A candle skewered transversely near its center of mass by a needle and balanced between two low-friction supports, when lit on both ends, will drip asymmetrically and begin to oscillate like a seesaw; these oscillations grow in time. We examine the onset of instability, and find that the candle does not oscillate quasi-stably until the vertical center of mass is lowered by the symmetrical melting of each end, creating a physical pendulum with a well-defined characteristic period. Additional asymmetric dripping below horizontal drives the pendulum, leading to linear growth in amplitude. The drop release becomes phase locked to the seesaw motion of the candle. We develop a small-angle analytical model that predicts the maximum growth rate when the dripping rate matches the seesaw frequency. Measurements of the motion, droplet phase, and melting rate confirm the validity of the model. We compare our results to earlier studies and make suggestions for the demonstrator.

## I. INTRODUCTION

I burn my candle at both ends.

It gives a lovely light.

Up and down, up and down,

It seesaws through the night. (Ref.1)

A candle is pierced by a needle transversely near its center of mass^{2} and balanced between two wine glasses (see Fig. 1). Assuming the alignment is not perfect, one end will be lower than the other. When both ends are lit, the lower end loses wax more rapidly than the upper, bringing both ends level. This leveling takes place on a timescale that is on the order of a minute or two. From this near-balanced condition, the wax begins to drip alternately from either end, and the seesaw starts to oscillate; these oscillations grow in time.

The first written records of the candle seesaw, or *moteur stéarique*, date to the nineteenth century,^{2–4} and the effect was included by the great Harry Houdini in his posthumous book of parlor magic tricks.^{5} Martin Gardner featured it as “Trick of the Month” in *The Physics Teacher*, September 1993,^{6} and later in *Magic Magazine*.^{7} A recent conference paper suggests that the system potentially becomes chaotic at high amplitude.^{8} For an extensive description of its history, including period illustrations, see Ref. 4.

Contemporary papers have proposed several explanations^{3,4,6,9} for the effect. Ehrlich^{3} fails to identify any mechanism for the growth of the oscillation, the most compelling feature of the seesaw.^{10} Theodorakis and Paridi^{4} neglect the departing droplets as part of the system,^{11} thus creating a phantom driver^{12} that increases in time.^{13}

Assuming conservation of momentum as the mechanism for the candle motion is a common misconception.^{14,15} Martin Gardner considers both energy and momentum conservation, posing these questions:^{6}

You might suppose that each time a candle end leaves a bit of wax on a plate it makes that end a trifle lighter, and that these changes in weight operate the little machine. But is this the case? Are the weight differences sufficient? Is there any way that the formation and dropping of the tallow can deliver a slight upward recoil? To find out you must burn your candle at both ends.

In this paper, we confirm the source of the driver as the mechanical energy delivered to the seesaw by the departing droplets, in much the way that a waterwheel^{16} operates. Combustion itself does not propel the candle; the loss of gravitational potential energy of the wax does. While the seesaw candle melts wax throughout its cycle, it releases the drops only below the horizontal. This release of gravitational potential is the energy source of the candle system. The work done by the departing droplets on the remainder of the candle provides power to drive and sustain the oscillations. We derive a linear small-angle analytical solution for the seesaw candle and predict growth rates of the instability that lead from random fluctuations to regular linear-growing oscillation. Simultaneous measurements of candle angle and drop occurrence substantiate the validity of the model. Under near-synchronous drip rate *ω _{d}* and seesaw oscillation frequency

*ω*the system exhibits phase locking

^{17,18}of the drip frequency. We identify a preferred position for the axle relative to the vertical center of mass of the candle, and show definitively that these “trifles” of changes in weight are indeed sufficient to drive the instability.

## II. THEORY

A long cylindrical candle of mass 2*M*_{0}, radius *r*, and length 2*R*_{0} is constrained to oscillate in a vertical plane about a horizontal axle. If the axle passes through its center of mass, there is no restoring torque; in the absence of friction the candle is free to spin indefinitely at constant rate if given an initial angular velocity. When lit at both ends and released at rest the candle drips randomly, without oscillation, melting away and hollowing out a portion of the wax above the axial center line near either wick. This initial melting creates a characteristic shape, roughly a trapezoid as seen in profile,^{19} reduces the total mass to 2*M*, shortens the cylindrical cross section of the candle to length 2*R*, and lowers the center of mass by distance *a* = *a*_{0} (see the Appendix) below the center line of the candle (see Fig. 2), permitting the system to behave as a physical pendulum. This trapezoidal profile is our assumed starting point.

If the candle is imperfectly balanced left/right, both ends will melt, but the faster drip rate of the lower end causes the candle to balance itself within a couple of minutes. If the wicks are now extinguished, the melted candle forms a physical pendulum with moment of inertia 2*I*, where

The candle thus oscillates with small-angle natural frequency

where *a* is the center of mass of the seesaw. If the candle is pierced through a chord whose length is less than its diameter, *a* increases, leading to faster oscillation, while having negligible effect on *I*, provided that $R\u226br$.

With both wicks relit, the candle drips more or less evenly on each end, but the system evolves into a regime where the drops depart preferentially from the lower wick, resulting in an oscillation of growing amplitude. Due to the cup shape near the wicks,^{20} the drops tend to be released when the wicks are below horizontal, near their lowest points. Under synchronous conditions (*ω _{d}* ≈

*ω*, wherein the rate of generation of molten wax approximately matches the seesaw oscillation frequency), a single wax droplet of mass

*m*drips off the lower wick in each half cycle of the candle motion, close to the lowest point.

One might be tempted to model this system using a time-dependent moment of inertia and applying Newton's laws. Such an approach, which causes a major flaw in Ref. 4, misses a basic point of physics: When analyzing forces or torques on an object, one must clearly identify the object. One might consider the candle plus the wax that drops as the system; alternatively one can consider the remaining candle as the system, and treat the wax about to drop as external.^{11} The latter case is what is followed here. This is done cycle-by-cycle, giving a map for the amplitude for a full cycle which is interpreted as a differential equation. We are, of course, assuming the length of the candle changes little compared to its total length during the time under consideration.

We obtain a quantitative expression for the maximum amplitude gain through the following simplified model, valid when a single droplet departs from each wick during a single oscillation: Assume that the growth in a single cycle is small so that the candles are oscillating with amplitude *θ*_{0} about *θ* = 0 with total energy $12(2I)\omega 2\theta 02$. We start our cycle with *θ* = *θ*_{0} and consider the system to be the solid candles without the attachments, i.e., the melting droplets are viewed as external to the system. We assume an identical release of mass *m* from the right and left on each cycle. As mass *m* is released the system becomes unbalanced, going from excess mass *m*/2 on the right to excess mass *m*/2 on the left, and this process repeats. In order to isolate the droplets that will be departing from the rest of the system, we start at initial amplitude *θ*_{0} with 3*m*/2 on the right, and *m* on the left, resulting in an average clockwise torque of approximately $(mg/2)(R+L/2)$. This torque drives the candle toward –*θ*_{0}, whereupon the right-hand droplet *m* is released, creating a counter-clockwise torque of the same magnitude. Due to the torque imbalance of $(mg/2)(R+L/2)$, as the candles rotate from *θ* = *θ*_{0} to *θ* = –*θ*_{0} average work $mg(R+L/2)\theta 0$ is done on the remainder of the oscillating system. Similarly, as *θ* goes from –*θ*_{0} to *θ*_{0} the left-hand droplet does additional work $mg(R+L/2)\theta 0$ on the system. We assume during the period that *I* and *a* remain constant, and that on average the system is in balance.^{21} Accounting for kinetic frictional loss^{22} during the cycle, $4\mu kWra\theta 0$, with *W* the weight of the seesaw, the net energy delivered to the candle oscillation is

where *μ _{k}* is the coefficient of kinetic friction and

*r*is the radius of the axle. During this cycle, the amplitude grows from $\theta 0\u2192\theta 0+\Delta \theta 0$, and the mechanical energy thus increases by

_{a}where Δ*θ*_{0} is the small variation of the amplitude for a single cycle. Equating the net energy supplied by the departing droplets to the change in total energy of the system implies

For notational simplicity, we define

Expressing this energy change as a smooth variation over a complete seesaw period (*T* = 2*π*/*ω*) the amplitude growth rate becomes

Note that this predicted growth rate assumes that the drops on average come off near the bottom of the motion on each side, so that the work done is maximal. If more than one drop departs on each side, they will not each have $\theta \u0307=0$, hence they will leave with some kinetic energy, and the predicted growth rate will be somewhat less.

One might expect an additional dependence of *m* on the amplitude,^{3,4} leading to exponential growth in the oscillation. Yet we observe only linear growth in the experiments that follow, and we find that the average melting rate of the candles is the same whether they are held horizontally or allowed to oscillate. The initial melting to achieve left/right balance does depend upon angle but occurs over a timescale too long to influence the oscillatory motion.

This derivation of $\theta \u03070$ assumes that the system is already oscillating. In practice, the initial motion is supplied by a random release of wax from one end before the other.

The synchronously driven case, when the generation rate of molten wax is such that the horizontal drop rate *ω _{d}* matches the natural frequency

*ω*of the seesaw oscillation, most simply illustrates the mechanism because the drops depart at the lowest point, and thus the torque provided by the departing drop does work over the largest possible angle. On the other hand, if the center of mass

*a*is reduced so that two or more drops come off on each side during a cycle, the system may need to shed the drops before and after the lowest point. Adjusting

*a*to give two drops per side, assuming that the drops nonetheless depart close to the bottom, and keeping

*I*and

*R*constant, Eq. (7) shows that if $m\u21922m$ as $\omega \u2192\omega /2$, in the absence of friction we expect four times the growth rate of the synchronous case.

## III. EXPERIMENT

The traditional axle of the seesaw candle is a darning needle that makes an exceptionally low-rolling-resistance bearing on the rims of two wineglasses. To measure the angle more readily, we mount a pair of paraffin candles of length 100 mm and diameter 12.7 mm to an aluminum crosspiece attached to a 6.35 mm diameter 175 mm long horizontal non-magnetic stainless steel shaft.^{24} Each shaft end is sharpened, ground to a cone, and held in a sapphire jewel bearing lubricated with watch oil (see Fig. 3).

Near one end of the shaft is a non-contact absolute quadrature rotary encoder (Broadcom AEDM-5815Z-06), equipped with a 5000 counts per revolution code wheel. A microcontroller (PIC HPC Explorer) decodes and accumulates the quadrature pulses on each edge, giving 20,000 counts per revolution, allowing angular position measurement with $0.02\xb0$ precision at 100 Hz acquisition rate. For small angles, the melting wax leads to droplet formation, rather than a steady stream. The droplets do not vary appreciably in diameter, nor does their mass differ by more than ±10% for static angles of $\xb130\xb0$. Hence the mass loss is monitored optically as the drops interrupt a pair of diode laser beams at fixed height below the axis, in the plane of oscillation. The light beams fall on amplified photodiodes and external comparators. Because the droplets' release velocity is small and known, and their final speed is much less than their terminal velocity, the time at which the droplets slide off is deduced from the angle, length, and velocity of the candle before the droplets cross the beams.^{25} Occasional satellite drops of negligible size^{26} are observed 40 ms after the main drop; the comparator thresholds are set to ignore these.

Candles are fitted into aluminum sleeves and mounted to the shaft with horizontal set screws. Diametrically opposed machine screw counterweights secure the candle holder to the shaft above and below the balance point, permitting adjustment of the vertical center of mass *a* of the system.

To reduce the sliding “dry” friction^{22} of the jewel bearings, the majority of the weight *W* of the seesaw is supported on axially magnetized passive magnetic bearings^{23} consisting of pairs of cylindrical stator magnets that each repel a shaft-mounted rotor magnet (K&J Magnetics, Inc.) (see Fig. 3). The two sets of stator magnets are adjusted vertically to support the weight of the candles and axially to minimize the thrust on the jewel bearings while maintaining the encoder wheel in the correct plane.

The average drip period of these candles when held level is 1.8 ± 0.5 s, shedding 0.020 ± 0.001 g per drop. Nearly all (>90%) of the mass is lost via dripping rather than combustion.^{27} The mass of the complete shaft without the candles is 79 g, and each candle adds 11 g before burning. Typical machine screw counterweights range from 1.1 to 4.1 g as needed to adjust the vertical center of mass of the system.

## IV. RESULTS

With unlit candles, the moment of inertia (2*I* = 1.00 ± 0.01 × 10^{−4 }kg m^{2}) was measured by plotting *ω*^{2} vs. the lower machine screw position. For comparison, the moment of inertia of the apparatus without candles was measured to be 1.6 × 10^{−5 }kg m^{2}. A typical damping curve when the candle is tuned for synchronous operation is shown in Fig. 4, indicating a regime of viscous damping due to air drag at high amplitude, followed by a regime of dry (sliding) damping due to residual friction in the bearings. The energy loss due to dry friction is constant per cycle, but does depend on frequency via Eq. (7). We measure the dry frictional decay rate under synchronous conditions to be 0.006–0.008 rad/s, depending upon the position of the magnetic bearings. Without the magnetic levitation the motion damps out within a dozen cycles due to dry friction in the bearings.^{28}

### A. Synchronous operation, ω_{candle} ≈ ω_{drop}

If the horizontal drip rate *ω _{d}* of each candle approximates the natural frequency

*ω*of the physical pendulum, the seesaw releases a drop on each side during a complete cycle, and tends to release the drops very near the lowest point.

Figure 5 shows a typical plot of *θ*(*t*) for candles that start at rest. With the candles level, the wicks are lit simultaneously, after which the droplets depart at times (▲ = *m _{L}*, and ▼ =

*m*) as shown. The drops appear randomly, but then adjust their phases to align with the low point on either side of the oscillation. The phase-space diagram in Fig. 6 overlays the angular velocity vs. angle for the system, along with the delay-corrected ($\theta ,\theta \u0307$) coordinates of the drops. This plot shows no evidence of an impulse upon the candle at the time of droplet departure, refuting the persistent notion that the seesaw candle is a form of kicked oscillator, i.e., one in which the departing droplets impart an impulse to the candle via Newton's third law as proposed in Ref. 3.

_{R}From this plot, we observe that the drops preferentially fall off when the end of the candle is below the horizontal, near its lowest point. Right-hand drops always occur at negative angle, and left-hand drops always occur at positive angle. This is not to say that the candle melts faster below the horizon; it is a statement that the drops slide off there. Surface tension of the wax in the cupped ends evidently holds the drops in place on the upper half of the cycle as well as for some considerable angle below level.

From Fig. 5, we obtain a linear growth rate of 0.0133± 0.0003 rad/s. In the absence of friction, Eq. (7) predicts 0.0177 ± 0.001 rad/s, with the drop-to-drop variation in mass *m* dominating the uncertainty. After subtracting the inferred *η*/*ω* = 0.0054 ± 0.0001 rad/s dry frictional loss rate term (*ω* increases by 5% between lighting the candle and the onset of growth), Eq. (7) predicts a growth rate of 0.012 ± 0.001 rad/s, which agrees within measurement error.

The seesaw frequency need not match the horizontal drip frequency exactly. If the pendulum period lies close (±0.3 s) to the horizontal period of droplet release, the pendulum phase locks the droplets,^{17,18} so that the drip rate conforms to the much more stable (albeit slowly varying due to melting) frequency set by the physical pendulum. Figure 7(a) shows the period of the seesaw motion, overlaid with the time between drops on either side. For comparison, Fig. 7(b) shows drip periods *T*_{0} for candles held steady. The seesaw motion in (a) phase locks the period of the droplets, narrowing their spread by resetting the phase of the droplets near the low point of the cycle. As the ends melt and the candles get shorter (at ≈ 0.62 ± 0.03 cm/min, regardless of whether the candle is oscillating at large amplitude or held level), the period of oscillation passes through a region wherein the droplets are pulled toward *T _{d}*:

*T*

_{0}= 1:1, but occasionally emit at 1:2, i.e., 2

*T*

_{0}. Similar pulling and locking occurs at 3

*T*

_{0}.

### B. $\omega candle\u224812\omega drop$

When the seesaw candle is slowed down by reducing the *a* parameter, more drops appear in each half cycle. The phase space plot of Fig. 8 shows the seesaw set so that two drops appear on each side, a cycle time of ≈4 s. The drops still appear below the horizontal on their respective sides, but now the drops are released before and after the low point on each half cycle. The growth rate in this case (0.05 radians/s) is so large that the seesaw has to be stopped and reset to the horizontal position, but is again expected from Eq. (7) with total mass of two drops at *ω _{d}*/2.

## V. NOTES FOR THE DEMONSTRATOR

### A. Where should the needle go? Tuning up your candle

The candle seesaw pierced with a needle can be given a range of axle position, from *a* = 0 (perfectly balanced, so the characteristic frequency goes to zero), all the way to *a* ≈ *r*, with *r* the radius of the candle, in which case the natural frequency is approximately

for a long thin candle of length $2R\u226br$.

Historic^{2,5} and contemporary^{3,4,6} descriptions of the candle seesaw prescribe pushing a needle through the exact center of the candle. This axle placement is not ideal because it leads to very large growth rate via Eq. (7) and the slow oscillations provide insufficient air drag force to reach a limit cycle. We can estimate the vertical center of mass *a*_{0} for a candle of length 2*R* and mass 2*M* that has been mounted through its exact center, then melts until it is shaped as in Fig. 2. Equation (1) is, to first order,

for a typical dinner candle of dimensions *R* = 14*r* and *L* = 2*r*. The vertical center of mass of the slanted section alone lies (see Eq. (A3))

below the origin, and so the center of mass of the whole candle, using Eq. (A5), becomes

For a candle of *r* = 1.0 cm, this yields *a*_{0} = 160 *μ*m. This small *a* parameter creates a very long seesaw period, via Eq. (2), of *T*_{0} = 2*π*/*ω* ≈ 12 s.

This seesaw period is about 10–15 times greater than the horizontal drip period of such a candle. At this remarkably low oscillation rate, any given drop will have a profound effect on the motion, which cannot be described as seesaw-like; the growth rate given in Eq. (7) becomes very large, due to the multi-drop mass *m* in the numerator, and the low frequency *ω* in the denominator. The first set of drops spilled on one side can take the pendulum past vertical. This central placement of the needle does not make for a compelling, reciprocating seesaw.^{29}

A more effective location of the axle is through the horizontal middle, but somewhat above the center of mass of the candle. Making *a* larger speeds the seesaw oscillations, allowing both a better match to the drip rate of the wax and greater velocity-dependent damping that stabilizes the seesaw at high amplitude. A position change of even two darning needle diameters (2.5 mm) will increase the natural seesaw frequency by a factor of four, yielding close to three or four drops per cycle.

The simplest and smoothest regime of operation (and highest growth rate per drop) is the synchronous regime (see Sec. IV A) when the horizontal drop rate matches the characteristic frequency of the pendulum. One need not be too careful about matching the frequencies of dripping and oscillation, because the candle naturally resets the phase of the drops whenever it releases them near the bottom of the motion. This mechanism permits the candle to drive itself to high amplitude, adjusting the drop release time even as the pendulum becomes non-linear.

Setting *ω _{d}* =

*ω*and solving Eq. (2) for

*a*gives

The needle must be able to pass through the candle body, $a\u2264r$, and so constrains the length of the candle

In practice, it is easier to make *a* slightly less than optimal and allow the candle to melt until *R* achieves the length to satisfy Eq. (13), or perhaps a 2:1 drop-to-pendulum frequency. A good starting point is to make *a* = *r*/2 for a typical long dinner candle.

### B. What is the purpose of the riders?

The decorative outboard riders and their support structure shown in Fig. 1, although not required to make the candle oscillate, do actually serve three purposes: they stabilize both the moment of inertia and vertical center of mass against changes in candle length upon melting, keeping the oscillation frequency fairly constant. In addition, they limit the amplitude of the system by providing additional viscous (air-drag) damping as the velocity of the candle increases. A seesaw equipped with riders can thus be tuned to reach a limit cycle that maintains synchrony with the drip rate for longer than would a bare candle.

### C. The role of friction for a demonstration candle

A cleaned nickel-plated steel needle axle resting atop wine glasses serves as an exceptionally low rolling-resistance bearing because both hard surfaces are ground or flame-polished flat and behave elastically.^{30,31} The needle is rigidly fixed to the candle body and thus the whole arrangement rolls without slipping; there is vanishingly small energy loss on each cycle when the candle moves at low amplitude, and so *η* can safely be ignored in Eq. (7). Another popular axle arrangement, such as shown in Refs. 9 and 24, uses sliding bearings; these may have too much friction to allow a synchronously tuned candle oscillation to start, and so a small *a* parameter, leading to high growth may be required. Whichever arrangement is used for the bearing, viscous damping due to air drag is important in achieving a stable limit cycle. This drag force limits the amplitude of the oscillations when the power provided by the departing drops matches the viscous drag at high amplitude.

## VI. CONCLUSIONS

The driving torque for the candle seesaw arises from the work done by the departing drops on the candle as they make their way from solid wax at horizontal to molten droplets that get released below the horizontal. We have analyzed the growth rate using a simple model that considers how much mass departs during each half cycle. This model considers only a fixed averaged melting rate, independent of angle, because measurements have shown that any angularly dependent dripping occurs on a timescale that is much slower than the typical seesaw period. The energy for the motion is supplied by the difference in gravitational potential between the masses' original horizontal position and the release points below the horizontal. Candle motion and drop release time indicate that the drops become phase locked to the dominant seesaw motion. Measurements of growth rate for the synchronous case, wherein the horizontal dripping rate matches the frequency of seesaw oscillation, confirm the model predictions.

## ACKNOWLEDGMENT

The authors would like to thank A. Kaufman for assistance with Fig. 3.

### APPENDIX: FINDING $a=a0$

A horizontally mounted candle melts until each of its two ends assumes a steady shape, which we approximate as a slant cut at angle *ψ* to the horizontal, as shown in Fig. 2. Figure 9 shows an end-on view of a slice through the cross section.

The missing mass forms a circular segment of area

with *ϕ* the opening angle from polar north. This is the net shaded area contained between a pie-shaped segment and the triangle AOB that lies below the chord. This circular segment has a first moment along the vertical axis of

which gives it a center of mass of

Parameterizing the opening angle *ϕ* with the axial distance *z* along the slant cut part of the candle, we note that the slant height *y* = –*sz*, with *s* a constant, for *z* going from –*L*/2 to *L*/2. Given that $y=r\u2009cos\u2009\varphi $, and $sdz=r\u2009sin\u2009\varphi d\varphi $, the center of mass of the entire slanted end of the candle becomes

which then gets subtracted from the center line of the candle.

Each side of the candle consists of cylindrical length *R* plus a slanted end of length *L*; the weighted effect is to lower the center of mass of the entire candle by

from the center line. This lowering of the center of mass by *a*_{0} provides the restoring torque for the physical pendulum about an axis that passes through the center of the candle.

## References

My candle burns at both ends

It will not last the night But ah, my foes, and oh, my friends

It gives a lovely light!

Reference 3, Eq. (8.5) is clearly wrong, because it does not follow from the earlier definition of the mass, and it has inconsistent units. The end result is an equation for a harmonic oscillator with no indication of how its amplitude grows.

Reference 4, Eq. (6) is not the angular momentum that would be conserved in the absence of torques, because it contains the mass and momentum of the departing mass. In the absence of torques, if the system has an initial angular velocity, the time dependence of the mass loss in Eq. (9) (which occurs throughout all phases of the motion) implies that the angular velocity increases as pieces of the system fall away. This is clearly incorrect. Consider a spinning ice skater shedding her mittens: her rotational velocity does not speed up if her hands remain at arms' length; the mittens merely carry away some of the system's angular momentum.

In the absence of the drops, the candle is balanced so that it oscillates around *θ* = 0. With drops present, the mass imbalance between the left and right sides is always $|m/2|$. The net effect in one cycle is for two drops to be moved on average from horizontal to a height $(R+L)\theta 0$ below horizontal. Some of this lost gravitational potential energy goes toward moving the droplets out to the rim; the rest is gained by the remainder of the candle.

For a spherical droplet of wax of mass 0.02 g, we estimate *v*_{term} = 8 m/s. Such a droplet released at 10 cm above the detectors reaches 1.4 m/s in the absence of drag force. Insofar as this velocity is $\u226a\u2009\u2009vterm$, the droplets can be tracked back to their departure time.

These 2 1/2-hr candles last 13 min when burned horizontally.

We estimate the shaft tip radius to be 1.5 × 10^{−4} m, which, along with *μ _{k}* = 0.15 for lubricated steel on sapphire, would give a minimum

*η*/

*ω*= 2.2 × 10

^{−2}s

^{–1}, exceeding our predicted growth rate in the absence of sliding friction.

We assume that the seesaw axle is of the low-rolling-resistance variety. If the axle slides, the increased frictional loss rate *η*/*ω* in Eq. (7) can be used to achieve a more manageable growth rate.

_{r}< 0.001