 0.2 rad/s^2 = 0.0312 Nm.
Therefore, the friction moment applied to the flywheel is approximately 0.0312 Nm.