A turbine wheel is positioned such that the center of the wheel sits 26 feet off the floor of an electric steam generating plant. The wheel has a radius of 18 feet and is turning at 1rev/8sec. Design a Trig Model that will predict the height of the sensor, located at the lowest point on the wheel at t = 0 seconds, from the floor of the plant. The Trig model should be in the form of h(t)=(a)cos((b)(t)+c)+d , where h(t) measures the height of the sensor from the floor, in feet, at any time t in seconds.
The wheel is turning at 1 rev/ 8 sec, so 1/8 rev/sec, i.e. pi/4 radians per second ( = 45 degrees per second) So the angle the wheel has rotated by at time T is T*pi/4 rad = 45*T degrees - we will denote the angle by x for clarity. The radius of the wheel is 18 feet, so after time T, the sensor will be 18-18cos(x) feet higher than it was when it started the rotation. To see this, draw the wheel having rotated through some angle y<pi/2 rad (=90 degrees), and use trigonometry to work out the vertical distance between the centre of the wheel and the sensor. Since you know the radius of the wheel, you can the work out how much height the sensor has gained in terms of the angle the wheel has rotated. Now, for any time t, we know that the sensor will be 18(1-cos(x)) feet higher than it started. We also know that at t=0 the sensor was at the lowest point of the wheel, which was 8 feet off the ground (since the centre of the wheel was 26 feet off the ground and the wheel has radius 18 feet) Hence, at any time t, the sensor is -18(cos(t*pi/4)-1) + 8 feet from the ground.
