Answer:
θt = 514.3 revolutions
Step-by-step explanation:
(1)The wheel accelerates uniformly from rest for 10 s and reaches the operating angular speed of 58rad/s.
The uniformly accelerated circular movement a circular path movement in which the angular acceleration is constant.
We apply the equations of circular motion uniformly accelerated
ωf = ω₀ + α*t Formula (1)
θ = ω₀*t + (1/2)*α*t² Formula (2)
ωf² = ω₀² +2*α*θ Formula (3)
Where:
θ : angle that the body has rotated in a given time interval (rad)
α : angular acceleration (rad/s²)
t : time interval (s)
ω₀ : initial angular speed ( rad/s)
ωf : final angular speed ( rad/s)
Number of revolutions made by the wheel from t = 0 to t = 10 s
Data
ω₀ = 0
t = 10 s
ωf = 58 rad/s
We replace data in the formula (1) to calculate α
ωf = ω₀ + α*t
58 = 0 + α*(10)
α = 58 /10
α = 5.8 rad/s²
We replace data in the formula (2) to calculate θ
θ = ω₀*t + (1/2)*α*t²
θ = 0 + (1/2)*( 5.8)*(10)²
θ₁ = 290 rad
(2)The wheel is run at that angular velocity for 30 s, and then power is shut off.
The movement of the wheel is circular with constant angular speed and the formula to calculate θ is:
θ = ω*t
ω = 58 rad/s , t= 30s
θ = (58 rad/s)*(30)
θ = (58 rad/s)*(30)
θ ₂= 1740 rad
(3)The wheel slows down uniformly at 1.4 rad/s² until the wheel stops.
ω₀ = 58 rad/s
α = -1.4 rad/s²
ωf = 0
We replace data in the formula (3) to calculate θ
(ωf)² = (ω₀)² + (2)*(α )*θ
0 = (58)² + (2)*(-1.4)*θ
(2)*(1.4)*θ = (58)²
θ = (58)² / (2.8)
θ = (58)² / (2.8)
θ₃ = 1201.42 rad
Total number of revolutions made by the wheel (θt)
θt =θ₁+θ₂+θ₃
θt = 290 rad+ 1740 rad + 1201.42 rad
θt = 3231.42 rad
1 revolution = 2π rad
θt = 3231.42 rad* ( 1revolution/2π rad)
θt = 514.3 revolutions