# Gyroscope Precession: Cycloramic.app for the iPhone 5

The recent Cycloramic app for the iPhone 5 was added to the iPhone App Store on December 21, 2012. It will spin the iPhone standing on its bottom edge while taking a panoramic video. Explanations of the mechanism, the actual physics, behind this spinning motion so far have been limited to just stating that the spinning is due to the vibration without any explanation as to how that vibration produces rotary motion. We collected a number of observations from various videos posted on the Internet, and we propose a physical mechanism that explains all of these observations (as of January 14, 2013).

- Sandra N. Shaefer and Craig G. Shaefer
Note: An Alternative Mechanism Not Relying on Precession has been added just in case the Proposed Mechanism is invalidated by future experimental evidence.
Note: The Cycloramic App's developer has verified that our proposed mechanism is valid, it agrees with their theory. See Note 9 below.

For a complete understanding of this mechanism, you may find the CM-Gyroscope_Nutation_pw107.pdf on the Gyroscope Precession and Nutation page useful. Here is the link to this wiki page: Gyroscope Precession and Nutation

## iPhone 5's Cycloramic.app: Proposed Mechanism

Now that you are an expert at explaining the operation of a gyroscope and its precessional motion, how about using this knowledge and understanding to explain one of the most interesting apps for the new iPhone 5? The app we are talking about is the Cycloramic.app''. This app was released on December 21, 2012 to the iPhone App Store and it uses the iPhone's vibration to take a $360\degree$ video. In the last month since its release, it has garnered a significant amount of attention, including the New York Times's Pogie Award for one of the Brightest Ideas of 2012. The question we ask is, what is the physical mechanism underlying its operation? Sure, we know that it uses the vibration of the iPhone, but how is this vibration turned into the rotation of the iPhone in a $360\degree$ circle in order to take a hands-off'' video?

Here is the PDF file giving a full derivation of Proposed Mechanism:
Here is the PDF file for the Bibliography of the Classical Mechanics Chapter (this Bibliography contains the citations for our Observations on the Cycloramic App):

The following video (Cycloramic_App_Proposed_Mechanism-short_version-1080p.mov) explains and demonstrates the Classical Mechanics Explanation for the spinning motions produced by the Cycloramic app. This explanation predicts, both qualitatively and quantitatively, the behaviors when the Cycloramic.app is run.
Click the following link to view the short version of the video on YouTube: http://youtu.be/slOpxFwgLx0

The following video (Gyroscope_Cycloramic_App_Proposed_Mechanism-detailed_version-1080p.mov) explains and demonstrates the Classical Mechanics Explanation for the spinning motions produced by the Cycloramic app. This explanation predicts, both qualitatively and quantitatively, the behaviors when the Cycloramic.app is run. This video contains a more detailed explanation than the previous one.

Click the following link to view the detailed version of the video on YouTube: http://youtu.be/86nOE_HYEfQ

(YouTube encodes the video at a number of different resolutions.)
Description of the video's experiments: This video discusses the Classical Mechanics Explanation for the proposed mechanism (gyroscopic precession) and demonstrates the nutational and precessional behaviors of a gyroscope.

Citations for the following observations are included in the CM-Gyroscope_CycloramicApp_pw112.pdf PDF file.

Looking at a number of YouTube videos as well as the videos on the cycloramic.com website, we notice a few things. First of all, nobody, including the app's designers, have a physical explanation of how this app causes the iPhone to rotate. Everything that we could find only states that the app uses the vibration of the iPhone in order to rotate, but nobody says anything beyond this. Of course, simply stating that it is the vibration of the iPhone that causes the phone to rotate is not a proper physics mechanism explaining how and why it does so. Now there are a few principle observations that we found in various videos (see the CM-Gyroscope_CycloramicApp_pw112.pdf for the references) and on various reviews of the Cycloramic.app, listed below:

1. Most obviously, the iPhone 5 rotates in a single direction when standing on its bottom edge with its vibration ringer operating.
2. The Cycloramic.app operates on the iPhone 5 but the developer states that it does not work on the iPhone 4/4S. Several individuals have left comments on YouTube demonstration videos stating that the Cycloramic.app does not work on the iPhone 4/4S.
3. The iPhone 5 uses a vibration rotary motor spinning an eccentric cam as its generator of vibration.
4. The iPhone 4/4S uses a linear oscillating vibrator, similar to a speaker cone and coil, as its generator of vibration.
5. One individual in a YouTube video states, but does not show in the video, that the iPhone 4S rotates only a partial way around a circle but not a full circle. Another app, the iTelekinect.app, will move but not rotate an iPhone 4S.
6. The Cycloramic.app will rotate an iPhone 5 even with the olloclip fisheye adapter attached, thus increasing the weight of the iPhone.
7. In another demonstration video, the iPhone is resting on a small and thin piece of glass sitting on top of a smooth wooden board. When the Cycloramic.app is started, the iPhone barely moves, but when the individual presses down on the glass so that it is held securely to the wooden board, the iPhone does rotate.
8. The Cycloramic App Hack Improves iPhone 5 Performance'' video demonstrates how a small, four-thicknesses, piece of Scotch transparent tape applied to the bottom of the iPhone 5 improves its spinning performance.

Given these observations, see if you can propose a physical mechanism that explains them all. [We note that neither the authors of the Cycloramic app nor any of the demonstration videos nor any of the review of this app on the Internet provide a physical mechanism to explain how it operates. Several mentions that it uses vibration, but none that we have found discuss just how vibration leads to a spinning of the iPhone 5. In addition, the author of the iTelekinect.app, which predated the Cycloramic.app by 44 days, indicates in a video that his was a lucky discovery of this motion produced by the vibration of the iPhone. This author also does not describe a physical mechanism behind this discovery. Since no physical mechanism has yet appeared (as of January 14, 2013), we put forward in the following Answer this gyroscopic precession mechanism as a proposed physical mechanism underlying the Cycloramic app's generated movement.]

### Caveat:This is a Proposed Mechanism'. It is based upon the currently available (January 14, 2013) information and observations, and it explains all currently known observations. Since this app is less than a month old at this time, there is only a relatively small amount of data available concerning the Cycloramic iPhone 5 app. When further data and observations become available, these may determine whether this mechanism is applicable or not to the Cycloramic app.

Item (1.) indicates that the iPhone 5 will rotate when its vibration ringer is activated. Our proposed mechanism is that this rotation is simply the result of the precessional motion of the eccentric cam rotor of the iPhone's vibration ringer motor.

This Figure illustrates the precessional, $\alpha$, and nutational, $\beta$, motions of the spinning top that result from a torque force, $mg$, being applied to the rotor's axle. The top's spin axis, denoted by $\bsym{L}_{\quat{k}}$, follows the thick red curve between the angles $\beta_1<\beta(t)<\beta_2$ as drawn on the surface of a sphere centered at the origin (fixed point of the top). Oftentimes the nutational motion dampens quickly because of friction in the pivot point. Demonstration of this precessional and nutational motions is included in the Video: Gyroscope-Nutation-1080p.mov video. (You can watch this video on YouTube: http://youtu.be/5Sn2J1Vn4zU )

In a nutshell, as shown in the above Figure, a torque, in this case due to the force of gravity acting on the Center of Mass of the rotor, applied to the spin axis of a rotating top results in both a precessional and nutational motion. The nutation in the $\beta$ direction usually dampens out quickly from friction leaving only the precessional motion around the $\bsym{\hat{z}}$ axis to continue for as long as the torque is applied to the rotor's axle. The precessional motion results because any applied torque, denote it by $\Gamma$, changes the angular momentum, $\bsym{L}_{\quat{k}}$, of the spinning top as given by $\Gamma=d\bsym{L}_{\quat{k}}/dt$. The direction of the changing angular momentum is perpendicular to the direction of the initial applied force, in this case, the gravity $mg$ force. Thus the change in the direction of $\bsym{L}_{\quat{k}}$ is in the $\quat{j}$ direction, and this motion is known as precession. This equation, derived directly from Newton's Laws, tells us that the greater the torque is, then the faster the angular momentum of the rotor changes direction.

This is a photograph of the Cycloramic iPhone 5 app preparing to take its $360\degree$ video. Touching the GO'' button will send its on its way.

From Item (3.) we see that the iPhone 5's vibration ringer is produced by an electric motor rotating an eccentric cam. Both the eccentric cam and the electric motor's armature both act as a spinning top whose rotation axis points horizontally in the standing'' iPhone (see the above Figure for a schematic diagram showing this orientation). With the spin axis horizontal, the force of gravity acting on the Center of Mass of the iPhone yields a torque being applied to the axle of the spinning cam. This torque causes the axis of rotation to precess producing the rotary motion of the iPhone when standing'' on its end.

Quantitatively, from the calculation of the average precession rate for a spinning heavy top with a single fixed point (see the derivation in the Classical Mechanics Chapter), we find that the rate is directly proportional to the torque applied to the rotors axle and inversely proportional to the rotational rate of the rotor. Applying this to the iPhone's vibrational motor, we find for the average precession rate of the vibrational motor's eccentric cam, denoted by $\overline{\dot{\alpha}}$, to be

where $\bsym{\Gamma}$ is the torque being applied to the axle of the vibrational motor. Notice that if the torque, $\bsym{\Gamma}$, increases then the rate of procession, $\overline{\dot{\alpha}}$, also increases. The denominator of this fraction for $\overline{\dot{\alpha}}$ includes $\lambda_{\quat{k}}$, the Moment of Inertia about the axle of the vibration motor, and $\omega_{\quat{k}}$, the rate of spin of the vibration motor's eccentric cam about its axis. Since it appears that the spin rate of the vibration motor is not being altered by the Cycloramic app, then both $\lambda_{\quat{k}}$ and $\omega_{\quat{k}}$ are constant.

And how does this torque on the vibration motor's rotor arise? It stems from the motion of the iPhone's housing as this motion is transferred to the vibration motor. The magnitude of this torque is directly proportional to the forces acting on the iPhone, in other words. So the greater the forces acting on the iPhone housing, the greater the torque on the motor's spinning cam and thus the faster the processional motion. This is the physical mechanism that we propose is behind the operation of the Cycloramic app.

This shows two schematic diagrams of an iPhone 5 executing the Cycloramic.app hands-off'' $360\degree$ video application. In Part $A$, the eccentric cam of the vibration motor is rotating into its lower half revolution thus lifting the corner of the iPhone off of the marble surface. In Part $B$ the eccentric cam is in its upper half thus forcing the iPhone body into the marble slab. See the text for an explanation.

Now there are really two possible alternatives to how this precessional motion is established. In the first, and perhaps most likely, case, the vibration of the iPhone 5 does not cause it to completely loose contact with the supporting surface. In other words, the vibration results in just one end of the bottom edge of the iPhone to rise above the surface and then repeatedly strike the surface, as shown in Parts $A$ and $B$ of the earlier Figure. The other case is that the vibration motor's cam is massive enough to cause the entire iPhone to rise off of the surface. We attempted to test which case was applicable by shining a strong light from the back and seeing if any light was transmitted through the gap during the hop'', but we could not observe any light and so this test failed. [We plan on obtaining a laser pointer and repeating the test with the brighter laser light.]

So, returning to the first case where the iPhone is always in contact with the surface, we see from Part $A$ of the earlier Figure that the force of gravity acting on the Center of Mass ($CM$) of the iPhone yields a small downward force, $\bsym{F_{\downarrow}}$, on the vibration rotor's axle. [This machanism, by the way, also works for the second case where the iPhone is wholly in the air at some point during the vibration. While the mechanism is still valid, I doubt that this case actually applies to the real iPhone.] This force generates a torque acting on the spin axis that produces a precessional motion in the clockwise direction, as shown in the diagram. When the iPhone strikes the surface, however, the surface must apply an upward force, $\bsym{F_{\uparrow}}$, on the iPhone housing that is transmitted to the vibration rotor's axle. This upwards force yields another torque being applied to the rotor's spin axis, but this torque is not opposite in direction to the original torque and thus generates a precessional motion in the counterclockwise direction, as illustrated in Part $B$ of the Figure.

Let's think about the qualitative motion of the iPhone's housing when the vibration ringer is on. If the iPhone were freely floating in outer space without a gravity source, then by Conservation of Angular Momentum when the vibration cam is rotating the iPhone's housing must be slowly rotating in the opposite direction to compensate for the cam's rotation. Likewise, the Center of Mass of the iPhone must be stationary by Conservation of Linear Momentum as the cam rotates. Since the mass of the eccentric cam moves in a circular motion as the cam spins, then the phone must also move in a circular motion to compensate for the cam and keep the Center of Mass fixed. Let's assume that there is something that restricts the iPhone housing from rotating, such as attached thin wires or a nearby surface.

This shows a schematic diagram of the motion of an iPhone 5 executing the Cycloramic.app hands-off'' $360\degree$ video application as well as a qualitative plot of the forces applied to the vibration motor's rotor by the housing of the iPhone. In Part $C$, the motion of the iPhone 5 is shown. If there were no marble surface, the iPhone body would execture a circular motion, but the marble surface intervenes causing the motion to remain above its surface. In Part $D$ the force, or torque, on the axis of the eccentric cam produced by the force of gravity and the force exerted by the marble slab when the iPhone is in contact with it is plotted. See the text for an explanation.

Next we assume that the iPhone is now standing'' upright on a hard surface. The diagram drawn in Part $C$ of the above Figure illustrates the circular motion of the iPhone's housing. For clarity, we describe this motion using the clockface numbering. As the housing moves from the $9$ o'clock position to the $3$ o'clock position, the torque on the cam is vertically down producing a clockwise precessional motion. But the net downward torque stems from the force of gravity acting on the phone. This is a relatively small torque leading to a relatively slow clockwise precession by the above Precession Rate Equation. Now without the hard surface, the housing would continue on its circular path. But because of the hard surface, the bottom edge of the iPhone's housing strikes the surface. The surface does not allow the housing to continue on its original circular path, rather it truncates the motion, keeping the housing always above the circular path, as shown in the red path drawn from the $3$ o'clock position around to the $9$ o'clock position. The hard surface must apply an upward force strong enough to keep the iPhone housing from entering'' the surface. This force is much greater than the gravity force, $\vert\bsym{F_{\uparrow}}\vert\gg\vert\bsym{F_{\downarrow}}\vert$, and, as such, it produces a much greater torque on the cam's axle and corresponding a much faster counterclockwise precessional motion. In words, the iPhone precesses faster in the counterclockwise direction while the iPhone strikes the hard surface as it follows its red path ($B$) than its slower clockwise precession while the iPhone is in the air following its blue path ($A$). [In more prosaic language, it is not the fall that kills you, rather it is the quick stop at the end'.] Part $D$ shows a qualitative plot of the forces, and thus the torques, being applied to the spin axle of the vibration motor as it rotates. We see that when the housing is in contact with the hard surface, the upwards force $\bsym{F_{\uparrow}}$ is quite large (red curve). While when the housing is floating above the surface the downwards force $\bsym{F_{\downarrow}}$ is quite small (blue curve). Since the larger torque along the red curve produces a faster counterclockwise precession while the smaller torque along the blue curve yields a slower clockwise precession, the net motion is a rotation of the iPhone in a counterclockwise direction. In fact, since the iPhone is in contact at its right-hand corner during the blue portion, this contact may produce enough friction to overcome totally the clockwise precession, so the clockwise precession may not even be observed in practice.

This is a screenshot of the GarageBand window showing an audio recording of the Cycloramic iPhone 5 app vibrating the iPhone on a glass surface. Counting the number of bumps'', we find a frequency of slightly less than $440\unit[\/]{Hz}$.

We recorded the sound produced by the iPhone with its vibrational ringer turned on as it rotates standing'' upright using the microphone of a MacBook Pro laptop computer and the GarageBand application. Using the timebase of GarageBand, we were able to compute the rotational frequency of the vibrational motor as slightly less than $440\unit[\/]{Hz}$ (see the above Figure). This frequency is high enough that we could not video record any clockwise motion due to the blue curve since our video recording was only $30\unit[\/]{fps}$. In order to observe any clockwise precessional motion would require a much faster frames per second recording rate than what we have available to us.

In summary, the precessional motion of the vibrational ringer's motor and cam accounts for the iPhone's rotation using the Cycloramic app.

This is a photograph of a gyroscope balanced by a counterweight employed to model the vibration motor in the iPhone 5. After the gyroscope's rotor is spun up to speed in a clockwise direction (looking down the axle of the rotor from the opposite side as the balancing pivot point), which direction will the gyroscope precess when it is tipped up or down vertically?

To better understand and to demonstrate this mechanism empirically, let's consider the balanced gyroscope suspended in a gimbal mount and shown in the above Figure.

A schematic diagram for this device is drawn in the following Figure. The gyroscope's rotor spins in the direction shown, establishing an angular momentum represented by the $\bsym{L}_{\text{gyro}}$ vector. From the pivot point for the gyroscope's axle is a second vector, denoted by $\bsym{\Gamma}$, that represents the torques applied to the gyroscope's axle by the forces $\bsym{F_{\uparrow}}\!\!$ and $\bsym{F_{\downarrow}}$. It is these forces that model the forces on the eccentric cam stemming from the iPhone floating above and then striking the hard surface as the eccentric cam rotates.

Now, as we see from the plot in the above Figure, the iPhone spends the same amount of time experiencing a $\bsym{F_{\uparrow}}$ as a $\bsym{F_{\downarrow}}$ since the cam rotates uniformly. But because the iPhone's housing comes in contact with the hard surface for a portion of the cam rotation, say the portion between $4$ o'clock and $8$ o'clock, then $\bsym{F_{\uparrow}}$ is greater in magnitude during this time than the corresponding magnitude of $\bsym{F_{\downarrow}}$. This produces a greater torque,
which yields a faster change in the angular momentum $\bsym{L}_{\text{gyro}}$ in one direction and thus an overall net spin in the counterclockwise direction. Recall from the Torque Equation that
and therefore the greater torque leads to a faster precession rate and thus a net precession in one direction. Thus the Cycloramic.app spin is nothing more than the usual precessional motion of a gyroscope experiencing a net torque being applied to is axle.

Now you might ask just what keeps the iPhone in contact with the hard surface during its $4$ o'clock to $8$ o'clock portion? Well, it is the fact that the cam is swinging in its upward directed arc during this interval, and thus the iPhone housing must swing in its downward directed arc in order to conserve linear momentum. This keeps the iPhone in contact with the hard surface and keeps the the force, and thus the torque, continually applied to the eccentric cam's axle.

This is a schematic diagram of a spinning gyroscope counterbalanced by a weight and supported on a pivot point. Forces $\bsym{F_{\uparrow}}$ and $\bsym{F_{\downarrow}}$ mimic the forces applied to the eccentric cam of the vibration ringer of the iPhone. The larger torque, $\vert\bsym{\Gamma_{\uparrow}}\vert\gg\vert\bsym{\Gamma_{\downarrow}}\vert$, means that the red counterclockwise precession is greater than the blue reverse clockwise precession. See the text for details.

In the more detailed and thus longer video, Gyroscope_Cycloramic_App_Proposed_Mechanism-detailed_version-1080p.mov, the larger force $\bsym{F_{\uparrow}}$ is supplied by striking quickly the counterweight with the edge of a ruler while the smaller force $\bsym{F_{\downarrow}}$ is supplied by a slower strike of the counterweight with the ruler. This mimics the iPhone bouncing against the hard surface as its eccentric cam rotates. As the video clearly demonstrates, this difference in the forces leads to a predominantly counterclockwise spinning of the gyroscope, modeling the spinning of the iPhone when it is vibrating. Also shown in this video is that if the gyroscope's rotor is spinning in the opposite direction, then striking the counterweight with the ruler produces the opposite, or clockwise, precession of the gyroscope.

Now how about the other observations listed above? Well, let's consider the fact that this app does not work with the iPhone 4/4S, as described in Item (2.). Not only does the developer state that the Cycloramic.app does not work with the iPhone 4/4S, but a number of the YouTube videos and comments on those videos [see the citations in the PDF file] also demonstrate that this app does not spin the 4/4S iPhones. Why? Well, as we note in Item (4.), the iPhone 4/4S does not use the same physical apparatus for its vibration ringer as the iPhone 5. In particular, the 4 and 4S employ a vibration ringer that moves a mass along a line instead of in a circle. Since the 4/4S vibration ringer is not spinning a mass, there is no gyroscopic precessional motion associated with this ringer. Hence there cannot be any spinning of the iPhone 4/4S when it vibrates in an upright standing'' position. Some individuals, and in particular the iTelekinect.app, state that the iPhone 4/4S does move, but the videos show that it does not ever complete a full revolution. This is important, for if it did complete a revolution then we would have to look for an alternative mechanism for the iPhone's rotation. The motion in the videos of the iPhone 4/4S show a rotation of less than $15\degree$. And this small rotation can easily be explained by an uneven surface. Basically, when the iPhone 4 is vibrating, its beating against the surface makes the interface between the iPhone and the surface be nearly frictionless, and thus any imperfections or irregularities in the surface allows the iPhone 4 to drift frictionlessly to a physically lower position, as described in the observation (5.). This drift could naturally involve a small rotation of the iPhone 4, but it would not involve a continuous rotation of the iPhone. Thus our proposed gyroscopic precessional motion as the physical mechanism of the spinning produced by the Cycloramic app is entirely consistent with its behavior on the iPhone 4/4S.

Now some have argued that the reason that Cycloramic.app spins the iPhone 5 but not the iPhone 4/4S is because the 4S is heavier than the 5. But this cannot be the reason because of the observation made in Item (6.) where the olloclip fisheye lens is attached to the iPhone 5 and it still spins are the same rate as it does without the lens attached. Looking up the technical specifications for the iPhones and the olloclip fisheye adapter, we see that the olloclip lens weighs $20\unit[\/]{g}$ while the iPhone 5 weighs $112\unit[\/]{g}$ and the iPhone 4 weighs $137\unit[\/]{g}$ with the 4S weighing $140\unit[\/]{g}$. Thus the iPhone 5 with the olloclip attached weighs nearly as much as both the iPhone 4 and 4S, but the extra weight does not stop or even slop the rotary motion as seen in the YouTube video [see the PDF file for the citations]. So the spinning of the slightly heavier iPhone 4/4S also should not be stopped simply because of its extra weight. Rather, we think that the iPhone 4/4S does not spin because its linear oscillating vibrator ringer cannot experience precessional motion like the vibration ringer motor of the iPhone 5.

In Item (7.) it is demonstrated that even when the iPhone 5 is resting on a very smooth surface, such as a Blu-ray disc box, the iPhone will not rotate. In addition, TheRykerDane demonstrates that even if a thin glass plate is used, the iPhone still will not rotate unless the glass plate is held firmly down. In other words, having a nearly frictionless surface is not the only criteria for spinning. The reason for this lack of spin on the thin glass plate can be explained by the fact that the glass plate is small and has a relatively smaller mass. Thus as the iPhone vibrates, the thin glass plate vibrates with it. This lessens the deceleration of the iPhone caused by it striking the glass plate. A lower deceleration means a smaller force, and a smaller force means less torque on the vibration motor's axle. Less torque means less precession thus explaining the lack of spinning. When the thin glass plate is held firmly down, then the glass plate does not vibrate with the iPhone and thus it serves as a hard surface yielding a fast deceleration, higher force, greater torque, and thus larger precessional motion. So, once again, our proposed gyroscopic precession mechanism explains this observation.

And lastly, how does our mechanism explain the Tape Hack'' listed in Item (8.)? All of the demonstrations and explanations from the developer indicate that as smooth of a surface as possible is needed for the app to spin the iPhone 5. The hard and smooth aluminum surface on the bottom edge of the iPhone is certain smoother than four thicknesses of Scotch tape which are both less smooth and also softer than the aluminum bottom itself. So what is happening here? We suspect that this very small strip of tape is actually increasing slightly the friction with the table surface. Now friction has two forms, static and kinetic, with the coefficient of static friction being much greater than the coefficient of kinetic friction. So the tape on the bottom edge of the phone increases the coefficients of friction between the phone and table. During the $A$ phase of the iPhone's motion when the torque is smaller, the increased static friction keeps the iPhone stationary. In other words, the iPhone does not break free during this reverse precession'' phase. But during the $B$ when when the torque is considerable larger, the precession does overcome the static friction and the iPhone rotates in its counterclockwise direction. Because the clockwise rotation is inhibited by the tape's added static friction while the counterclockwise rotation breaks free and so is not inhibited by the much lower kinetic friction, the net result is that the iPhone rotates better with the tape hack than without it, even though this hack increases the friction between the iPhone and the table.

In conclusion, our gyroscopic precession mechanism explains all of the observations made to date of the actions of the Cycloramic.app, including the observations in Items (1.) to (8.).

## Alternative Mechanism Not Relying on Precession

Are there other possible mechanisms for the Cycloramic.app? As we have already mentioned, of course there are other possibilities, but we are particularly fond of the gyroscopic precession mechanism because precession is such a strong effect and would produce exactly the rotation motion displayed by the Cycloramic.app. But let's now consider alternative mechanisms.

By all indications, this app requires the hard surface to be both quite hard and immobile (it does not flex or move), but also quite smooth. Perhaps the smoothness is required in order to distinguish between static and kinetic friction (see the Nonconservative Systems: Friction'' Subsection 43.3.4.2 starting on page 4064 of the Classical Mechanics Chapter for a full discussion of the distinction between static and kinetic friciton).

The basic idea is that because of the rotary motion of the eccentric cam, the downward force of the iPhone on the hard surface varies. If this variation in downwards force takes the iPhone from the domain of static friction to the domain of kinetic friction, then the iPhone might move by repeatedly switching between static and kinetic friction. All that is needed is a delay in the switch between static and kinetic friction in one direction compared to a faster switch in the opposite direction. This would produce a net motion in one direction. This net movement, if there is a separate pivot point that attempts to fix one point of the bottom edge to the hard surface, then converts into a rotary motion.

Let's see how this static/kinetic friction concept operates first, and then apply it to the iPhone.

In Part $A$ of this figure, the static friction opposes and exactly cancels the tension in the rope: $F_{\text{s}}=T=w$. The mass $m$ does not move. In $B$ the maximum static friction, $F_{\text{s,max}}=\mu_{\text{s}}N$ has been reached ($F_{\text{s,max}}=T=\omega$), the object is on the verge of moving. In Part $C$ we have $T=W>F_{\text{s,max}}$ and thus the mass $m$ begins to slide. Now the friction reduces to the sliding friction $F_{\text{k}}=\mu_{\text{k}}N$. [Notice that the resultant of the normal force $N$ and the frictional force $F$ must pass through the $CM$ of the mass $m$ since the tension $T$ also passes through the $CM$.]

There is a maximum beyond which static friction cannot go. This maximum is given by
where $F_{\text{s,max}}$ is the maximum static frictional force, $N$ is the normal force holding the two surfaces together, and $\mu_{\text{s}}$ is called the coefficient of static friction. The coefficient is a constant that depends upon the physical characteristics of the two surfaces, such as from what material they are composed, how rough the surfaces are, and whether there is any lubricant present.

Thus we see that the static friction is always less than or equal to this maximum,
Part $B$ of the above Figure shows this maximum static friction situation where $F_{\text{s,max}}=T=\omega$. The mass $m$ is on the verge of moving, and any increase in the tension will produce motion of the mass.

This figure illustrates the idealized characteristic nature of frictional forces. In the left-hand graph, we see that initially the frictional force increases linearly with the external tension pulling the body. During this interval the body remains at rest since the friction force exactly cancels the external force. Once the frictional force reaches its maximum at $F_{\text{s,max}}=\mu_{\text{s}}N$, the body begins to slide and the sliding, or kinetic, friction takes over: $F_{\text{k}}=\mu_{\text{k}}N$. The right-hand graph shows the net force acting on a body as the external force is increased slowly.

Once the two bodies begin to slide past one another, the frictional force decreases, the static friction disappears and is replaced by a sliding friction. And the sliding frictional force is given by
where $F_{\text{k}}$ is the sliding, or kinetic friction and $\mu_{\text{k}}$ is called the coefficient of sliding, or kinetic, friction and is a different constant that depends upon the physical characteristics of the two surfaces. In general (but not always),

[Also notice that the coefficients of friction are dimensionless quantities.]

Part $C$ of the sliding block Figure illustrates the sliding friction. Now $F_{\text{k}}< T = W$ and so there is a net force on the mass $m$ which produces an acceleration of $m$ to the right. The mass moves off to the right, in other words.

This Figure shows eight schematic side cutaway diagrams of an iPhone 5 executing the Cycloramic.app hands-off'' $360\degree$ video application. Each diagram represents one snapshot of the repetitive motion of the eccentric cam $R$, that is, ${a}\rightarrow {b}\rightarrow {c}\rightarrow {d}\rightarrow {e}\rightarrow {f}\rightarrow {g}\rightarrow {h}\rightarrow {a}\rightarrow$. Even though the iPhone is resting on the marble surface, for clarity we have drawn the marble surface slightly below the bottom surface of the iPhone. See the text for a full explanation.

So, in summary, if the normal force holding two surfaces together is greater, then the static friction force before the surfaces break apart'' is also greater. Once the break'' occurs, and the system converts to the much smaller kinetic friction, then the two surfaces can move relative to one another.

If we apply this to the iPhone sitting on its bottom edge, and viewed from the side as a cutaway, as shown in the above drawing, we see the eccentric cam drawn in a revolving schematic view $R$. Consider that the iPhone, in an effort to conserve its momentum, has a force imposed opposite in direction to that of the cam. The horizontal component of this force is drawn as the magenta vectors $\bsym{F_R}$. The normal force applied to the bottom of the iPhone by the marble surface is denoted as a blue $\bsym{F_{<}}$, a green $\bsym{F_{mg}}$, or a red $\bsym{F_{>}}$. The vector $\bsym{F_{mg}}$ denotes the normal force due simply to gravity on the bottom surface of the iPhone. Because of the motion of the cam, $\bsym{F_{<}}<\bsym{F_{mg}}<\bsym{F_{>}}$. These schematic diagrams are ordered in a repetitive cycle,${a}\rightarrow {b}\rightarrow {c}\rightarrow {d}\rightarrow {e}\rightarrow {f}\rightarrow {g}\rightarrow {h}\rightarrow {a}\rightarrow$. Let's assume for a moment that at the point in the rotation of the cam shown in Part $a$, the iPhone breaks away'' from the marble surface, that is, this is the point where the surfaces switch from static friction to kinetic friction because the horizontal force applied by the iPhone's housing is greater than that supported by static friction. The iPhone thus begins to move to the left. Almost immediately, the cam is in the position illustrated in Part $b$, and immediately after this the iPhone's housing moves to the right. It continues moving to the right through Parts $c$ and $e$ where the net movement is the greatest. Between Part $e$ and $f$ the normal force $\bsym{F_{>}}$ increases allowing the friction between the bottom of the iPhone and the surface to increase. The iPhone stops is movement because of this increased friction at the point shown in Part $f$. The iPhone stays fixed in place by static friction from Part $f$ through $h$ and on to Part $a$ where once again it breaks loose because of the decreasing normal force between the two surfaces. As you can see, this process of staying fixed in place during a phase of higher normal force and thus greater static friction and then releasing and sliding under kinetic friction during a phase of less normal force produces a net movement of the iPhone to the right. Since the iPhone 5's rotary virbratory motor and cam are off-center, towards the left-hand side of the iPhone as shown in the earlier Figure, then this motion in conjuction with a point on the iPhone's bottom edge that acts like a pivot point causes the iPhone to spin around this pivot point.

This Alternative Mechanism, by the way, may provide an explanation as to why the micro tripod' tape hack, the observation in Item (8.), where the tiny piece of tape added to the bottom of the iPhone, works because the tape acts as a pivot point around which the iPhone can more easily rotate.

While this Alternative Mechanism is potentially a viable mechanism, we do not like it as well as the gyroscopic precession because it relies on a relative fine tuning of the static and kinetic coefficients of friction. At the same time, since it appears that the operation of the Cycloramic app is actually quite sensitive to the physical characteristics of the hard surface, then perhaps indeed what is required is for the surface's static and kinetic coefficients of friction be such as to allow the mechanism shown in the above Figure. Gyroscopic precession, on the other hand, is propelled by a relatively strong force and it always occurs whenever a body is spinning and a torque is being applied to the spin axis. Thus we believe that the more likely mechanism is the gyroscopic precession mechanism, but we put forward this alternative proposal if the precession mechanism is invalidated by future experimental evidence.

## Solution PDF file:

Here is the PDF file giving a full derivation of Proposed Mechanism:
Here is the PDF file for the Bibliography of the Classical Mechanics Chapter (this Bibliography contains the citations for our Observations on the Cycloramic App):

The following videos demonstrate the Cycloramic app's spinning of the iPhone, describes the gyroscopic precession mechanism for the iPhone, and employs a laboratory gyroscope to demonstrate this mechanism in a physical experiment, including the fast that the precession is faster in one direction than the other.

The following video (Gyroscope_Cycloramic_App_Proposed_Mechanism-detailed_version-1080p.mov) explains and demonstrates the Classical Mechanics Explanation for the spinning motions produced by the Cycloramic app. This explanation predicts, both qualitatively and quantitatively, the behaviors when the Cycloramic.app is run.
Click the following link to view the short version of the video on YouTube:
http://youtu.be/slOpxFwgLx0

The following video (Gyroscope_Cycloramic_App_Proposed_Mechanism-detailed_version-1080p.mov) explains and demonstrates the Classical Mechanics Explanation for the spinning motions produced by the Cycloramic app. This explanation predicts, both qualitatively and quantitatively, the behaviors when the Cycloramic.app is run. This video contains a more detailed explanation than the previous one.

Click the following link to view the detailed version of the video on YouTube: http://youtu.be/86nOE_HYEfQ

(YouTube encodes the video at a number of different resolutions.)
Description of the video's experiments: This video discusses the Classical Mechanics Explanation for the proposed mechanism (gyroscopic precession) and demonstrates the nutational and precessional behaviors of a gyroscope.

## Notes:

Note 1: We have been unable to find any explanation or discussion of a physical mechanism behind the spinning produced by the Cycloramic or the iTelekinect apps. [Mike Schramm, in his Tuaw.com review, states But somebody out there discovered that when the iPhone 5 vibrates while standing up, it spins around at a steady rate, and thus you have Cycloramic.'', implying that this was a serendipitous discovery. It may be that the developers themselves do not fully understand the physical mechanism behind their app. For instance, Rostami, the developer of the iTelekinect.app, in one of his videos says that he discovered' the rotation of his iPhone 5 when it was on its vibration ringer.] Thus we put forward our gyroscopic precession mechanism that explains all of the observations obtained so far. Further experiments with high-speed video equipment might be employed to further verify our mechanism, but we do not have such equipment available to us.

Note 2: From the direction of spinning of the iPhone, we can ascertain from the simple Newton's Law explanation the direction of the rotation of the vibration motor's cam.

Note 3: Is it possible that our proposed mechanism is not the operative one? Yes, of course, this is possible. But a few things are working in our favor. Firstly, our gyroscopic precessional mechanism explains all of the current observations of the Cycloramic.app, including the fact that it works on the iPhone 5 but not the iPhone 4/4S and it requires not only a smooth surface but also a hard surface. Secondly, gyroscopic precession is a physically valid argument. Thirdly, the mechanism does not rely on the fine structure of the bottom of the iPhone 5, that is, it is not due to the orientation'' of grooves or filaments in the aluminum bottom surface. This we know because of the Tape hack!'' to improve the rotary performance of the iPhone on softer surfaces. Fourthly, gyroscopic precession is a strong effect, in the sense that the resulting force is quite large. As we have discussed in the Classical Mechanics Chapter's section on Precession and Nutation, the gyroscopic action essentially turns'' the torque force $90\degree$ so that a vertical force becomes a horizontal force because of gyroscopic precession. For our proposed precessional mechanism, further evidence will either verify or refute our proposed mechanism. For instance, if a Cycloramic.app is developed that fully rotates the iPhone 4/4S, then we will have to reconsider our proposed mechanism. Also, if the iPhone does not rock'' due to the rotation of the eccentric cam, then we would have to modify our proposal. [In an attempt to observe the iPhone's rocking'' motion, we shown a flashlight from behind the iPhone as it was rotating, but we could not see' any light passing underneath the iPhone so we concluded that our light source was not strong enough to visually observe since we are fairly certain that the iPhone is indeed bouncing'' on the hard surface from the enhanced sounds that it produces when vibrating. We shall repeat this experiment but with a laser to see if we can visualize the laser beam underneath the iPhone and ascertain how the iPhone moves or rocks'' when it is rotating.]

Note 4 (added January 26, 2013): We see that the Cycloramic.app was updated from version 1.0 to version 2.0 on January 24, 2013. While the new version adds an iPhone 4/4S mode, it is for guided'' panoramic pictures. We believe that this means that the Cycloramic.app still does not rotate the iPhone 4/4S, rather the new guided'' mode must somehow help guide the user when manually rotating the iPhone 4/4S into correct positions for taking panoramic photographs. In other words, the new version does not provide the automatic rotation for the iPhone 4/4S as it does for the iPhone 5. Therefore our proposed mechanism is still safe -- the new version 2.0 does not invalidate our proposed mechanism. Our mechanism still stands.

Note 5 (added January 27, 2013): We see that the developer of the Cycloramic App (CycloramicApp) has left a comment on Sandra's proposed mechanism YouTube video (short version) on January 26, 2013 stating, Great explanation, thank you!''

Note 6 (added January 27, 2013): Now we see that the developer of the Cycloramic App (CycloramicApp) has replied on his/her Cycloramic v2 video to a request for comment on Sandra's Proposed Mechanism YouTube video on January 27, 2013 stating, I think she did an excellent job!!''

Note 7 (added January 27, 2013): The developer of the Cycloramic App (CycloramicApp) left a comment on this wiki page saying that we did an excellent job with our proposed mechanism.

Note 8 (added February 2, 2013): The developer of the Cycloramic App (CycloramicApp) now appears to have been partially successful with what he calls a micro tripod' in getting the iPhone 4S to rotate. The rotation is not nearly as strong as it is for the iPhone 5. We also note that the micro tripod' is in the center of the bottom edge. Either this result invalidates our gyroscopic precession mechanism, or the iPhone 4S rotation is from a different mechanism. Since the micro tripod in the center of the bottom edge would allow the iPhone 4S to see-saw from one corner touching the hard surface to the other corner touching, it may be that through a see-sawing and tipping motion of the iPhone 4S, first one corner's edge and then the other corner's opposite edge is striking the surface leading to an unsymmetric (chiral) repetitive striking that produces the rotation. With this result, more empirical evidence is needed for both the iPhone 5 and the iPhone 4S.

A further experiment that would be very interesting is if reversing the direction of the eccentric cam rotation in the iPhone 5 also reverses the direction of the rotation? And, even better, would be to know which direction the eccentric cam is rotating and whether one corner of the iPhone 5 is always in contact with the hard surface while the opposite corner under the vibrator motor bounces? These data would help validate or invalidate the gyroscopic precession mechanism.

Note 9 (added March 15, 2013): On January 28, 2013, in responding to Bruno F.'s request that he left on my funphysicsfacts.dyndns-wiki.com website's webpage for the Proposed Mechanism of the Cycloramic app, I attached a PDF file of this Proposed Mechanism of the Cycloramic app for the iPhone 5'' Example and sent it to the app's developer. In mid-March the Cycloramic App developer verified that our proposed mechanism was valid, or, at least, it agrees with their theory on how the App works. In particular, on March 12, 2013, Bruno Francois of Egos Ventures Inc. (developer of the Cycloramic App) replied in an email to me, ... sorry it took so long to reply, we had not got a chance to thouroughly review you[sic] paper. Overall it does make sense and agrees with our theory. Thank you. I will keep you updated.'', while on March 15, 2013 Bruno Fran\c{c}ois added in another email the comment, All our team here is very impressed with your daughter's work!''

Important: We should also mention that there is some potential confusion over what are the vibration motors in iPhone 4s, for example, iFixit.com states in their teardown of the iPhone 4S that Out comes the vibrator motor. It appears that Apple elected to go with the linear oscillating vibrator that we found in the Verizon iPhone 4 as opposed to the rotational electric motor with counterweight in the AT\&T version.'' This means that any particular iPhone 4 may have a linear motor or a rotary motor as its vibration ringer, depending upon which version of the iPhone 4 it is. This may explain the apparent contradictory comments in YouTube videos, with some individuals claiming that the iPhone 4/4S does not rotate (the linear vibrator version?) while others claiming that the iPhone 4/4S does rotate (the rotary vibrator version?) some. So, the data so far is still not conclusive, and we await further empirical evidence. What we really need is high resolution, high speed video to record the actual motion that the iPhone undergoes as it rotates on a hard surface using the Cycloramic app.

This Figure shows eight schematic side cutaway diagrams of an iPhone 4/4S executing the Cycloramic.app hands-off'' $360\degree$ video application. Each diagram represents one snapshot of the repetitive motion of the linear vibratory motor $R$, that is, ${a}\rightarrow {b}\rightarrow {c}\rightarrow {d}\rightarrow {e}\rightarrow {f}\rightarrow {g}\rightarrow {h}\rightarrow {a}\rightarrow$. Even though the iPhone is resting on the marble surface, for clarity we have drawn the marble surface slightly below the bottom surface of the iPhone. See the text for a full explanation.

While the gyroscopic precession mechanism is ruled out if there is no rotary motion in the vibratory ringer of the iPhone, we notice that the Alternative Mechanism Not Relying on Precession'' starting on page 4338 is still potentially a valid mechanism, under one condition. If the actual axis of the linear vibration is tilted slightly away from being completely horizontal, then the linear motion of the vibrating mass will cause the normal force between the iPhone's bottom and the hard surface to fluctuate in a fashion similar to that discussed in the previous Alternative Mechanism where the rotary motion of the eccentric cam caused this normal force between the surfaces to vary. If the normal force varies, then once again there is the possibility that when the normal force is less then static friction is too small and is overcome by the horizontal force caused by conservation of momentum. The iPhone's housing thus slides in this case. But when the linear vibratory mass is moving in the other direction, the normal force is increased allowing the static friction to rise also. This will allow a delay in the breaking away'' of the static friction to the situation of kinetic friction caused by the horizontal force. This delay produces a net movement preferentially in one direction. If there is a pivot point, including an artificial pivot produced by a micro tripod' of tape, then this movement will yield a rotational motion of the iPhone.

The above Figure shows the corresponding diagrams and argument for how the linear vibratory motor of the iPhone 4/4S might produce a rotary motion. See the discussion in the Alternative Mechanism Not Relying on Precession'' for details.

Once again, while this is a viable mechanism under the current experimental facts, it is our belief that the gyroscopic precession mechanism is still the most viable one because precession is produced by a relatively strong force and it occurs at all times when a body is spinning and a torque is being applied to the spin axis. However, if future empirical evidence invalidates the precessional mechanism, then this static versus kinetic friction mechanism will still be a viable alternative proposal.

This Proposed Mechanism of the Cycloramic app for the iPhone 5' Example should be cited as `Sandra N. Shaefer and Craig G. Shaefer, Math for the Motivated: Classical Mechanics: Precession: Proposed Mechanism of the Cycloramic App for the iPhone 5, 2013 ed., 4433-4450, (self-published), Rapid City, SD, 2013.