A ceiling fan is spinning at 45 revolutions per minute when it is switched to a higher speed. It accelerates uniformly, and 2.0 seconds later it is spinning at 110 revolutions per minute. Through how many radians did it rotate during the transition of speeds

θ = 16.2 rad

Explanation:

First we find the angular acceleration by using first equation of motion in angular form:

ωf = ωi + αt

where,

ωf =final angular speed = (110 rev/min)(2π rad/1 rev)(1 min/60 s) = 11.5 rad/s

ωi =initial angular speed = (45 rev/min)(2π rad/1 rev)(1 min/60 s) = 4.7 rad/s

α = angular acceleration = ?

t = time = 2 s

Therefore,

11.5 rad/s = 4.7 rad/s + α(2 s)

α = (6.8 rad/s)/(2 s)

α = 3.4 rad/s²

Now, we use 2nd equation of motion:

θ = ωi t + (1/2)αt²

where,

θ = rotation = ?

Therefore,

θ = (4.7 rad/s)(2 s) + (1/2)(3.4 rad/s²)(2 s)²

θ = 9.4 rad + 6.8 rad

θ = 16.2 rad

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