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 200 mph; (iii) the lowest point on the ride should be c = 3 feet above the level ground.

a) R = 186.742 ft

b) θ = 0.7854 rads

c) θ = 14.14 rads

Explanation:

Given:

- The linear tangential velocity v = 200 mph

- The angular frequency a = 15 RPM

- The distance of bottom point from ground c = 3 ft

Find:

- The radius R of the ferris wheel

-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 1.5 seconds to reach the  top of the ride. Find the angle θ. (Round your answer to two decimal places.)

-  Poor engineering causes Tiff's seat to fly off in 9 seconds. Describe where Tiff is located (an angle description)  the instant she becomes a human missile. (Give your answer as an angle measured counterclockwise from her starting point P.)

Solution:

- The relationship between linear v and rotational speed w is given by:

                                    v = R*w

                                    200*5280 / 60 = R*15*2*π

                                    R = 200*5280 / 60*15*2*π

                                    R = 186.742 ft

- the angle swept from point P to angle θ above horizontal is given in terms of angular speed:

                                    π/2 + θ = a*2π*1.5 / 60

                                    θ = 15*2π*1.5 / 60 - π/2

                                    θ = 0.7854 rads

- Similarly calculate the angle θ swept entirely till 9 seconds

                                    θ = a*2π*9 / 60

                                    θ = 15*2π*9 / 60

                                    θ = 14.14 rads

- The total angle swept from initial position is θ = 14.14.

b. a lightning bug, plug of grass, and a worm