To derive a sine function that represents the vertical position $V$ of a rider on the Ferris wheel relative to the ground level over a complete revolution, starting from position $B$, let’s analyze the given details.

Given Information

1. Diameter of the Ferris Wheel: 18 meters, so the radius $R = 9$ meters.
2. Period of Rotation: The Ferris wheel completes one revolution every 40 seconds.
3. Height of the Wheel’s Center Above Ground: 12 meters.
4. Starting Position $B$: At position $B$, we assume the rider is at the leftmost horizontal point, which would correspond to the phase where the vertical position is at the midpoint of its oscillation (since it's horizontally aligned with the center).

Sine Function Form

A sine function representing the vertical position of a point on a Ferris wheel can be expressed as: $V(t) = A \sin\left(\omega t + \phi\right) + D$ where:

• $A$ is the amplitude (equal to the radius of the wheel, $R$).
• $\omega$ is the angular frequency.
• $\phi$ is the phase shift.
• $D$ is the vertical shift.

Step-by-Step Derivation

1. Amplitude $A$:

   • The amplitude is the radius of the Ferris wheel, $A = 9$ meters.

2. Angular Frequency $\omega$:

   • Since one revolution takes 40 seconds, the angular frequency $\omega$ is given by: $\omega = \frac{2\pi}{40} = \frac{\pi}{20} \, \text{radians per second}$

3. Vertical Shift $D$:

   • The center of the wheel is 12 meters above the ground, so this is the baseline around which the rider oscillates. Thus, $D = 12$.

4. Phase Shift $\phi$:

   • Since the rider starts at position $B$, which is aligned horizontally with the center of the Ferris wheel, this corresponds to the phase where the sine function equals zero. In the sine function, this position corresponds to the point where $\sin(\omega t + \phi) = 0$. Therefore, $\phi = -\frac{\pi}{2}$ to ensure the function starts at zero.

Final Sine Function

Putting it all together, the function for the vertical position $V(t)$ of the rider relative to the ground over time $t$ (in seconds) is: $V(t) = 9 \sin\left(\frac{\pi}{20} t - \frac{\pi}{2}\right) + 12$ This function represents the vertical position of the rider on the Ferris wheel, starting from position $B$.

Would you like further explanation on any part of the derivation, or assistance with other parts of the question?

Related Questions

1. How would the equation change if the Ferris wheel had a different diameter?
2. What would the function look like if the wheel completed one revolution every 60 seconds instead?
3. How can we determine the maximum and minimum heights reached by the rider using this function?
4. If a rider starts at the lowest point (position $A$), what would the phase shift be?
5. How does the value of $\omega$ affect the speed of rotation in the sine function?

Tip

For trigonometric functions representing circular motion, remember that the amplitude represents the radius, the vertical shift represents the distance from the ground to the center, and the period of the function corresponds to the time taken for a full revolution.