TRIG WHEEL 1
Part One
A water wheel shown above has a diameter of 8 m. The wheel turns through 1 revolution (cycle) in 60 seconds. Represent the motion of the water wheel as a sine function with respect to time at the starting point indicated.
Part 2
How long does it take for the wheel to reach a height of 2m?
At what time does the wheel drop below 2m?
Firstly sketch a diagram like the one above to get a feel for the problem. Add in the time values along the x axis, 0 at 0, 15, 30 and 60 seconds followed by their equivalent names in terms of π/2, , π, 3π/2 and 2π at the end.
A water wheel shown above has a diameter of 8 m. The wheel turns through 1 revolution (cycle) in 60 seconds. Represent the motion of the water wheel as a sine function with respect to time at the starting point indicated.
Part 2
How long does it take for the wheel to reach a height of 2m?
At what time does the wheel drop below 2m?
Firstly sketch a diagram like the one above to get a feel for the problem. Add in the time values along the x axis, 0 at 0, 15, 30 and 60 seconds followed by their equivalent names in terms of π/2, , π, 3π/2 and 2π at the end.
Amplitude (A)
The amplitude is the same dimension as the radius of the wheel which is 4 m.
A more formal way of doing this is:
A more formal way of doing this is:
The Frequency (B)
The wheel completes one revolution in 60 seconds so it has a period of 60 seconds. We use a formula to work out the value of B. The frequency B is equal to the distance of 2π divided by the time for one period, which is:
Because the period is 60 seconds we substitute that value into the formula.
horizontal (Phase) shift (C)
Because this is a normal sine wave function there is no shift to be concerned with so C = 0
Vertical Shift (D)
Because the waveform is on the 'x axis' it has no vertical displacement neither up or down.
Putting it all together
on the calculator
Enter the equations into the calulator, you will need to adjust the view window to see the graph properly. Use the brackets if you have trouble entering the sine equation. I have used the abc key to represent B as π/30. You can also use 2π/60 as its the same. As long as Xmin is at least 60 you will be OK and Ymin and Ymax set to at least +4 and -4. The graph will draw two functions, the sinewave and the linear line. Dont worry about the dot setting.
To answer the question about the time to reach 2m, Go into GSOLVE and intersect (ISECT), hit the cursor key to find the intersecting point where the curve and the line intersect. The calculator then indicates it takes 5 seconds. Hit the curser key again to 'jump' to the next point of intersection which is at 25 seconds. If we were asked to say how long was the water wheel bucket in question above 2m then we just subtract 5 sweconds off 25 to get 20 seconds.
creating the graph with desmos
graph using geogebra
Geogebra uses a different formula because it needs to be told where to draw the graph other wise it would keep repeating the waveform over and over which is fine for a water wheel but not so good for Ferris wheel passengers. That's why it says "If the graph is restricted to the zone between 0 and 60" then draw the graph there. Notice that 2π/60 is the same as π/30 as in above.
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