Let me first list all of the possibilities I considered that I later rejected. This is far from exhaustive, and I'm looking forward to seeing other people's creativity.
Bad Ideas
- Sit on a tire swing with the fan pointing to the side. Point the fan up, measure speed of rotation of the system on the tire swing.
- Get a laser or collimated flashlight. Point it into the fan at a single point. On a wall on the other side of the fan, you will have a dot of light that is flashing, measuring the rate of the flashing could be an easier problem.
- Attach springs to the tip of all the blades. Observe how far out the acceleration deforms them, apply Hooke's Law to find acceleration, convert to angular speed.
Now, I am almost sure that the experimenter doesn't have the tools to execute any of these methods very well. Not even one. I'll try to break this down as to why.
Firstly, do we know the moment of inertia of the fan? No. Do we know the moment of inertia of the tire swing with a person and a fan on it? No. Is it even constant? No. I'm not saying we can't figure these things out, but it's an absurdly inferior method that will get terrible data.
On to the laser method. How are we going to measure the flash rate of the laser? I thought endlessly about this problem. Generally, a reference would be good, or if you could use electronics you could nail down the speed almost exactly and very easily. But I don't think anything is available that will work.
Now, the spring idea.. where to begin? The measurement of the deformation length is error-prone. The weight of the springs themselves will affect the speed of the fan. What spring do you have with appropriate characteristics anyway?
My best proposal
I'm hoping you can take off the cover. If you can't take off the cover, I hope you can take off some part of it, so that you can get a protruding shaft. I think the best way to make this measurement with household stuff would be to:
- Get a part of the shaft isolated
- Tape or attach the end of a thin string to it
- Turn the fan on with a stopwatch
- Time how long it takes for the fan to completely wrap the entire string around the shaft with a stopwatch
- Count the number of times the string has wrapped around the shaft
There you go, you have a number of turns per some amount of time. Ideally you would use a very light string that offered little to no resistance when pulled, as in, have it loosely laid out on the floor. Fishing wire could possibly be very good. You will want the acceleration time to be small compared to the entire measured time.
Some other (not terrible) possibilities
It occurred to me that acoustic methods might have some merit here. Get something that the blades can smack against, like when you take a pencil and stick it in the fan. Open up Windows sound recorder (accessories -> entertainment -> sound recorder), or a program like Audacity. Use some sound editing program and zoom in really tightly on the sound. See if you can identify a periodic shape that corresponds to a single hit. Count the peaks over a given time frame. Once again, you have number of rotations (or 1/3rd rotations) per unit time. If you already have an educated guess as to the frequency, then identifying the acoustic pattern from individual hits might not be very bad, not to mention, there is a lot of design flexibility in this experiment and computers should have a sufficiently high sample rate.
I think the ideal would be some kind of visual timing mechanism like the OP suggests. I'd imagine that a mechanical reference could be of use. Like if you had another fan that you knew the speed of, you could place it in front of the unknown fan and adjust its speed until you saw some patterns that indicated they were in sync. Yes, I'm lacking a lot of what's required to do this effectively, but maybe someone else can offer better advice.
The Experiment
Half of the papers on my desk are blown away. I'm getting complaints about the wretched sound of pen on fanblades, and people in my office are not too happy with me right now. But this is all in the name of physics! I am editing to present my experimental results. I used the acoustic method to determine the speed of my fan.
Firstly, my experimental apparatus is the Galaxy 20 inch model 4733 fan. It has 5 blades. I can't find any shopping results for you, but maybe someone else can. Here is a pretty good quality demo of the Galaxy 20" fan on youtube. And this video specifically states they have the 4733 model that I'm using. Why do people upload youtube videos of these things?! Do you have to "unbox" every single thing you buy??
Ok, moving on. I'm using the Audicity program and the microphone from a Microsoft Lifechat headset.
The fan has 3 settings, plus 0 for off. Setting 1 is the slowest and setting 3 is the highest. It produces quite a good breeze and has served me well. To start off, I'll share a waveform I recorded with it on setting 3 and setting 1 with me doing nothing else to it.
As you can see, this is not too useful. It makes a sound, but there's no way to distinguish peaks. Maybe it has a frequency that reflects the speed of the fan, but I can't be sure (and I haven't had much luck with the spectrum visualizations). You can see how the sound it makes is different between the two, and the 3rd setting is obviously louder, and the frequency is obviously different, but we want actual numbers.
So I put a plastic pen in it (the butt of the pen). Now, you might not want to try this at home (like I just did), but I kind of had to play with the angle to get it to not miss blades. It's very easy for it to jump and miss one, which would mess up the count. I had to press kind of hard and it was rather loud. But I got results. Here are the waveforms for 0.5 seconds, and my markup in order to count the "hits". I also provide the actual count in the image.
You can check my work for the count itself. I'm also happy to upload some mp3 files, but I'm not sure where I'll host them right now. The above image was made with the high-tech research software MS Paint. I'll give answer denoted $rpm_i$, where $i$ is the number of the setting, and the number of hits above will be denoted $hits$. I'll take the error in each hit count to be $\pm 1$ hit. The formula and reported results are as follows. Remember, it has 5 blades.
$$rpm_i = \left( hits_i \pm 1 \right) \times \frac{turn}{5 hits} \times \frac{1}{0.5 s} \times \frac{60 s}{min}$$
$$rpm_1 = 456 \pm 24 rpm$$
$$rpm_2 = 624 \pm 24 rpm$$
$$rpm_3 = 864 \pm 24 rpm$$
Power Consumption (addendum)
I used my KILL A WATT device to record power consumption for all the different speed settings. I'll denote this with $P_i$ but I need to explain a little about the difficulty in making this measurement. I believe my KILL A WATT to be fairly reliable and it gives stable power measurements for devices with constant power consumption. This is not quite true for the fan. When I first turn on the fan it consumes more power than after I leave it running for some time. The largest swing I observed was $50.5 W$ at max and $45.3 W$ at minimum for setting 1. This gives you another possible source of error. Since the experiment was performed with the fan on for a good while I'll report the lowest readings I have.
$$P_0 = 0 W$$
$$P_1 = 45.3 W$$
$$P_2 = 65.1 W$$
$$P_3 = 97.1 W$$
Now, I want to take just one quick second to apply the physics concepts of friction here. Dynamic friction between two solid bodies is often taken as a constant, and fluid friction is often taken as a power law, as in, $v^n$ where $n$ is most commonly from 1 (fully laminar) to 2. We can apply that here! I converted the previous speeds to rad/s and plotted the speed versus power consumption for all 3 "on" settings. Then, I guessed a certain offset and subtracted this value from the power consumption and applied a power fit to what was left. I realized after I did this that a constant power consumption does not correlate to a constant force (that would be linear), so I'm really just assuming some base power consumption for the device, and this yields a more perfect power fit for what's left just due to the mechanics of the motor. I found that I could get a better power fit by making this subtraction. The power fit had $R^2=1.0000$ for a constant offset from $8 W$ all the way to $13 W$ so I took the middle ground of $10.5 W$, which is to say, I made an educated guess that a constant power loss accounts for $10.5 W$ out of the consumed power. The power fit follows a satisfying $1.5$ power law, which is about what I expect for fluid friction.
$$P_i = 10.5 W + 0.1343 \omega_i^{1.4374} W$$
Lastly, I want to report the general intensity of the sound in my mp3 files. I need to put a disclaimer with this that it might not be accurate in any physical sense. I would want to ask an audio engineer about this issue - I don't know if the dB of an audio file represents the physical dB of the sound at the point the measurement was made. My guess is that this will depend on the recording device. Anyway, I want to give a dB measurement (I'll denote $A$) for the average peak for the pen hits on the fan blade, as per the audio file.
$$A_1 = -8 dB$$
$$A_2 = 0 dB$$
$$A_3 = -3 dB$$
Did I have the microphone in different locations when I took these measurements? Yes, I did. If I had to guess, I would say that I had it about 3-4 inches away from the pen contact point when I did setting 1 & 2 and closer to 7-8 inches when I did the 3rd setting. Obviously the 3rd setting was louder to my ear, and I had to set the headset down when taking that measurement because holding both was getting difficult.
I offer this data because with my guesstimates you could potentially calculate the energy released in the sound wave on a hit (assuming the dB measure is a 'real' measurement). Then with a conversion efficiency (from mechanical to sound), estimate the energy dissipated in a hit. You could also take some generic values for fan motor efficiency to relate power consumption to friction forces. You could then use a tailored mathematical form for power consumption (like above) for the friction losses and apply the energy dissipation rate from the pen hits and estimate the speed loss due to the pen. It's just something good to keep in mind for future experiments, so that you can show that the process of measuring isn't affecting what is being measured too much. With my 20" fan I don't think it matters too much, but repeating the experiment on a smaller fan could benefit from these calculations, and in order to do so you should have the microphone located in a fixed position for all measurements (unlike what I did).
Comments
It's possible that the pen contact was slowing down the fan some. In fact, this is almost surely the case, but this is a rather large fan. It is also an old fan. I would expect these speeds to be less than someone with a newer one. I've taken power measurements that could be used for some other investigation if desired. One use would be to guesstimate the impact the application of the pen on the blades has on the fan speed. I have already taken a shot at putting together a picture of where the friction comes from and developed a formula for power consumption as a function of speed based on a breakdown between static and fluid friction.
