# Find angle in radians on a Ferris Wheel

John has been hired to design an exciting carnival ride. Tiff, the carnival owner, has decided to create the world's greatest ferris wheel. Tiff isn't into math; she simply has a vision and has told John these constraints on her dream: (i) the wheel should rotate counterclockwise with an angular speed of a = 15 RPM; (ii) the linear speed of a rider should be 250 mph; (iii) the lowest point on the ride should be c = 4 feet above the level ground.

(b)Once the wheel is built, John suggests that Tiff should take the first ride. The wheel starts turning when Tiff is at the location P, which makes an angle θ with the horizontal, as pictured. It takes her 1.4 seconds to reach the top of the ride. Find the angle θ (radians).

So for this part I tool 1.4/(60/15) and got 0.35 of a revolutions. Then I have no idea where to go from here.

(c) Poor engineering causes Tiff's seat to fly off in 6 seconds. Describe where Tiff is located (an angle description) the instant she becomes a human missile.

I would assume for this one it would be just like b but using 6 sec. Again I just dont know what direction to go.

• Where is the picture? Anyway, once you've found the 0.35 revolutions (that's correct) that the wheel makes, you need to convert that to radian measure. 1 full revolution is $2\pi$ radians. That's the total angle the wheel turns through in a counterclockwise rotation to get to the top. To get the angle theta, I think you have to subtract off the right angle $\frac{\pi}{2}$ if the "picture in my head" is correct. But you really should show the picture to be complete. – Deepak May 21 '14 at 0:54
• I added a picture – Sondra May 21 '14 at 1:00

Good that you added the picture. To get $\theta$, work out:

$$(0.35)(2\pi) - \frac{\pi}{2}$$

since you need to subtract off the right angle as I mentioned in my comment.

For the next part, you need to figure out her angular position at 6s. You know that the wheel makes 0.25 (a quarter) revolution per second. So in 6 seconds, it makes 6*0.25 = 1.5 revolutions.

1.5 = 1 + 0.5, and 1 full revolution brings her back to her starting position. Another half revolution will take her to a diametrically opposed position. Just work out the angular position here, and for clarity, illustrate your answer with a diagram.

• so the answer would 3 pi? – Sondra May 21 '14 at 1:16
• The answer to neither part is $3\pi$. Can you show your working? – Deepak May 21 '14 at 1:17
• so you take 1.5 rev and multiply it by 2pi to get radians. Is there another step? – Sondra May 21 '14 at 1:21
• You're asked to find her final position. She travels 1.5revs before flying off. 1 rev brings her back to exactly where she started, so discount that. You only need to count the other half rev (0.5rev). That brings her to a point diametrically opposed to where she started from. Let O be the centre of the circular wheel. Extend the radial line PO past the centre until it intersects the circumference on the other side. Call that point P'. That denotes her final position. How would you describe it? She would still make an angle of $0.2\pi$ radians with the horizontal, but "above" it not below it. – Deepak May 21 '14 at 1:26