Hi there!
We can first find the angular displacement of the first 20 seconds by finding the angular acceleration.
Begin by converting 120 rev/min to rev/sec:
120 rev/min × 1 min/60 sec = 2 rev/sec
ω = αt
2 = 20α
α = 0.1 rev/sec²
θ = ωit + 1/2αt²
The fan starts from rest, so:
θ = 1/2αt²
θ = 1/2(0.1)(20²) = 20 revolutions
Now, for the next 40 seconds, we can find the fan's deceleration:
ωf = ωi + αt
0 = 2 + αt
-2 = 40α
α = -0.05 rad/sec²
Use the above equation:
θ = ωit + 1/2αt²
θ = 2(40) + 1/2(-0.05)(40²) = 80 - 40 = 40 rev
θ = 4800 - 2400 = 2400 revolutions
Add the two:
θtotal = 20 + 40 = 60 revolutions