Question

5. A fan has three thin blades that spin to

produce a breeze. The diameter of the fan is
16 inches.
a. Determine the length of an arc between
two consecutive fan blades.
b. What is the area of each sector formed
by the radii passing through the center of
consecutive fan blades?
c. What is the angular velocity if the tip of
each blade in the fan moves 8m radians
in 2 seconds?

Answer 1

a) The length of an arc between two consecutive fan blades is 16.76 inches

b) The area of each sector is 67.02 inches²

c) The angular velocity is 4m rad/sec

Explanation:

* Lets explain how to solve the problem

- The fan has three thin blades that spin to produce a breeze

- The diameter of the fan is 16 inches

- The three blades divided the circle into three equal parts

- The circumference of the circle is 2πr

a)

∵ The diameter of the circle = 16 inches

∵ The radius of the circle is half the diameter

∴ The radius (r) = 1/2 × 16 = 8 inches

∵ The length of the circle = 2πr

∴ The length of the circle = 2π(8) = 16π

- The length of an arc between two consecutive fan blades is 1/3

the length of the circle

∴ The length of the arc = 1/3 × 16π = 16.76 inches

* The length of an arc between two consecutive fan blades is

16.76 inches

b)

- The area of a sector in the circle =
`(x)/(360)(\pi r^(2))`

where x is the central angle of the sector and r is the radius

of the circle

∵ The angle between each two consecutive blades = 360°/3

∴ x = 360°/3 = 120°

∵ r = 8 inches

∴ The area of each sector =
`(120)/(360)(\pi )(8^(2))=67.02`

* The area of each sector is 67.02 inches²

c)

∵ The angular velocity = Ф rad ÷ t, where Ф is the central angle

with radian measure and t is the time in seconds

∴ ω = Ф/t radian/second

∵ Ф = 8m radians

∵ t = 2 seconds

∴ ω = 8m ÷ 2 = 4m rad/sec

* The angular velocity is 4m rad/sec