# Ferris Wheel Problem: Find r, ω, Times for 80ft Height

In summary, the conversation was about finding the radius and angular speed of a rotating Ferris wheel and using parametric equations to determine the rider's coordinates at different times. The problem also involved finding the time when the rider's height is 80ft, which can have multiple solutions due to the periodic nature of the sine function.

## Homework Statement

A Ferris wheel of radius 100 feet is rotating at a constant angular speed ω rad/sec counterclockwise. Using a stopwatch, the rider finds it takes 5 seconds to go from the lowest point on the ride to a point Q, which is level with the top of a 44 ft pole. Assume the lowest point of the ride is 3 feet above ground level.

Let Q(t)=(x(t),y(t)) be the coordinates of the rider at time t seconds; i.e., the parametric equations. Assuming the rider begins at the lowest point on the wheel, then the parametric equations will have the form: $$x(t)=rcos(ωt-π/2)$$ and $$y(t)=rsin(ωt -π/2)$$, where r,ω can be determined from the information given. Provide answers below accurate to 3 decimal places. (Note: We have imposed a coordinate system so that the center of the ferris wheel is the origin. There are other ways to impose coordinates, leading to different parametric equations.)

Find r, find ω. During the first revolution find the times when the riders height is 80ft.

## Homework Equations

$$x(t)=rcos(ωt-π/2)$$
$$y(t)=rsin(ωt -π/2)$$

## The Attempt at a Solution

The radius is obviously already given to me. To find ω I drew a triangle with vertices at the origin, Q, and at (0, 44). The hypotenuse was 100 ft, and the adjacent side to the angle I was trying to find was (103-44) or 59 ft. To find the angle I need I found the inverse cosine of (59/100) and divided it by 5. The angular speed rounded to 3 decimal places is .188

To find the times when the rider's height is 80ft is where I'm having problems. According to the problem the origin is the center of the wheel does that mean I set y=(80-103)?

does that mean I set y=(80-103)?
Looks right to me.

Thank you, turns out I was on the right track. Here's what I did.

$$-23=100sin(.1879474972t-π/2)$$
$$-.23=sin.1879474972t-π/2$$
$$-.2320776829=.1879474972t-π/2$$
$$t=7.123$$

Now the problem is I'm not sure how to find the second time.

When you invert the sine function you get, in principle, infinitely many solutions. What are all the values of arcsin(sin(x))?

All I knew is this.
$$arcsin(sin.1879474972t-π/2)=1879474972t-π/2$$

Did I do something wrong or what should I do?

The sine function is periodic, i.e. as x increases y=sin(x) keeps taking the same set of values again and again. Therefore 'the angle whose sine is y' has many possible values. The arcsin function is defined, by convention, to take the value between -pi/2 and +pi/2, but there is an infinity of other x values.
Draw a circle. Starting at the bottom, go around it clockwise until you have gone through some angle theta < pi. Draw a horizontal line through that point. Now continue around until you hit that line again on the right hand half of the circle. What angle have you gone through now (in total)?

## 1. What is the Ferris wheel problem?

The Ferris wheel problem involves finding the radius (r) and angular velocity (ω) of a Ferris wheel, as well as determining the times at which the wheel reaches a height of 80 feet.

## 2. Why is this problem important?

This problem is important because it allows us to calculate the motion of a Ferris wheel, which can have practical applications in designing and operating amusement park rides. It also helps us understand circular motion and its mathematical representation.

## 3. How do you find the values of r and ω?

To find the values of r and ω, we can use the formula h = r + rcos(ωt), where h is the height of the wheel at time t. By setting h = 80 feet and solving for r, we can find the radius of the wheel. Then, by using the formula ω = 2π/T, where T is the period of the wheel, we can find the angular velocity.

## 4. How do you determine the times at which the wheel reaches a height of 80 feet?

To determine the times at which the wheel reaches a height of 80 feet, we can use the formula t = (1/ω)arccos((h-r)/r), where h is the height of the wheel and r is the radius. By plugging in the values for h and r, we can calculate the times at which the wheel reaches 80 feet.

## 5. Can this problem be applied to other circular motion scenarios?

Yes, this problem can be applied to other circular motion scenarios. The same formulas and methods can be used to find the radius, angular velocity, and times for any circular motion where the height is known at a certain time. This includes amusement park rides, rotating objects, and celestial bodies in orbit.

Replies
1
Views
3K
Replies
2
Views
3K
Replies
3
Views
1K
Replies
6
Views
2K
Replies
1
Views
2K
Replies
9
Views
13K
Replies
1
Views
6K
Replies
4
Views
21K
Replies
2
Views
23K
Replies
3
Views
6K