A cooling fan is turned off when it is running at 870 rev/min. It turns 1200 revolutions before it comes to a stop.
What was the fan's angular acceleration, assumed constant?
How long did it take the fan to come to a complete stop?
870 rpm = 91.1 rad/s
The average angular velocity during decleration is half that, or
w,av = 45.6 rad/s
The time it takes to stop is
2400*2*pi/45.6 = 165 seconds
If the want the angular acceleration in radians/sec^2, use
alpha = (91.1 rad/s)/165 s
How did you get 91.1 rad/s?
To find the angular acceleration of the fan, we need to use the formula:
ω^2 = ω0^2 + 2αθ
Where:
ω = final angular velocity (0 rad/min, since the fan comes to a stop)
ω0 = initial angular velocity (870 rev/min, convert to rad/min: 870 rev/min * 2π rad/rev * 1 min/60 s = 290π rad/min)
α = angular acceleration (unknown)
θ = angular distance (1200 revolutions, convert to rad: 1200 rev * 2π rad/rev = 2400π rad)
Plugging in the values, we can solve for α:
0 = (290π)^2 + 2α(2400π)
0 = 84100π^2 + 4800πα
Simplifying:
α = -84100π^2 / (4800π)
α ≈ -553/3π rad/min^2
Therefore, the fan's angular acceleration is approximately -553/3π rad/min^2.
To find the time it took for the fan to stop, we can use the equation:
ω = ω0 + αt
Where:
ω = final angular velocity (0 rad/min)
ω0 = initial angular velocity (870 rev/min, convert to rad/min: 870 rev/min * 2π rad/rev * 1 min/60 s = 290π rad/min)
α = angular acceleration (-553/3π rad/min^2, negative sign indicates deceleration)
t = time (unknown)
Plugging in the values, we can solve for t:
0 = 290π - (553/3π)t
Simplifying:
(553/3π)t = 290π
t ≈ 1740/61 s
Therefore, it took approximately 1740/61 seconds for the fan to come to a complete stop.
To find the angular acceleration, we can use the formula:
ωf = ωi + αt
where:
ωf is the final angular velocity (0 rev/min, as the fan has come to a stop)
ωi is the initial angular velocity (870 rev/min)
α is the angular acceleration (unknown)
t is the time it takes for the fan to stop (unknown)
We are given that the fan turns 1200 revolutions before coming to a stop. A revolution is defined as one complete rotation of 360 degrees, which is equivalent to 2π radians. Therefore, we can convert the number of revolutions to radians:
θ = (number of revolutions) * (2π radians/revolution)
θ = 1200 rev * 2π radians/rev
θ = 2400π radians
We can now solve for the angular acceleration (α) using the formula:
ωf^2 = ωi^2 + 2αθ
0^2 = (870 rev/min)^2 + 2α(2400π radians)
Simplifying:
0 = 756900 rev^2/min^2 + 4800πα radians/min^2
To find the angular acceleration (α), we need to express it in terms of radians/minute^2:
1 revolution/min = 2π radians/min
Taking the derivative of both sides:
1 revolutions/minute^2 = 2π radians/minute^2
Substituting this into the equation:
0 = 756900 rev^2/min^2 + 4800πα radians/minute^2
0 = 756900 rev^2/min^2 + 9600πα rev^2/min^2
Now we can equate the coefficients:
9600πα = -756900
α = -756900 / (9600π)
α ≈ -7.89 radians/minute^2
So, the fan's angular acceleration is approximately -7.89 radians/minute^2.
To find the time it takes for the fan to come to a complete stop, we can use the equation:
ωf = ωi + αt
0 = 870 rev/min + (-7.89 radians/minute^2) * t
Solving for t:
7.89t = 870
t = 870 / 7.89
t ≈ 110.19 minutes
Therefore, it takes approximately 110.19 minutes for the fan to come to a complete stop.