A flywheel has an angular speed of 1200 rev/min when its motor is turned off. The wheel attains constant decelerations of 1.5 rad/s2 due to friction in its bearing. Determine the time required for the wheel to come to rest and the number of revolutions the wheel makes before it comes to rest
2π•f(fin)= 2π•f(init)-εt
f(fin)=0, f(init) = 1200 rev/min =20 rev/s.
2π•f(init)=εt
t=2π•f(init)/ ε= 2π•20/1.5=83.8 s.
2π•N= 2π•f(init) •t - εt²/2.
N= f(init) •t - εt²/4π=
=20•83.8 – 1.5•(83.8)²/4•π =
= 838 rev.
To determine the time required for the flywheel to come to rest, we need to use the equation of angular motion:
ω = ω₀ + αt
Where:
ω is the final angular velocity (0 rad/s, since the wheel comes to rest)
ω₀ is the initial angular velocity (1200 rev/min, which needs to be converted to rad/s)
α is the constant deceleration (-1.5 rad/s²)
t is the time required for the wheel to come to rest
First, let's convert the initial angular velocity from rev/min to rad/s:
ω₀ = (1200 rev/min) × (2π rad/1 rev) × (1 min/60 s)
ω₀ = 40π rad/s
Now, rearranging the equation to solve for t:
0 = 40π rad/s + (-1.5 rad/s²) × t
-40π rad/s = -1.5 rad/s² × t
t = (-40π rad/s) / (-1.5 rad/s²)
t ≈ 84.8 s
Therefore, the time required for the flywheel to come to rest is approximately 84.8 seconds.
To determine the number of revolutions the wheel makes before coming to rest, we can use the equation:
θ = θ₀ + ω₀t + 0.5αt²
Where:
θ is the total angular displacement (revolutions)
θ₀ is the initial angular displacement (0 revolutions)
ω₀ is the initial angular velocity (40π rad/s)
α is the constant deceleration (-1.5 rad/s²)
t is the time required for the wheel to come to rest (84.8 s)
Plugging in the values:
θ = 0 + (40π rad/s) × (84.8 s) + 0.5 × (-1.5 rad/s²) × (84.8 s)²
Evaluating the expression:
θ ≈ 337.9 revolutions
Therefore, the flywheel makes approximately 337.9 revolutions before it comes to rest.