A ceiling fan with 80-cm-diameter blades is turning at 60 rpm. Su...

Question

A ceiling fan with 80-cm-diameter blades is turning at 60 rpm. Suppose the fan coasts to a stop 25 s after being turned off. (b) Through how many revolutions does the fan turn while stopping?

Accepted Answer

Hi, everyone. In this practice problem, we are being asked to calculate the number of revolutions that the wheel will make before it actually comes to a stop. We will have a grinding wheel of a radius of eight centimeter rotating clockwise at a constant speed of 650 R pm. Due to power outage the wheel completely stops turning in 12 seconds. And we're being asked to calculate the total number of revolutions that it will take for the wheel to actually finally comes to a stop. The options given are a 65 revolutions. B 74 revolutions, C 408 revolutions and D 76 revolutions. So in order for us to actually solve this problem, we need to actually calculate the angular displacement, which is going to be theta F minus theta I. So this is what we're interested in calculating. But in order for us to actually calculate this, what we wanna do is to actually use the kinematic equation which is going to be theta F equals half alpha T squared plus omega I T plus theta I. This is the kinematic equation that we are going to use, which we are going to finally rearrange to be theta F minus theta I equals half alpha T squared plus omega I T. However, from the problem statement, we still do not know a bunch of different things. The first one is going to be the alpha and also the Omega I right now is still in 650 R PM. So we wanna do, we wanna convert that into radiant per second. So what we need to find first is the alpha or essentially the angular acceleration. So we will assume that the wheel actually accelerates at a constant rate. And the rate of change of angular velocity of a particle rotating at an angle in a circular orbit is going to be called the angular acceleration or alpha. So essentially alpha will be equals to omega two minus omega one, all of that divided by delta T. So what we are given in the following statement is that our Omega two is going to be zero radiance Per seconds because um it comes to a complete stop after 12 seconds. But our Omega I or Omega one is going to be 650 R pm, which I am going to change that into or convert that into radiant per seconds just so that it's easier for us to actually use in the following calculation. So 6 50 R PM equals 6 50 revolution per minute. I want to multiply that by two pi radium per one revolution And also change the minute in two seconds. So I'm gonna multiply that with one minute per, every 60 seconds. And they will give us an omega one of 68.1 radium per seconds. So now we have Omega two and omega one. We also know that the delta T or the amount of time that it takes for it to stop turning is 12 seconds. So we can actually substitute all of those information into our angular acceleration equation. So we wanna substitute that. So alpha is going to then equals to omega two minus omega one, all of that divided by delta T and omega two is going to be zero radiance per seconds. Omega one is going to be 68.1 radium per second Divided by Delta T which is going to be 12 seconds and they will give us an angular acceleration of negative 5.68 radiance per second squared just like. So, so looking at this kinematic equation that we um recalled previously, we can actually start inputting all of our known values into that equation. So looking back at the kinematic equation, we have theta F minus theta I equals half alpha P squared plus omega I or in this case, that will be omega one or the angular velocity before it actually starts stopping. So we wanna substitute all our uh known information. So half alpha alpha is going to be negative 5.68 radium per second squared, D squared is going to be 12 seconds squared plus Omega I is going to be Omega one And that will be 68. radiance per seconds. And the time is also going to be 12 seconds just like. So that will essentially give us the theta F minus theta I to be 408.24 radiance. And what we are interested at is going to be the number of revolutions. So we want to convert this radiance into a revolution. So we know that one revolution will be two pi radiant. So in order for us to convert the radium into revolution, we want to multiply that by one revolutions and divide it by two pi radiant and they will give us theta F minus theta I value to be 65 revolutions, which is going to be the answer to this particular practice problem. So it will take the wheel 65 revolutions from 650 R PM for it to actually come to a full stop or a complete stop. So the wheel will turn 65 revolutions which will correspond to option A in our problem statement. So answer A is going to be the answer to this particular practice problem. And that will be all for this particular video. If you guys still have any sort of confusion on on this particular topic, please make sure to check out our other lesson videos and that'll be all for this one. Thank you.

A ceiling fan with 80-cm-diameter blades is turning at 60 rpm. Suppose the fan coasts to a stop 25 s after being turned off. (b) Through how many revolutions does the fan turn while stopping?

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