I need help figuring out how to solve this

Question

A gasoline and air mixture is ignited by spark plugs, which forces the pistons to move up and down, rotating the crankshaft. The animation on the left shows a single piston moving up and down. The green connecting rod spins the crankshaft.   

  

1. If the radius of the circle is 3 inches, and the frequency is 2000 revolutions per minute, create a function model for this situation, assuming you are starting from the 180 degree position on the unit circle. Insert your equation below.  

 

2. If the distance from the top of the engine to the center of the circle is 10 inches, write a new function that shows the movement relative to the top of the engine.  

Answer 1

The function model will be an oscillating wave, either a sine or cosine wave. The function model will show the position of the crankshaft away from the horizontal center. At 0-degrees and 180-degrees, the crankshaft is in-line with the horizontal center and the position away from this center would be 0. As such, the wave we will use is the sine wave. If the starting position was 90-degrees or 270-degrees, the crankshaft would be perpendicular with the horizontal center and we would use a cosine wave instead.

y = m*sin(ax+b)

The amplitude for this function is 3, since the crankshaft's circle's radius is 3 inches. Therefore, m = 3. There are 2,000 revolutions per minute. Therefore, a = 2,000. Finally, we know we are starting from the 180-degree position. Therefore, b = 180/pi (converted degrees to radians).

y = 3sin(2000x+180/pi)

If we change our relative position from the horizontal center to the top of the engine, this means the crankshaft's horizontal center is 10m below the top of the engine. Which means when the horizontal center is at zero, our function should output -10.

y = 3sin(2000x+180/pi) - 10