Ceiling Fan - A ceiling fan of radius 0.60 m begins at rest and is then spun up

Question

Ceiling Fan - A ceiling fan of radius 0.60 m begins at rest and is then spun up to a high speed by applying a constant angular accleration of 0.25 rad/s6^2 for 20 s. The fan the revolves at a constant rate after the 20 s mark.

Part A - How long will it take the fan to complete its first full revolution?

Part B - What is the final angular velocity of the fan?

Part C - What are the final velocity and acceleration of a point at the tip of the fan's blades?

Part D - When the fan is revolving at a constant rate (after 20 s), how long will it take to comple a full revolution?

Explanation / Answer

s = ut + 0.5 * at^2

since fan begins from rest so u = 0

s for first revolution = 2 * pi

ao,

2 * pi = 0 + 0.5 * 0.25 * t^2

time it'll take to complete its first revolution = 7.0898 sec

v = u + at

v = 0 + 0.25 * 20

v = 5 rad/sec

final angular velocity of fan = 5 rad/sec

linear velocity = angular velocity * radius

linear velocity = 5 * 0.6

final velocity at the tip of the fan = 3 m/s

acceleration at tip of fan = v^2 / r

acceleration at tip of fan = 3^2 / 0.6

acceleration at tip of fan = 15 m/s^2

speed = distance / time

5 = 2 * pi / time

time = 1.2566 sec

time it'll take to complete one revolution at constant rate = 1.2566 sec

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