When switched off, a spinning ceiling fan slows down with an angular acceleration of 1.2 rad/s2. If the fan was spinning at the rate of 10π rad/sec before it was turned off, how many revolutions did it go through before coming to the stop?
φ=ω₀t-εt²/2
ω=ω₀-εt
ω=0
ω₀=10π rad/s
ε=1.2 rad/s²
φ=2πN
0=ω₀-εt => t = ω₀/ε
2πN = ω₀t-εt²/2=
=ω₀²/ε - ε ω₀²/2ε²=ω₀²/2ε
N= ω₀²/4πε
To solve this problem, we can use the equation of rotational motion:
ω^2 = ω0^2 + 2αθ
Where:
- ω is the final angular velocity (0 rad/sec since the fan comes to a stop)
- ω0 is the initial angular velocity (10π rad/sec)
- α is the angular acceleration (-1.2 rad/s^2 since it is slowing down)
- θ is the angular displacement (we are trying to find this)
Substituting the given values into the equation, we have:
(0)^2 = (10π)^2 + 2(-1.2)θ
Simplifying this equation, we get:
0 = 100π^2 - 2.4θ
Rearranging the equation, we have:
2.4θ = 100π^2
θ = (100π^2) / 2.4
θ ≈ 1309 revolutions
Therefore, the spinning ceiling fan went through approximately 1309 revolutions before coming to a stop.