Yes we/you can.
I recall seeing a famous video of a homemade version of the Cavendish torsion balance experiment from the early 1960's, made I think for the PSSC high school course. Basically, the physicist hung a torsion balance from a high ceiling by a long (>10 m?) piece of computer data tape (chosen because it would not stretch). He carefully minimized air currents. The torsion masses were two .5 kg bottles of water on a wooden bar (no magnetic interference). Mass, in the form of boxes of sand, say 20kg was piled around on the floor as static mass and then reversed in position with respect to the suspended masses. There was a clear plastic box around the balance (with a hole in its top for the suspending tape to pass through) also to minimize the effect of air currents, since the lateral force on each bottle is about G*m1*m2/r^2 = (6.7e-11)*0.5kg*20kg/(0.1m)^2 N ~ 6.7e-8 N, i.e. a lateral force on each bottle equivalent to that generated by a weight of about 7 micrograms, about that of a 1 mm^3 grain of sand. This is visible to us because the long arms of the torsion balance convert this small force into a torque on the suspending filament, and the restoring torque is itself very small.
I found an Italian dubbed version of the video on Youtube. See http://www.youtube.com/watch?v=uUGpF3h3RaM&feature=related and a slightly longer version at http://www.youtube.com/watch?v=V4hWMLjfe_M&feature=related. I believe the demonstrator was Prof. Jerrold Zacharias from MIT and the PSSC staff. If anyone can point me to the original undubbed black and white film loop, I'd appreciate it.
It looked really crude but qualitatively it worked. The mirror moved upon reversal of the mass positions. Yeah, experimental physics!! Calculate it out. Use your laser pointer. Glue mirrors. Calibrate. Give it as an experiment in class. Make a (music?) video. Put it on Youtube and embed it here. Social physics.
I also found some other do it your self experimenters with crude equipment, experimental tips (try fishing line) and different masses.
See http://funcall.blogspot.com/2009/04/lets-do-twist.html
http://www.hep.fsu.edu/~wahl/phy3802/expinfo/cavendish/cenco_grav.pdf
and http://www.fourmilab.ch/gravitation/foobar/, which uses a ladder, some cobblestones, monofilament fishing line and has videos. For the experiment in this last reference, you don't need mirrors, since you can see the balance masses move directly because their excursion is so large. See also http://www.youtube.com/watch?v=euvWU-4_B5Y
For all these experiments there is no calibration of the restoring force of the twisted filament (which Cavendish did from the free torsion period of the balance), the balance beam of one appears to be styrofoam, (so I would worry about subtle charge effects), and the beam hits the support of the fixed masses so that it bounces and we do not see the harmonic angular acceleration we might expect. This last problem is apparently well known to amateur experimenters in this field.
Another exposition and video is at http://www.juliantrubin.com/bigten/cavendishg.html
The best summary and historical exposition I found is at https://en.wikipedia.org/wiki/Torsion_bar_experiment . I did not realize that the experiment was originally designed by John Michell, a contemporary, whose designs and apparatus passed to Cavendish upon his death. See https://en.wikipedia.org/wiki/John_Michell. Newton had considered the deviation from vertical that a stationary pendulum would have near a terrestrial mountain in the Principia (1686). Although he considered the deviation too small to measure, it was measured 52 years later at Chimborazo, Ecuador in 1738, which was the first experiment showing that the Earth was not hollow, apparently a live hypothesis at the time. The same experiment was repeated in Scotland in 1774. See https://en.wikipedia.org/wiki/Schiehallion_experiment . Mitchell devised the torsion balance experiment in 1783, and started construction of a torsion balance. Cavendish did his experiment in 1797-1798. To me this is all quite inspiring.
Editorial (I'll move this positive rant to meta soon) - given the obviously widely varied audience on this site, I would very much like to see more questions like this one relating to amateur or home experiments. The analysis of the data and possible sources of errors in these experiments is often subtle, and is very instructive. To have real physicists and other clever students publicly criticize some aspect of an experiment provides something that many students may never get otherwise. The social network framework will help many newcomers from different countries learn what real science is in a way that yet another dose of imperfectly understood theory never will. And it's fun too.
I don't know if it qualify as home experiment, but you can use the internet to get access to thousands of kilometres of optical fibres for free. It allows you to measure a lower bound for the speed of light in the fibres, which is $c/n$, where $n$ is the refractive index of glass, typically around 1.5. This corresponds to $2\times 10^8 \text{m/s}$. Using ping, you measure a round trip time, that is it should correspond to 100 km/ms of round trip.
From Paris, I ping the website of Columbia, in New-York, I have
fred@sanduleak2:~$ ping www.columbia.edu
PING www.columbia.akadns.net (128.59.48.24) 56(84) bytes of data.
64 bytes from www-csm.cc.columbia.edu (128.59.48.24): icmp_req=1 ttl=113 time=125 ms
64 bytes from www-csm.cc.columbia.edu (128.59.48.24): icmp_req=2 ttl=113 time=116 ms
....
64 bytes from www-csm.cc.columbia.edu (128.59.48.24): icmp_req=16 ttl=113 time=112 ms
^C
--- www.columbia.akadns.net ping statistics ---
17 packets transmitted, 16 received, 5% packet loss, time 16023ms
rtt min/avg/max/mdev = 108.585/118.151/132.156/7.728 ms
The minimum round trip time is 108 ms, which would correspond to 10,800 km instead of 5839 km. Off by a factor of 2, but the correct order of magnitude, due to delays in switches etc., which is why we said this is a lower bound.
If one looks more precisely the trajectory of my packets to New York with tracepath
fred@sanduleak2:~$ tracepath www.columbia.edu
1: sanduleak2 0.266ms pmtu 1500
....
3: pioneer.ens-cachan.fr 1.072ms
....
6: vl172-orsay-rtr-021.noc.renater.fr 28.747ms asymm 9
7: te0-1-0-5-paris1-rtr-001.noc.renater.fr 20.931ms
8: renater.rt1.par.fr.geant2.net 30.307ms asymm 9
9: so-3-0-0.rt1.lon.uk.geant2.net 33.780ms asymm 10
10: so-2-0-0.rt1.ams.nl.geant2.net 36.570ms asymm 11
11: xe-2-3-0.102.rtr.newy32aoa.net.internet2.edu 127.394ms asymm 12
12: nyc-7600-internet2-newy.nysernet.net 128.238ms
13: columbia.nyc-7600.nysernet.net 135.948ms
14: ....
We see that the packets travel around (Paris, London, Amsterdam) and cross the Atlantic between Amsterdam (10) and New-York (11) in 127-37=90 ms (roundtrip). This still gives us a 9000 km distance, way too long. I don't know if it is due to the cable trajectory, electronic delays, to small sampling by tracepath or an error on my calculation.
Related to this ping delay, you have the funny 500 miles bug.
Another in-the-lab experiment using cheap material and computers is in the arXiv paper speed of light measurement using ping. However, their measurement is indirect (they measure the propagation inside CAT5 cables), but it should also be doable with optical fibres.
Edited to add: My idea of using tracepath probably comes from Measuring the Earth with Traceroute. In this paper they are more lucky than I was (only 20% slower, instead of 100% !)
Best Answer
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
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:
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.