That big grandfather clock in the library might be an impressive piece of mechanical ingenuity, and an even better example of fine cabinetry, but we’d expect that the accuracy of a pendulum timepiece would be limited to a sizable fraction of a minute per day. Unless, of course, you work at CERN and built  “the most accurate pendulum clock on the planet.”

While we’re in no position to judge [Daniel Valuch]’s claim, we’re certainly inclined to believe him, mainly because the 1950s-era Czechoslovakian pendulum clock his project was based on, the Elektročas HH3, was built specifically as a master clock for labs, power plants, and broadcast use. The pendulum of this mid-century beauty is made of the alloy invar, selected for its exceptionally low coefficient of thermal expansion. This ensures the pendulum doesn’t change length with temperature, but it still only brings the clock into the 0.1 second/day range.

Clearly that’s not good enough for a clock at CERN, the European Laboratory for Nuclear Research, where [Daniel] works as an RF engineer. With access to a 10-MHz timebase from a cesium fountain atomic clock — no less a clock than the one that’s used to define the SI second, by the way — [Daniel] looked for ways to sync the clock up to it. Now, we know what you’re thinking — he must have used some kind of PLL to give an electromagnetic “kick” to the bob to trim the pendulum’s period. Good guess on the PLL, but the trimming method is a little cruder — [Daniel] uses a stepper motor attached to the clock’s frame to pay out or retract a length of fine chain into a cardboard dish attached to the pendulum’s rod. The change in mass changes the pendulum’s center of gravity, which changes its effective length, and allows the clock to be tuned a couple of seconds per day.

It seems like [Daniel] is claiming that his chain-corrected clock won’t drift more than a second from the cesium clock for 158 million years. Again, we’ll take his word for it, but it’s a wonderfully ad hoc approach to tuning the clock, and we appreciate its simplicity.

Newton’s cradles consist of a series of suspended spherical masses, and are normally started by pulling one ball back. The outer balls then click back and forth for an interesting distraction. 

To make things even more interesting, “TecnoProfesor” made his own version using ping pong balls and RGB LEDs. As the outer balls sway, they light up in sequence, along with the inner three balls that stay largely in one place.

Power here isn’t provided by kinetic energy, but everything moves via a pair of servo motors. An Arduino Mega is used to control the light/motion simulator, while a button and potentiometer allow the user to change between two modes and variable swing frequency.

It's just an egg, placed in a vulnerable position. "I thought it would be fun to pit a large metal machine against a small fragile object," says Mendoza.

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You have to admit [Dylan Rush’s] clock is a real swinger. Literally. You’ve seen the desk novelties where an arm with leds mounted on it sweeps out a message? [Dylan] did the same thing to make a clock but instead of drawing numbers, he actually draws an analog clock face. Y’know one of those round things with arms?

Behind the clock is an Arduino driving a MAX7219 LED controller. Using the MAX7219 was a challenge because it expects a grid of LEDs while the clock needs a linear array. [Dylan] used a line of individual LEDs wired to match what the controller wanted. A rotary encoder tells the processor the position of the arm so the Arduino sketch can determine which LEDs should be lit to show the time and clock face.

What’s even more amazing is [Dylan] created this before clocks became infamous.

Swing over to the video after the break.

In today’s lab we derived the formula for the period of a simple pendulum (assuming the small-angle approximation), , then measured both circular and simple pendulums.  For the circular pendulum we measured the radius of the cone on the first orbit and the last orbit, the length of the string (the slant height of the cone), and approximated the period by timing 10 or 20 periods and dividing.  For the simple pendulum, we used the photogate setup described in More on pendulums, to get very precise and repeatable measurements of the period.  The hardest part for us was measuring the length of the pendulums, since the center of mass for the bob was not obvious and the exact position of the pivot was not obvious—these uncertainties probably resulted in length measurements being ±5mm, making a large contribution to inaccuracy.

Here is a table of the measurements (and calculated g) we made for the circular pendulum:

The range of estimates for g is larger than I would like.  I think that the decay of the oscillation of the pendulum makes quite a difference.  The average of all the estimates of g is 967 gm/sec^2, which is rather low.

And for the simple pendulum:

The pendulum ticked reliably for quite a while, and the periods were remarkably consistent.  The estimates of g from the simple pendulum are good to about 0.5%, which is the limitation of accuracy on our pendulum length measurements and close to the limit of the accuracy of the small-angle approximation.  The average of the 5 measurements looks good to about 0.2%, which seems pretty good to me, since we certainly weren’t measuring the lengths that accurately.

I looked up the gravitational field in Santa Cruz on Wolfram Alpha’s gravitational fields widget:

total field | 9.7995 m/s^2  (meters per second squared)
angular deviation from local vertical | 0.00322°  (degrees)
down component | 9.79945 m/s^2  (meters per second squared)
west component | 3.4×10^-4 m/s^2  (meters per second squared)
south component | 0.0316 m/s^2  (meters per second squared)
(based on EGM2008 12th order model; 11 meters above sea level)

While the lab was running, one of the students wrote a Python script (using numpy for mean and standard deviation) to read the data and compute the numbers in the table.  We could have talked directly to the Arduino, but it was simpler to cut the numbers from the Arduino serial monitor and paste them into a file for the script to read. That allowed us to keep the Arduino running throughout, and just cut and paste the good numbers, discarding the junk from starting or stopping the pendulum.

I’m quite pleased with the photogate setup, which was very simple to build and worked reliably during the experiment. Crudely wrapping tape around the string made a lumpy opaque object, whose rotation probably contributed to the standard deviation of the  period—having a smoother cylinder for the optical blocker would probably make the period measurement much more consistent.  But that would not improve the mean estimates much since errors in adjacent period measurements cancel.  I believe that our mean periods are much more accurate than the standard deviations suggests, with errors less than 1 per thousand.

I had to make one change in the Arduino code during the lab to accommodate all the different pendulum lengths—I had a dead time before recognizing the next pulse, to prevent getting 2 pulses per period as the string passed through the beam twice.  I started with a dead time of 1 second, which as a bit too long for the smallest pendulum.  Reducing the dead time to 500 msec for that pendulum made it count reliably.  Note that for the 2nd and 3rd pendulum, we measured for about 3 minutes without a bad time measurement, and could have gone longer if we had had the patience.

In Newton’s measurement of g, I described a failed experiment to measure g with a motorized circular pendulum. Further experimentation on my own lead me to adopt for this week’s lab the standard approach using an unpowered circular pendulum.  The cone formed by the string can be described as having height , base radius , and hypotenuse , the length of the string.  If the circular pendulum has period , then (derived in the Newton post).  If we make the string long and push the pendulum with the right speed to get a nearly circular (rather than elliptical) motion, then is nearly constant for many orbits, and we can estimate the period with just a stopwatch by counting 20 or 30 periods.  Using a large enough mass means that neglecting air resistance is now reasonable (which it was not for the tiny mass I started with).

Thanks to John Burk for suggesting that I forget about the motor—that seems to be the best approach, even though I then can’t use the photogate to time the period.  I’m hopeful that we can measure the height and the period accurately enough to get within about 2% of the right value for .

This week in addition to doing the circular pendulum right, I wanted to do simple pendulums.  I’ve assigned problem 4.P.89 in Matter and Interactions, which seems to be the only place in the book that simple pendulums are done.  It is a computational problem, since there isn’t an analytic solution (though the small-angle approximation works pretty well up to about 45°).  I hope the students have done that by tomorrow!

I wanted to measure the period of the pendulum directly (not averaging over many periods), to demonstrate that the amplitude does not matter much.  Unfortunately, I’ve not yet built a sensor that works for this. I tried using the photogate, but I could not hit the 1 cm gap consistently, even with a shorter pendulum.

I also tried using a magnetic sensor (using the circuit I used for the speed-of-sound lab) with a magnet for the pendulum weight, but that triggered at random times as the magnet came close.  Even 20cm away the field was enough to trigger the detector, and I got almost random timings.  A magnetometer was no better than the coil and comparator, as the magnetic field varied chaotically (from movements of the magnet other than the simple pendulum swing, such as twirling on the string).  The magnetometer was usable as a compass, though, which is good, because I originally bought it for the robotics club to use as a compass.  There are some tricky points to using it as a compass, which I’ll talk about in a different post.

I then tried marking the top of the string with a bit of electrical tape and using the photogate there.  That was the most successful so far—if I hold the photogate steady enough, I can get readings repeatable to ±20msec, which is much better than I can do with any other approach I’ve tried.  For one pendulum hanging from the edge of my desk, I either got  two pulses at about 1.11 and 0.45 seconds or one pulse every 1.56 seconds, depending on whether the marker on the string passes all the way through the beam or blocks it continuously at the end of the swing. The random variation I get is probably because of holding the sensor by hand (to align with the string).

If I had a more rigid way to mount the sensor, I should be able get more consistent readings, so my main engineering task was to get a rigid pivot point on the ceiling beam (without making any holes) and mount the photogate in a rigid, but adjustable, way.  Of my two standard mechanical engineering techniques, duct tape and Lego, I chose Lego:

A view of the photogate mounted on the Lego beam next to the pendulum string.

Closeup of the photogate, showing the breakout board and sensor wedged between a plate and a beam, with a 2-plate spacer.

Having come up with a nice way to grip the photogate and still be able to swing a pendulum string into the gap, I connected the beam holding the photogate to the same right-angle platform that we had used last week for the motorized pendulum. This left a little gap that I could rest the Arduino board in, so that there was no tension on the wires to the photogate.

I was a bit worried that I might have to put my laptop on top of a ladder, since the USB cord is not very long, but I have a spare pair of USB-to-Cat5 converters (one set is for the robotics project), so I was able to make an extension cord out of a flexible Cat-5 Ethernet cable, giving me enough length to put my laptop safely on the desk.

The same Lego that holds the photogate can also support the Arduino, so I don't need to hold anything in my hands.

I had two other ideas I haven’t tried: using one of the ultrasonic range finders to track the pendulum motion and using a video camera to time the motion.  These require interpolation of position data to estimate the period, so I’d rather avoid them for now. The top-of-string photogate will work (I think) for the simple pendulum, and the circular pendulum can be timed with a stopwatch averaged over many periods.  (I could even use the photogate timer as a stopwatch, though the resolution of the stop watch on my Casio wristwatch is 0.01 seconds, and human reflexes make anything less than 0.1 second pretty much noise.)