gravitron question

Question

Chapter ProblemA popular ride at many theme parks is the Gravitron, which pins riders
to the walls of the ride as it spins and then
drops the ?oor to leave them suspended by
friction and centrifugal effects. Suppose that
such a ride is designed with a radius of 4 m
and a period of 2 s, with the origin at the
centre of rotation.

a)Model the x-position as a function of time,
t, using a sine function.
b)Model they-position as a function of time,t,
using a cosine function.
c)Show that x^2 + y^2 =16 for all values of t.
d)Explain why the relation in part c) is
always true.

Answer 1

In general, f(t) = A sin(t*2pi/P) where A is the amplitude and P is the period of the sine wave.  For this problem, A = 4 m and P = 2 s:

 

x(t) = 4 sin(pi*t)

 

y(t) = 4 cos(pi*t)

 

x² + y² = 4² sin²(pi*t) + 4² cos²(pi*t) = 16 (sin²(pi*t) + cos²(pi*t))= 16

 

x² + y² = 16 is the equation of a circle  centered at the origin with radius √16 = 4 m.  The riders travel along this circle during the ride.