What is the speed of the tip of a blade 10 secs after the fan is turned off?

Question

A ceiling fan with 83-cm-diameter blades is turning at 64 rpm. Suppose the fan coasts to a stop 28 sec after being turned off.

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

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

Answer 1

Assuming the fan slows down in a linear fashion, the deceleration of the fan is 64rpm / 28s or 2.28571 rpm/s.

Knowing this, we can determine the rotational speed of the fan after 10 seconds:

64 rpm - 10s * (2.28571rpm/s) = 41.14286 rpm

To convert this to a tip speed, we need to convert rpm to rad/s.  This is done by knowing that 1 rev = 2*pi rads and also 1 min = 60 seconds.

41.14286 rev/min * 2*pi rads/rev *(1min/60s) = 4.30847 rad/s

The tip speed is equal to the rotational speed multiplied by the radius, so that would give us 4.30847 rad/s * 83/2 cm = 178.8 cm/s or 1.788 m/s.

To determine the total revolutions, you will have to integrate the deceleration equation given above for time interval given:

Note 64 rpm = 1.06667 rev/s and 2.28571 rpm/s = 0.03810 rev/s^2 (by dividing both by 60s/min).  This is done so the units match as we will be multiplying this by seconds in the next step.

#revs=integral from t=0s to t=28s of (1.06667 - (0.03810) t )dt.  This yields the result of [1.06667 t - 0.5 * 0.03810 * t^2] evaluated at t=0 and t=28.  Subbing in the boundary conditions we get 1.06667(28)-0.5*0.03810*(28^2) = 14.93333 revs in 28 seconds.

(Note: the integration could be skipped by knowing the formula d = Vit+0.5at^2... which was the result of the integration)

Answer 2

The angular acceleration is negative because it goes from a high rpm to a low rpm.

Converting to radians/sec the intiial angular speed is found as follows;

64 rev/min X 2*pi radians/1 rev X 1 min/60 sec= 6.70 rad/sec

angular acceleration= change in ang speed/change in time

                                    =( 0 - 6.70)/28= 0.239 rad/s^2

the angular speed after 10 seconds = 6.70-0.239t

                                                               = 6.70-.239(10)

                                                                = 4.31 rad/s

regular velocity = angular speed*r

                           = 4.31(0.83/2) = 1.787 m/s

note:I converted the diameter to meters

                                                              

Number of revolutions = 1/2 angular accel *t^2

                                       =.239/2*28^2= 93.88 radians

                                      

93.88 radians* 1 rev/(2*pi rad)= 14.9 rev

Answer 3

Whoever made up that question is apparently assuming constant deceleration, which is pretty unlikely in the real world of ceiling fans. They slow down mainly because of torque due to fluid resistance and induced current (because the motor acts as a generator). Both of those torques are roughly proportional to speed, so the motion decreases exponentially. If not for sliding friction, which is constant, the fan would never stop completely. To get a real-world correct answer, you would have to know much more than what is given in the question.

However, you teacher probably wants you to play the game and assume constant deceleration. In that case, the speed after 10 sec would be 18/28 x 64 = 41 rpm; the tip speed is 41 rpm x 2π x 83 cm = 21,382 cm/min =356.4 cm/sec.

The average speed while stopping is half of the initial speed, so