How do I find the equation of a sinusoidal curve from just knowing the lowest point and highest point of it?
For example:
A weight attached to the end of a long spring is bouncing up and down. If we do not worry about friction, its distance from the floor varies sinusoidally with time. You start a stopwatch. When the stopwatch reaches 0.3 seconds, the weight reaches a highpoint for the first time at 60 cm. above the floor. The next time the weight reaches a lowpoint is 1.8 seconds later and that lowpoint is at 40 cm. above the floor.
The distance between the high and low points is 60-40 or 20 cm, so the amplitude of the curve will be 10
The time between the high and the low points is 1.8-.3 or 1.5 seconds, so the period of the curve is 3 seconds.
If we had y = 10cos(2pi/3)x we would fulfill those two conditions, but our first highpoint would be at 0 sec.
So we need a horizontal shift of .3 sec to the right.
that would be 9/(20pi)
so lets see what we have so far
y = 10cos(2pi/3)[x - 9/(20pi)]
(check: phase shift = (2pi/3)*9/(20pi) = .3)
Finally we have to move the curve vertically so that its centre line runs along y = 50, the midway between 40 and 60
so we would have
y = 10cos(2pi/3)[x - 9/(20pi)] + 50
You could do the same thing with a sine curve, but the phase shift would be different. I chose the cosine curve because .3 was closer to the y-axis, and the cosine curve starts with y=1 at x = 0, so my shift would be smaller.
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To find the equation of a sinusoidal curve given the highest and lowest points, you will need to determine the amplitude, period, vertical shift, and phase shift of the curve.
1. Amplitude: The amplitude is the vertical distance between the highest and lowest points of the curve. In this case, the amplitude is (60 cm - 40 cm)/2 = 10 cm.
2. Period: The period is the time it takes for the curve to complete one full cycle. In this case, the time between the highpoint and the next lowpoint is 1.8 seconds. Since the motion repeats itself, the time for a full cycle is twice the time between the highpoint and the next lowpoint. Therefore, the period is 2 * 1.8 seconds = 3.6 seconds.
3. Vertical Shift: The vertical shift represents any vertical displacement from the midpoint of the curve. In this case, since the midpoint is the average of the highest and lowest points, the vertical shift is (60 cm + 40 cm)/2 = 50 cm.
4. Phase Shift: The phase shift represents any horizontal displacement of the curve. Since the stopwatch is started at 0.3 seconds and the highpoint is reached at that time, there is no phase shift. Thus, the phase shift is 0.
Putting all of this information together, the equation of the sinusoidal curve is:
y = amplitude * sin(2π/period * (x - phase shift)) + vertical shift
Substituting the values we found:
y = 10 cm * sin(2π/3.6 s * (x - 0)) + 50 cm
Simplifying further:
y = 10sin(1.74533x) + 50
Therefore, the equation of the sinusoidal curve is y = 10sin(1.74533x) + 50.