The flywheel of a steam engine begins to rotate from rest with a constant angular acceleration of 1.25 rad/s2. It accelerates for 26.7 s, then maintains a constant angular velocity. Calculate the total angle through which the wheel has turned 49.9 s after it begins rotating.
To solve this problem, we need to break it down into two parts: the acceleration phase and the constant velocity phase.
First, let's find the angular velocity at the end of the acceleration phase using the formula:
ω = ω₀ + αt
where
ω = final angular velocity
ω₀ = initial angular velocity (which is zero in this case)
α = angular acceleration
t = time
Substituting the given values, we have:
ω = 0 + (1.25 rad/s²)(26.7 s)
ω ≈ 33.375 rad/s
Now, let's find the angle turned during the acceleration phase using the formula:
θ = ω₀t + (1/2)αt²
where
θ = angle turned
ω₀ = initial angular velocity
α = angular acceleration
t = time
Substituting the given values, we have:
θ = 0(26.7 s) + (1/2)(1.25 rad/s²)(26.7 s)²
θ ≈ 223.8375 rad
During the constant velocity phase, the angular acceleration is zero, so we can simply use the formula:
θ = ωt
where
θ = angle turned
ω = angular velocity
t = time
Substituting the given values, we have:
θ = (33.375 rad/s)(49.9 s)
θ ≈ 1662.79125 rad
Finally, to find the total angle through which the wheel has turned 49.9 s after it begins rotating, we add the angles turned during the acceleration and constant velocity phases:
Total angle = angle during acceleration + angle during constant velocity
Total angle ≈ 223.8375 rad + 1662.79125 rad
Total angle ≈ 1886.62875 rad
Therefore, the total angle through which the wheel has turned 49.9 s after it begins rotating is approximately 1886.63 radians.