The flywheel of a steam engine begins to rotate from rest with a constant angular acceleration of 1.39 rad/s^2. It accelerates for 30.3 s, then maintains a constant angular velocity. Calculate the total angle through which the wheel has turned 69.1 s after it begins rotating.
2010 rad
To calculate the total angle through which the wheel has turned, we can use the equations of rotational motion. We know that the flywheel starts from rest, so its initial angular velocity (ω0) is zero. It also has a constant angular acceleration (α) of 1.39 rad/s^2.
First, we need to find the final angular velocity (ωf) after the flywheel accelerates for 30.3 s. We can use the equation:
ωf = ω0 + αt
Substituting the given values, we find:
ωf = 0 + 1.39 * 30.3
= 42.177 rad/s
Next, we can calculate the angle (θ1) covered during the acceleration period using the equation:
θ1 = ω0*t + 0.5*α*t^2
Substituting the values, we have:
θ1 = 0.5 * 1.39 * (30.3)^2
= 626.26385 rad
Finally, we can calculate the angle (θ2) covered during the period of constant angular velocity using the equation:
θ2 = ωf*t
Substituting the values, we get:
θ2 = 42.177 * (69.1 - 30.3)
= 1,652.3767 rad
To find the total angle through which the wheel has turned, we add θ1 and θ2:
Total angle = θ1 + θ2
= 626.26385 + 1,652.3767
= 2,278.64055 rad
Therefore, the flywheel has turned a total angle of approximately 2,278.64 radians after 69.1 seconds of rotating.