Question

ONLY NEED ANSWER TO PART (A + B)

Some amusement parks have a double Ferris wheel, which consists of two vertically rotating wheels that are attached to each other by a bar that also rotates. There are eight gondolas equally spaced on each wheel. Riders experience a combination of two circular motions that provide a sensation more thrilling than the classic single Ferris wheel. In particular, riders experience the greatest sensation when their rate of change in height is the greatest. • Each of the two wheels is 6 m in diameter and revolves every 12 s. • The rotating bar is 9 m long. The ends of the bar are attached to the centres of the wheels. • The height from the ground to the centre of the bar is 8 m. The bar makes a complete revolution every 20 s. • A rider starts seated at the lowest position and moves counterclockwise. • The bar starts in the vertical position. Consider the height of a rider who begins the ride in the lowest car. a) Write a function f(t) that expresses the height of the rider relative to the centre of the wheel at time t seconds after the ride starts. Write a second function g(t) that expresses the position of the end of the bar (the centre of the rider's wheel) relative to the ground at time t seconds. b) Explain how the sum of these two functions gives the rider's height above the ground after t seconds. c) Use technology to graph the two functions and their sum for a 2-min ride. d) What is the maximum height reached by the rider? When does this occur? e) What is the maximum vertical speed of the rider? When does this occur? f) Design your own double Ferris wheel. Determine the position function for a rider on your wheel. What is the maximum speed experienced by your riders? Is there a simple relationship between the dimensions of the Ferris wheel and the maximum heights or speeds experienced?

Answer 1

P = at t= 6s at t as Now Ist consider the ferris wheel. In this wheel rider starts at the lowest point. In this point height of the rider with respect to wheel center is aR where R= Radius of the wheel. So, at t=0, h =-R at t=3s, h = or If at t=3s R at t=65 h=R pattoos at t=95 h=OR at t=125 h=-R again h = - Ros wt where w = 211 2 = 7/8 or h=-R68 (t) = -363(2 t) = f(t) now the bar starts in the vertical position at t=0 So at to the height of the center of wheel with Respect to ground ground g(t) = (8-4.5)m. at t-o om at t=55 (8+4.5 m at t=105 = 8m at t= 155 so 4.5m 4.5m 87 [: 20s is the time period (8-45) at t= 20s again So, 8(t) - 8-4.5Cos (w't) = 18-4.565 (11 So, g(t) is the position of the wheel center from ground with respect to time and f(t) is the position of a rider from center of wheel with respect to time. So sum of this two function will give the position of vider to from ground at a time z. F(t) = g(t) + f(t) = 8-4.5 CM (7+) - 363(4+)