A 20m diameter Ferris wheel rotates at an angular speed of π/30 radians per second. The time for one full revolution is calculated to be 60 seconds. Cosine and sine functions of the x-coordinate in terms of time are determined, with the cosine function being 10cos(πt/30 + 3π/2) and the sine function being 10sin(πt/30). A graph of these functions over time is drawn.

1. JennyGu
11779148
Physics 101 Learning Object
Consider a Ferris wheel with a diameter of 20m. Imagine that is plotted on an axis, with the
centre of the wheel at the origin (see diagram below). The angular speed of the Ferris wheel is
𝜋
30
sec-1.
a. Define angular speed in your own words in relation to a Ferris wheel.
b. How long does it take the Ferris wheel to make one full revolution?
c. Determine a cosine function of the x-coordinate in terms of time, assuming the starting
point is at the lowest point of the Ferris wheel.
d. Determine a sine function of the x-coordinate in terms of time, assuming the starting
point is at the lowest point of the Ferris wheel.
e. Draw the sine/cosine function of the x-coordinate in terms of time.
Solution:
a. The angular speed (ω) is the rate at which the angle between an initial position and final
position is increasing along the path of the Ferris wheel (i.e. the edge of a circle), with
the vertex of the angle at the centre of the wheel. In this case, the angle is increasing by
a rate of π radians per 30 seconds, or approximately 0.105 radians per second.

2. JennyGu
11779148
b. Time for one revolution: T =
2𝜋
𝜔
T =
2𝜋
𝜋
60
= 60 sec.
Another way to approach this problem is to think about it logically, if it takes 30 seconds
for a compartment on the Ferris wheel to move with a change in angle by π radians, or
half a circle, it would take double the time, 60 seconds, to have a change in angle by 2π
radians, or a complete circle.
c. Radius (R) = 10m
Angular speed (ω) =
𝜋
30
sec-1
Starting angle (φ) =
3𝜋
2
x(t) = Rcos(ωt+φ)
x(t) = 10cos(
𝜋
30
t +
3𝜋
2
)
d. Because the starting angle is
3𝜋
2
, the cosine graph has been shifted to look like a sine
graph with x(0) = 0. Therefore, x(t) can also be written as:
x(t) = 10sin(
𝜋
30
t)
e.
x-axis = time
y-axis = x-coordinate
Because time cannot be negative (unless it is made relative to a certain initial time), all
values of x(t) for t<0 can be disregarded.