A wind turbine rotates at 15.1 rpm and has an angular acceleration of 0.0659 rad/s2. If the wind turbine takes 24.0 s to come to a complete stop, how many revolutions will this take?
THREE WAYS: A, B, and C
omega at start = 15.1 revs/min * 2 pi rad/rev * 1 min/60 s
= 1.58 rad/s
if acceleration is constant then omega is linear and
average omega = 1.58/2 rad/s
1.58/2 * 24 = 19 radians
so
A) 19 radians * 1 rev/6.28 radians = 3.02 revs to stop
or just
15.1 / 2 = 7.55 rpm average = 7.55/60 revs/second
so
B )7.55/60 * 24 = 3.02 revs to stop
Of course you could do it a harder way
angle = omega initial* t - (angular accelertion/2) t^2
= 1.58 * 24 - (0.0659/2)24^2
= 38 - 19 = 19 radians
so
C) 19 /6.28 = 3.02 revs
To find the number of revolutions the wind turbine will take to come to a complete stop, we first need to find the initial angular velocity of the turbine.
Given:
Angular velocity (ω) = 15.1 rpm
We know that 1 revolution is equal to 2π radians. Therefore, we can convert the given angular velocity from rpm to radians per second (rad/s) using the following conversion:
1 rpm = (2π radians) / 60 seconds
So, the initial angular velocity of the wind turbine in rad/s is:
ω = (15.1 rpm) * (2π radians / 60 seconds)
Next, we need to find the time it will take for the wind turbine to come to a complete stop.
Given:
Angular acceleration (α) = 0.0659 rad/s²
Time (t) = 24.0 seconds
Using the equation:
ω = ω₀ + αt
Where:
ω = final angular velocity (which is 0 since it comes to a complete stop)
ω₀ = initial angular velocity (which we need to find)
α = angular acceleration
t = time
Substituting the values into the equation:
0 = ω₀ + (0.0659 rad/s²) * (24.0 seconds)
Simplifying, we get:
ω₀ = -(0.0659 rad/s²) * (24.0 seconds)
Now, we have the initial angular velocity (ω₀) of the wind turbine.
To find the number of revolutions it will take for the wind turbine to come to a complete stop, we can use the following formula:
Number of revolutions = |ω₀| / (2π radians)
Substituting the value of ω₀, we get:
Number of revolutions = |-(0.0659 rad/s²) * (24.0 seconds)| / (2π radians)
Calculating this expression, we find:
Number of revolutions ≈ 0.643 revolutions (rounded to three decimal places)
Therefore, the wind turbine will take approximately 0.643 revolutions to come to a complete stop.
To find the number of revolutions, we need to first calculate the final angular velocity of the wind turbine.
Using the formula for angular acceleration:
angular acceleration (α) = change in angular velocity (Δω) / change in time (Δt)
We can rearrange the formula to solve for Δω:
Δω = α * Δt
Substituting the given values:
Δω = 0.0659 rad/s² * 24.0 s
Δω = 1.5786 rad/s
Next, we need to find how many revolutions this corresponds to. Since we know the initial angular velocity (15.1 rpm), we can use the formula:
Number of revolutions = final angular velocity (rad/s) / (2 * π)
Converting the initial angular velocity from rpm to rad/s:
Initial angular velocity = 15.1 rpm * (2π rad/1 min) * (1 min/60 s)
Initial angular velocity = 15.1 * 2π / 60
Initial angular velocity = 0.5π rad/s
Now, we can calculate the number of revolutions:
Number of revolutions = Δω / (2 * π)
Number of revolutions = 1.5786 rad/s / (2 * π)
Number of revolutions ≈ 0.2507 revolutions
Therefore, it will take approximately 0.2507 revolutions for the wind turbine to come to a complete stop.