Ferris Wheel Loading Window and Safety Stop
A riverside observation wheel has 16 identical gondolas equally spaced around a circle. The center is 14 m above the boarding platform, and the wheel radius is 11 m. A gondola starts at the lowest point at t=0 and rotates counterclockwise at a constant angular speed, completing one turn every 48 seconds. Its height in metres above the platform is modeled by h(t)=14+11 sin((π/24)t−π/2). Riders board only when the gondola is at or below 4 m above the platform. A safety rule says the wheel may not stop with a gondola between 22 m and 25 m above the platform unless an operator is present. Determine the loading window, the first time the height reaches 22 m, and whether a proposed 10-second stop at t=18 s needs an operator.
1. Find the time intervals during the first 48 seconds when a gondola is at or below 4 m, so riders may board.
Read the worked answer
Set 14+11 sin((π/24)t−π/2)≤4. Then 11 sin(...)≤−10, so sin(...)≤−10/11.
Let α=arcsin(10/11)≈1.141 rad, and let θ=(π/24)t−π/2. In the first revolution, θ runs from −π/2 at t=0 to 3π/2 at t=48. The inequality sin θ≤−10/11 holds on [−π/2,−α] and on [π+α,3π/2].
Converting back: the first interval is t from 0 to (π/2−α)·24/π≈(1.571−1.141)·7.639≈3.28 s. The second interval starts at (π+α+π/2)·24/π≈(4.712+1.141)·7.639≈44.71 s and ends at 48 s. Thus the loading windows are about 0 to 3.28 s and 44.71 to 48 s.
2. Find the first time after t=0 at which the gondola height is 22 m.
Read the worked answer
Solve 14+11 sin((π/24)t−π/2)=22, so sin((π/24)t−π/2)=8/11≈0.7273. The first angle after t=0 with this sine value is arcsin(8/11)≈0.8148 rad.
Thus (π/24)t−π/2≈0.8148, so (π/24)t≈0.8148+1.5708=2.3856. Then t≈2.3856·24/π≈18.22 s. The first time the gondola reaches 22 m is about 18.2 s.
3. At the proposed stop time t=18 s, determine the stopping height and decide whether an operator is required under the stated rule. Mention any margin-of-error concern.
Read the worked answer
At t=18 s, θ=(π/24)·18−π/2=3π/4−π/2=π/4. Therefore h=14+11 sin(π/4)=14+11·0.7071≈21.78 m.
The restricted band is 22 m to 25 m. Since 21.78 m is below 22 m, the proposed stop does not by itself require an operator. However, the margin is only about 0.22 m, so a small timing error or rounding in the control system could place the gondola inside the restricted band. The safe judgement is to verify the actual stopped height, not rely only on the nominal 18 s.