A fan with a diameter of 10cm is spinning at 25 revolutions per minute clockwise. The fan's reference blade starts pointing directly upwards. Determine an equation for a sinusoidal function that represents the fan's reference blade's vertical distance from the center (a negative value being accepted). State the period, amplitude and vertical shift for this function.
I need help please. I have to finish this question.
since the curve starts at the top when t=0,
y = a cos(kt)
the diameter is 10, so the radius is 5:
y = 5 cos(kt)
The period is 1/25 min, so 2pi/k = 1/25, making k = 2pi/25
y = 5 cos(2pi/25 t)
Not sure what you mean by the blade's distance from the center. I think you meant the tip of the blade.
To determine an equation for a sinusoidal function that represents the fan's reference blade's vertical distance from the center, we need to consider the properties of a sinusoidal function: amplitude, period, and vertical shift.
1. Amplitude: The amplitude refers to the maximum distance from the center that the reference blade reaches. In this case, the fan's blades create a circular motion, so the maximum vertical distance from the center is equal to the radius of the fan, which is half its diameter. Therefore, the amplitude is 10cm/2 = 5cm.
2. Period: The period of a sinusoidal function is the length of one complete cycle. In this case, since the fan completes one full rotation in 60 seconds (25 revolutions per minute), the period is 60 seconds.
3. Vertical Shift: The vertical shift represents any vertical displacement of the sinusoidal function. Since the reference blade starts pointing directly upwards when the fan is at its initial position, there is no vertical shift. Therefore, the vertical shift is 0.
Now, let's define the equation for the sinusoidal function representing the fan's reference blade's vertical distance from the center. We'll use the general form of a sinusoidal function:
y = A * sin(B * (x - C)) + D
where A represents the amplitude, B represents the frequency (2π divided by the period), C represents a horizontal shift, and D represents a vertical shift.
In this case, the equation becomes:
y = 5 * sin((2π / 60) * x)
The equation above represents the fan's reference blade's vertical distance from the center over time. The amplitude is 5cm, the period is 60 seconds, and there is no vertical shift.
Note: The equation assumes that x represents time in seconds, but it's important to consider the units you are using and adjust the equation accordingly.