Question

A fan is designed to last for a certain time before it will have to be replaced (planned obsolescence). The fan only has one speed (at a maximum of 675 rpm), and it reaches the speed in 4.0 s (starting from rest). It takes the fan 8.0 s for the blade to stop once it is turned off. The manufacturer specifies that the fan will operate up to 1 billion rotations. Andre lives in a hot climate, works outside of the home from approximately 8:00 am to 5:00 pm, Monday through Friday, does not own an air conditioner, and can't sleep with the fan running. Estimate how many hot days ?hot Andre will be able to use the fan, rounded to the nearest day.

Answer 1

given
wo = 0 rad/s
w = 675 rpm

= 675*2*pi/60

= 70.7 rad/s

angular angular acceleration during speeding up, alfa1 = (w -wo)/t

= (70.7 - 0)/4

= 17.675 rad/s^2

angular displacement during speeding up,
theta1 = (w^2 - wo^2)/(2*alfa1)

= (70.7^2 - 0^2)/(2*17.675)

= 141.4 radians

t2 = 9 hours - 12 s

= 9*60*60 - 12

= 32388 s

angular displacement while rotates at maximum speed,

theta2 = w*t2

= 70.7*32388

= 2289832 radians

angular acceleration while slowing down, alfa3 = (0^2 - 70.7)/8

= -8.8375 rad/s^2

angular displacement during slow down, theta3 = w*t3 + (1/2)*alfa*t3^2

= 70.7*8 + (1/2)*(-8.8375)*8^2

= 282.8 radians

angular displacement of fan in one hot day, theta = that1 + theta2 + theta3

= 141.4 + 2289832 + 282.8

= 2290256 radians

= 2290256/(2*pi)

= 364505 revolutions

no of hot days the fan can used, N_hot = 1*10^9/364505

= 2743 days <<<<<<<<-------------Answer