Answer:
it makes N= 43.493≈ 43 revolutions
Explanation
there are 3 stages
1) steady acceleration from rest to 580 rpm in 1.0 seg
2) constant angular speed for 3.0 s
3) steady acceleration to halt in 2.0 s
knowing that
θ final =θ initial +ω initial * t + 1/2 α * t²
where θ angle respect with the horizontal axis , ω= angular velocity α= angular acceleration
since ω initial = 0 , θ initial=0 , α =(ω final - ω initial)/t = ω final /t
θ final = 1/2 ω final * t
since f = ω/(2π) and N=(θ final- θ initial)/(2π) , where f=frequency and N= number of revolutions
N1= 1/2 f final * t
N1= 1/2 * 580 revolutions/minute * 1 s * 1 min/60 sec= 4. 833 rev
for 2) θ final - θ initial = ω initial + ω final * t
θ final = ω final * t
N2= f final * t = 580 revolutions/minute * 3 s * 1 min/60 sec= 29 rev
for 3)
θ final =θ initial +ω initial * t + 1/2 α * t²
since ω final = 0 , α =(ω final - ω initial)/t = -ω initial /t
θ final- θ initial = ω initial * t + 1/2 α * t² = ω initial * t - 1/2 ω initial * t = 1/2 ω initial * t
θ final- θ initial = 1/2 ω initial * t
therefore
N3= 1/2 f initial * t = 1/2 * 580 revolutions/minute * 2 s * 1 min/60 sec= 9.66 rev
therefore
N total = N1 +N2 +N3 = 4. 833 rev + 29 rev + 9.66 rev =43.493 revolutions
