The flywheel of a steam engine begins to rotate from rest with a constant angular acceleration of 1.41 rad/s2. It accelerates for 25.5 s, then maintains a constant angular velocity. Calculate the total angle through which the wheel has turned 65.5 s after it begins rotating.
V = a*t = 1.41 * 25.5 = 35.96 rad/s.
d=V*t = 35.96 * (65.5-25.5) = 1438 rad.
1438
To calculate the total angle through which the flywheel has turned, we need to determine two things: the angle covered during the acceleration phase and the angle covered during the constant velocity phase.
During the acceleration phase, we can use the equation:
θ = ω₁t + (1/2)αt²
Where:
θ = angle covered
ω₁ = initial angular velocity (0 since it starts from rest)
α = angular acceleration (1.41 rad/s²)
t = time (25.5 s)
Plugging in these values, we get:
θ₁ = (0)(25.5) + (1/2)(1.41)(25.5)²
Simplifying the equation:
θ₁ = (1/2)(1.41)(25.5)²
Now, during the constant velocity phase, the angular velocity remains constant, so the angle covered can be calculated using the formula:
θ = ωt
Where:
θ = angle covered
ω = angular velocity
t = time (65.5 s - 25.5 s = 40 s)
Now, we know the angular acceleration is zero, so the flywheel maintains a constant angular velocity. Hence, we can calculate θ₂ as follows:
θ₂ = ω(40)
To find ω, we'll use another formula using the initial angular velocity (ω₁) and the angular acceleration (α), given that the final time during the acceleration phase is 25.5 s:
ω = ω₁ + αt
Plugging in the values:
ω = 0 + (1.41)(25.5)
Simplifying:
ω = 1.41(25.5)
Finally, we can calculate θ₂:
θ₂ = (1.41)(25.5)(40)
To get the total angle through which the wheel has turned, we add θ₁ and θ₂:
Total angle = θ₁ + θ₂
Substituting the values we found earlier, we can calculate the desired answer.