1. ## Sinusoidal function help

A small windmill has its center 10 meters above the ground and blades 4 meters in length. In a steady wind, a point P at the tip of one blade makes a rotation in 24 seconds.

If the rotation begins at the highest possible point, determine a function that gives the height of point P above the ground at a time T

2. Originally Posted by imakeitrain
A small windmill has its center 10 meters above the ground and blades 4 meters in length. In a steady wind, a point P at the tip of one blade makes a rotation in 24 seconds.

If the rotation begins at the highest possible point, determine a function that gives the height of point P above the ground at a time T
1. Draw a sketch.

2. According to my sketch the point P has the coordinates P(x, 10+y).

3. Since the argument of the sine function is an angle measured against(?) the horizontal axis the argument here must be $\displaystyle \frac \pi2 + \mu$ when you calculate the x-coordinate.

4. According to the text of the question a blade of the mill makes a full turn in 24 s. If T is measured in seconds the angle $\displaystyle \mu$ is calculated by:

$\displaystyle \mu=\frac{2 \pi}{24} \cdot T =\frac{ \pi}{12} \cdot T$

5. Since the argument of the cosine function is an angle measured against(?) the horizontal axis the argument here must be $\displaystyle -\frac \pi2 + \mu$ when you calculate the y-coordinate.

6. Therefore you have:

$\displaystyle f(T)=\left\{\begin{array}{l}x=4 \cdot \sin\left(\frac \pi2 + \frac{ \pi}{12} \cdot T\right) \\ h=10+y=10+4 \cdot \cos\left(-\frac \pi2 + \frac{ \pi}{12} \cdot T\right)\end{array}\right.$

7. Since you are only interested in the height above ground you'll get:

$\displaystyle h(T)=10+4 \cdot \cos\left(-\frac \pi2 + \frac{ \pi}{12} \cdot T\right)$

3. Nicely done, earboth!