Suppose that after you are loaded into a FerrisWheal car, the wheel beguins turning at 3 revs/min.The wheal had a diameter of 20m and the bottom of the seat of the wheel is 2m above the ground. Express the hight of the seat above the ground as a function of time (t seconds), after it begins turning.
The ferris wheel angle of the car from the lowest position is
A = 2 pi f t where f = 3/60 = 0.05 rev/second and t is the time it passes the lowest position
The height of the seat above the ground has a minimum value of 2 m.
At other times, the height h is
h = 2 + 20 (1 - cos 2 pi f t)
Period = 20 sec
2pi/k = 20, k = pi/10
amplitude = 10
so how about:
height = 10sin pi/10(t-5) + 12
check:
t=0, height = 2
t=5, height = 12
t=10, height = 22
t=15 height = 12
t= 20 height = 2 , checks out.
I assumed the radius was 20, but it was the diameter. 20 in my equation should be 10.
To solve this question, we need to understand the relationship between the height of the seat above the ground and the time it takes for the Ferris Wheel to complete a revolution.
First, let's determine the time it takes for one complete revolution. We know that the Ferris Wheel is turning at a rate of 3 revs/min, which means it completes 3 revolutions in one minute. Therefore, the time it takes for one complete revolution is 1/3 minutes.
Now, let's express the height of the seat above the ground as a function of time. To do this, we'll need to use a trigonometric function, specifically the sine function, since the height of the seat will vary sinusoidally as the Ferris Wheel rotates.
The height of the seat above the ground at any given time is given by the equation:
h(t) = A * sin(B * t + C) + D
In this equation, A represents the amplitude (the maximum height above the ground), B represents the frequency (which is related to the time it takes for one complete revolution), C represents the phase shift (which determines the starting position of the seat), and D represents the vertical shift (which is the average height of the seat above the ground).
In our case, the amplitude A is equal to the radius of the Ferris Wheel, which is half the diameter, so A = 20/2 = 10m. The frequency B is determined by the time it takes for one complete revolution, which is 1/3 minutes. However, we need to convert this to seconds, so B = (2π) / (1/3 * 60) ≈ 37.699 radians/second. The phase shift C is zero since we want to start at the bottom of the wheel. Finally, the vertical shift D is equal to the height of the seat when the wheel is at the bottom, which is 2m above the ground.
So, the formula for the height of the seat above the ground as a function of time (t seconds) becomes:
h(t) = 10 * sin(37.699 * t) + 2
This equation will give you the height of the seat above the ground at any given time after the Ferris Wheel begins turning.