Question:
A wind turbine has a total height of 50m, with 3 equally spaced blades that are 16m in diameter.
1) Create equations modelling the height of points on the tips of blades One, Two and Three. Graph these three curves on one graph.
[Answers: Blade 1: y=8sin(x)+42
Blade 2: y=8sin(x+120)+42
Blade 3: y=8sin(x-120)+42]
2) Maintenance can only be done on the blades when the points on blades Two and Three are at the same height. Identify the rotation angles when this would occur as well as the height of the points at that time.
3) What is the height of the tip of Blade Two, when the tip of Blade Three is 1m below its maximum height?
4) On a particular day the blades are rotating at a rate of 60 per second. How fast are the tips of the blades travelling? If you wanted to throw a tennis ball through the blades and not be struck, how long is the gap of time between blades?
I'm just so freaking lost on how to answer 2, 3 and 4. I have the equations for the first question, but have no idea how to find the answers for the three others. I've been stuck on this for over an hour. I think I'm just overcomplicating how to answer them, which is why I'm stuck, and if anyone could please point me in the right direction, I'd be super thankful.
#2 When blade 1 is vertical, blades 2 and 3 are at angles 30 and 150
#3 when blade 2 is 7m above the axle, then its angle from the vertical is arccos(7/8). Blade 3 is 120 degrees from that angle.
#4 If you mean the blades are rotating at 60°/s = π/3 rad/s, then since s = rθ, their tips are moving at 8π/3 m/s
Assuming the ball is thrown far from the axle, so its size is relatively small compared with the gap, then since the blades are 120° apart, they take 2 seconds to move from one position to the next.
See what you can do with this info...
Let's break down each of the remaining questions step-by-step:
2) To find the rotation angles when the points on blades Two and Three are at the same height, we need to set the equations for the heights of the points on Blade Two and Blade Three equal to each other.
Using the equations derived earlier, we have:
8sin(x+120) + 42 = 8sin(x-120) + 42
Simplifying this equation, we get:
sin(x+120) = sin(x-120)
To find the rotation angles, we need to find the values of x that satisfy this equation. Since sin(x) = sin(180 - x), we can rewrite the equation as:
x+120 = 180 - (x-120)
2x = 240
x = 120
So, the rotation angle when the points on blades Two and Three are at the same height is 120 degrees.
To find the height of the points at that time, we substitute x = 120 into any of the blade equations. Let's use the equation for Blade Two:
y = 8sin(x+120) + 42
y = 8sin(240) + 42
y ≈ 50.8
Therefore, at a rotation angle of 120 degrees, the height of the points on blades Two and Three is approximately 50.8m.
3) To find the height of the tip of Blade Two when the tip of Blade Three is 1m below its maximum height, we need to set the equation for the height of Blade Three equal to its maximum height minus 1m, and then solve for the value of x.
Using the equation for Blade Three:
y = 8sin(x-120) + 42
Setting y = 50 - 1, we have:
50 = 8sin(x-120) + 42
8sin(x-120) = 9
sin(x-120) = 9/8
To find x, we can take the inverse sine (sin^-1) of both sides:
x - 120 = sin^-1(9/8)
x ≈ 154.57
Therefore, at a rotation angle of approximately 154.57 degrees, the tip of Blade Two will have a height that corresponds to Blade Three being 1m below its maximum height.
4) To determine how fast the tips of the blades are traveling, we can find the derivative of the equations for the heights of the points on each blade with respect to time (assuming the angle x is a function of time).
The general formula for the instantaneous rate of change (velocity) of a point on a rotating object can be expressed as:
v = ω * r
Where v is the velocity, ω is the angular velocity, and r is the distance from the center of rotation.
In this case, the distance from the center of rotation is the radius of the wind turbine, which is half the diameter of the blades, or 8m.
The angular velocity can be found by considering that the given rotation rate is 60 rotations per second. Therefore, one rotation occurs in 1/60 seconds. The angular velocity (ω) is then 2π divided by the time it takes to complete one rotation.
ω = 2π / (1/60) = 120π
Now, substituting the radius and angular velocity into the velocity formula:
v = (120π) * 8 = 960π
Therefore, the tips of the blades are traveling at a speed of approximately 960π meters per second.
To calculate the gap of time between the blades, we need to find the time it takes for one blade to rotate 120 degrees, which is the angle between each blade.
Using the given rotation rate of 60 rotations per second:
Time for one rotation = 1 / 60 seconds
Time for 120 degrees = (120 / 360) * (1 / 60) = 1/ 180 seconds
Therefore, the gap of time between the blades is 1/180 seconds.
I hope this step-by-step explanation helps you understand how to answer questions 2, 3, and 4. If you have any further questions, feel free to ask.
To answer questions 2, 3, and 4, we'll need to use the equations we derived in question 1. Let's go through each question step by step:
2) To find the rotation angles when the points on blades Two and Three are at the same height, we can equate the two height equations and solve for x.
y = 8sin(x + 120) + 42 (equation for Blade 2)
y = 8sin(x - 120) + 42 (equation for Blade 3)
Setting these two equations equal to each other:
8sin(x + 120) + 42 = 8sin(x - 120) + 42
Now, we can simplify by canceling out the 42 on both sides:
8sin(x + 120) = 8sin(x - 120)
Dividing both sides by 8:
sin(x + 120) = sin(x - 120)
Since sine is an odd function, sin(x + 120) = sin(x - 120) would imply x + 120 = 180 - (x - 120). Solving for x:
x + 120 = 180 - x + 120
Simplifying:
2x = -120
x = -60
Therefore, the rotation angle when the points on blades Two and Three are at the same height is -60 degrees.
To determine the height of the points at that time, we can substitute -60 into any of the height equations (for example, Blade 2):
y = 8sin(-60 + 120) + 42
y = 8sin(60) + 42
y = 8√3/2 + 42
y ≈ 46.39
So, at a rotation angle of -60 degrees, the height of the points on blades Two and Three would be approximately 46.39 meters.
3) To find the height of the tip of Blade Two when the tip of Blade Three is 1m below its maximum height, we need to determine the corresponding rotation angle.
First, we need to find the maximum height of Blade Three. We know that the general equation for Blade Three's height is y = 8sin(x - 120) + 42, where the maximum height occurs when sin(x - 120) = 1 (since the maximum value of sine is 1).
So, we have 8sin(x - 120) + 42 = 8(1) + 42 = 50, which is the maximum height of Blade Three.
Now, we want to find the rotation angle (x) when Blade Three is 1m below its maximum height. This means the height would be 50 - 1 = 49m.
Setting up the equation:
8sin(x - 120) + 42 = 49
Subtracting 42 from both sides:
8sin(x - 120) = 7
Dividing both sides by 8:
sin(x - 120) = 7/8
We can take the inverse sine (sin^-1) of both sides to find the rotation angle:
x - 120 = sin^-1(7/8)
x - 120 ≈ 47.81
x ≈ 167.81
So, at a rotation angle of approximately 167.81 degrees, the height of the tip of Blade Two would be the same as the height of Blade Three when it is 1m below its maximum height.
4) To determine how fast the tips of the blades are traveling, we need to take the derivative of the height equations with respect to the rotation angle (x).
Let's start with Blade Two's height equation: y = 8sin(x + 120) + 42
To find the derivative, we'll use the chain rule:
dy/dx = 8cos(x + 120)
Similarly, we can find the derivative for Blade Three's height equation:
dy/dx = 8cos(x - 120)
Now, substituting the rotation angle at which the blades are rotating (60 per second) into the derivative equations:
dy/dx = 8cos(60 + 120) = 8cos(180) = -8
Therefore, the tips of the blades are traveling at a speed of -8m/s (negative since we're looking at the downward direction).
To calculate the time gap between the blades, we need to determine the time it takes for one complete rotation (360 degrees).
The time for one full rotation (T) is given by T = 360 degrees / 60 rotations per second = 6 seconds.
Therefore, the time gap between the blades is 6 seconds.
I hope this explanation helps you understand how to approach and solve these questions. If you have any further questions, feel free to ask!