Question

A ceiling fan with 90-cm-diameter blades is turning at 68 rpm . Suppose the fan coasts to a stop 29 s after being turned off.

part A) What is the speed of the tip of a blade 10 s after the fan is turned off?

part B) Through how many revolutions does the fan turn while stopping?

Answer 1

Diameter of the blades of the ceiling fan = D = 90 cm = 0.9 m

Radius of the blades of the ceiling fan = R = D/2 = 0.9/2 = 0.45 m

Initial rotational speed of the fan = N₁ = 68 rpm

Initial angular speed of the fan = ₁

₁ = 7.121 rad/s

Final angular speed of the fan = ₂ = 0 rad/s (Comes to a stop)

Time taken by the fan to come to a stop = T = 29 sec

Angular acceleration of the fan =

₂ = ₁ + T

0 = 7.121 + (29)

= -0.2456 rad/s²

Angular speed of the fan 10 sec after the fan is turned off = ₃

Time period = T₁ = 10 sec

₃ = ₁ + T₁

₃ = 7.121 + (-0.2456)(10)

₃ = 4.665 rad/s

Speed of the tip of a blade 10 sec after the fan is turned off = V₃

V₃ = ₃R

V₃ = (4.665)(0.45)

V₃ = 2.1 m/s

Angle through which the fan rotates while stopping =

= ₁T + T²/2

= (7.121)(29) + (-0.2456)(29)²/2

= 103.23 rad

Number of revolutions made by the fan while stopping = n

n = 16.43 rev

A) Speed of the tip of a blade 10 sec after the fan is turned off = 2.1 m/s

B) Number of revolutions the fan turns while stopping = 16.43 rev