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The blades on a model windmill take 4 seconds to make a full rotation...........?
The blades on a model windmill take 4 seconds to make a full rotation. At the lowest point, the tip of the blade is 0.5 m off the ground, and at the highest point the tip of the blade is 2.5 m off the ground.
a) sketch a graph of height against time to show the distance between the tip of the blade and the ground for two complete rotations. At time zero, the blade is pointing straight down (closest to the ground). Clearly label the scale on both axes
b) Determine an algebraic equation relating the height (h) in metres of the tip of the blade above the ground at time (t) in seconds.
- 8 years agoFavorite Answer
The low point is 0.5m; the high point is 2.5m. That means the windmill has a diameter of 2m and hence a radius of 1m.
a) Make a table of points on the graph.
t | h
0 | 0.5 <----- Tip at bottom.
1 | 1.5 <----- Windmill rotated 90deg in 1 sec, so height is at center (0.5 + radius)
2 | 2.5 <----- Highest point after 1/2 rotation.
3 | 1.5 <----- 270deg rotation after 3sec; tip is on way back down.
4 | 0.5 <----- One complete revolution after 4sec; back down to low point.
5 | 1.5
6 | 2.5
7 | 1.5
8 | 0.5
Plot those points on a graph and then connect the dots with a smooth curve. As the other answerers have implied, you'll get something that looks like a sine wave. Well, actually it doesn't just look like a sine wave, it _is_ a sine wave, but shifted 1.5m upward & 1sec to the right.
b) The height is actually a function of the cosine, not the sine (although you could convert it to a sine function via an identity if you so desired). Here's how you determine the height function.
Draw a circle on a sheet of paper. This represents the path of the windmill blade's tip. At time t=0, the tip is at the very bottom, so draw the radius from the center of the circle to the bottom point. Call the center Point C and the bottom point A. Now suppose the windmill rotates (counterclockwise) by an angle of theta (say 45deg). Go ahead and draw a second radius from the center to the 4:30 point on the circle. Call this new point B; it's the new position of the blade tip. The interior angle is theta. Now draw a horizontal line from point B to radius AC. Call the point of intersection D. Note that triangle BCD is a right triangle. The height of the blade tip from its initial position is ||AD||:
ht = ||AD||
ht = ||AC|| - ||CD||
ht = r - r cos(theta) <----- r is radius of circle, ||AC|| = r = ||BC||
ht = r(1 - cos(theta))
But don't forget... the initial height is 0.5m off the ground in your problem, so you need to add that in.
ht = r(1 - cos(theta)) + 0.5
And since r=1m in your problem, this reduces even further to...
ht = (1 - cos(theta)) + 0.5
ht = 1.5 - cos(theta)
That's your function for the height... except for one thing. The above equation is expressed in terms of theta, and your question asks for the height in terms of time! So you need to find an expression for theta in terms of time.
The windmill makes one complete revolution (2pi rads) in 4 seconds. That means theta increases by 2pi/4 radians per second. So...
theta = (2pi/4)t = (pi/2)t
So now you can come up with your height function expressed in terms of time.
h(t) = 1.5 - cos((pi/2)t)
Let's check this against the table we made in part 'a' above.
t | h(t)
0 | 1.5 - cos(0) = 1.5 - 1 = 0.5
1 | 1.5 - cos(pi/2) = 1.5 - 0 = 1.5
2 | 1.5 - cos(pi) = 1.5 - -1 = 1.5 + 1 = 2.5
3 | 1.5 - cos(3pi/2) = 1.5 - 0 = 1.5
4 | 1.5 - cos(2pi) = 1.5 - 1 = 0.5
and so on. Looks pretty good to me. :-)
I said earlier that you could convert the function from a cosine function into a sine function. If you expand sin(theta - pi/2) you'll arrive at the conclusion sin(theta - pi/2) = -cos(theta). Now substitute into h(t) and you have an equivalent function expressed in terms of the sine.
H(t) = 1.5 + sin((pi/2)t - pi/2)
H(t) = 1.5 + sin((pi/2)(t - 1))
Go ahead and try it out for yourself; i.e. build another table and check it against what we already have.
- Anonymous8 years ago
a) sin(x) graph with maximum point 2.5 (x2 graphs within 0<x<360)
y axis = hight
x axis = time
b) h=sin(t)Source(s): a level mathematics
- Irv SLv 78 years ago
Hint: think sine wave + Ht to center.