Rotation of a ring around a horizontal rod

A simple experiment is described where a metal ring was rotated in a vertical plane on a horizontal rod. The ring rotated about 100 times before coming to a stop, so the friction force on the ring remained very small. However, measurements of the rotation frequencies of the ring around the rod and around its centre of mass indicated that the ring was sliding rather than rolling, with an unusually low coefficient of sliding friction.


Introduction
Spinning objects such as an egg [1,2] or a spinning disk [3,4] provide interesting demonstrations of precession in the classroom but can present a challenge for undergraduate students since theoretical explanations are usually quite advanced and experimental measurements can be difficult.A simplified spinning experiment is described in the present paper where a thin metal ring is spun in a vertical plane on a horizontal metal rod.The centre of mass of the ring rises and falls during each revolution, so the rotation speed of the ring varies with time during during each revolution.As a first approximation, the rotation speed can be calculated by assuming that the total energy is conserved.The basic physics is similar to that in a loop-theloop experiment where a ball can reach the top of the loop if it has sufficient speed at the bottom of the loop [5,6].In the present experiment, the ring rotated about 100 times around the rod before coming to a stop, implying that the friction force acting on the ring was very small.
The main challenge in the present experiment is to account for the fact that the ring does not actually roll around the rod.Measurements show that contact points on the ring do not remain at rest but slide on the rod as the ring rises and falls.In that case, it might be expected that sliding friction should bring the ring to a rapid stop.In fact the ring can continue to rotate for 20 s or more, depending on the initial rotation speed and the coefficient of friction.A typical experimental result is shown in supplementary video Ring.movwhere a 68 mm diameter aluminium bracelet was spun by hand on a 4 mm diameter steel rod.The centre of mass of the ring rotates in a circular path around the rod and the ring itself rotates at a different frequency around its centre of mass.
The geometry is shown in figure 1.The centre of mass of the ring, G, rotates in a circular path of radius R at angular velocity Ω = dθ/dt, where θ is the angular displacement of G from its lowest point directly below the rod.The linear velocity of G is v G = RΩ.The rod has radius r.The ring rotates about a horizontal axis through G at angular velocity ω which varies with time due to the torque acting on the ring.The total force on the ring includes its weight, mg, a tangential friction force, F, acting at the contact point, and the normal reaction force, N, exerted on the ring by the rod.The assumed positive directions of Ω, ω, F and N are as shown in figure 1.

Equations of motion
As the ring rises, v G decreases with time according to the relation while the centripetal force on the ring is given by both equations expressing Newton's second law.The torque acting about the centre of mass is given by  ( ) q q =which is the same as the relation for a simple, undamped pendulum of length L = 2R.However, measurements of Ω and ω with metal rings on a metal rod indicate that the rolling condition is satisfied only for brief instants of time during each revolution of the ring.Numerical solutions of equations (1)-( 3) were therefore obtained by assuming that F = ±μ k N, the sign of μ k depending on the sliding direction, where μ k is the coefficient of sliding friction.The numerical solutions were obtained using a predictor-corrector method with suitably small time steps.The contact point, P, on the ring slides on the rod at speed v P = (R + r)ω − RΩ.The sign of v P was monitored to determine the sign of μ k at each integration step, and the value of μ k was chosen to give a good fit to the experimental data.When the magnitude of v P decreased to a value less than 0.005 m s −1 then rapid changes in the sign of F were needed to maintain v P at a very small value, indicating that the ring actually rolled without sliding for a short time during each rotation cycle.
If the ring slides, then equations (1) and (2) which is similar to equation (4), apart from (a) a missing factor of 2 in the first term and (b) the additional damping term which changes sign depending on the sliding direction.The sign change results in nonlinear behaviour of the ring as it rotates around the rod.

Experimental results
Three different rings were spun on three different rods to compare the results.One of the rings was the 68 mm diameter aluminium bracelet, and the other two were stainless steel rings with inner diameters of 28 mm and 41 mm.The three rods were a 2 mm diameter steel rod, a 4 mm diameter steel rod and a 9 mm diameter copper tube.Each ring was filmed end-on at 300 frames s −1 and analysed with Tracker motion analysis software to measure Ω and ω versus time, by digitising marks on each ring at diametrically opposite points.Typical results for the 28 mm ring on the 2 mm steel rod are given in figure 2, showing the last seven revolutions before the ring lost contact with the rod.Each of the rings on each of the rods behaved in a similar manner, indicating that the Ω/ω ratio varied with time during each revolution, both at the start and at the end of the spinning phase.Only one of the rings on one of the rods is presented in this paper to avoid repetition.Additional results are presented in [7].
If the ring in figure 2 rolled without slipping then Ω/ω = 1 + r/R = 1.077.In fact, the Ω/ω ratio varied from about 0.8-1.2during each cycle, indicating that the ring was sliding on the rod most of the time.Otherwise, the ring behaved as expected, with Ω increasing to a maximum at the lowest point of the ring and decreasing to a minimum at the highest point.Eventually, the normal reaction force, N, decreased to zero and then the ring lost contact with the rod.
The trajectories of two different points on a ring are shown in figure 3 for the bracelet rotating on a copper tube.The two points approach the tube at an angle of about 60°to the normal, and also move away from the tube at an angle of about 60°to the normal.That result differs from the behaviour of contacts points on the circumference of a ball or wheel that rolls without sliding on a horizontal surface, which approach and leave the surface in the normal direction.In the latter case, those points do not have a tangential velocity component so there is no sliding motion.In the present case, each contact point approaches the surface with a tangential velocity component and also departs with a similar tangential velocity component.Consequently, contact points on the bracelet, and the other rings, slide on the surface rather than roll without sliding.A numerical solution of equations (1)-( 3) for the ring in figure 2 is shown in figure 4, assuming that v G = 0.75 m s −1 and ω = 53 rad s −1 when the ring is at its lowest point, and that μ k = ±0.1.The solution is in good agreement with the experimental results in figure 2, showing that Ω varies with time from about 35 to about 55 rad s −1 as the ring rises and falls, and that Ω/ω varies from about 0.8 to 1.2 during each cycle, rather than maintaining a rolling without slipping condition where Ω/ω = 1.077.The oscillation periods of Ω and ω are the same, but Ω and ω are out of phase, Ω being a maximum before ω is a maximum.Consequently, the Ω/ω ratio varies with time but with an average value of about 1.077.The numerical solution also shows that the minimum value of Ω decreases after each cycle, as observed, due to frictional energy losses.
The numerical solution indicates that there is a short period of time where v P = 0 and the ring rolls without slipping when it reaches its lowest point.The rolling stage is not evident in the experimental data in figure 2(b) which indicates that the ring slides the whole time, either forwards or backwards depending on the magnitude of Ω/ω and hence on the sign of v P .The difference might arise if μ k varies with sliding speed, since the numerical solution shows that the duration of the rolling phase decreases when μ k decreases.However, the Ω/ω ratio also decreases when μ k decreases.On the other hand, numerical solutions indicate that rolling with Ω/ω = 1.077 ± 0.003 persists for each whole cycle if μ k > 0.2, which is a more commonly observed value of the coefficient of sliding friction for metal surfaces, but it is not consistent with the experimental results.
Qualitative features of the numerical solution are indicated in figure 5, showing the positions of the ring at which sliding commences or reverses direction.Starting at t = 0 when the ring is at its lowest position and where v G and Ω are a maximum, the ring rises upwards, with a large decrease in Ω.There is a smaller decrease in ω since the rate of change of ω depends on the friction force which is proportional to N which decreases as the ring rises.The ratio Ω/ω therefore decreases so the ring starts sliding with v P > 0 and F < 0. After the ring passes its highest point, Ω increases and then the friction force reverses direction, with the result that ω also increases.An additional effect introduced by sliding is that the ring took about 10% longer to rise from its lowest to its highest position than to fall from its highest to its lowest position.The results in figure 5 were obtained by plotting F/N and v P versus θ, as shown in figure 6.

Conclusion
The experiment described in the present paper would be suitable as either a student experiment or a lecture demonstration to complement the well known loop-the-loop experiment.The main features can be explained in terms of energy conservation, but the details involve the fact that the rings were observed to slide around each rod rather than roll without sliding.The coefficient of sliding friction was found to be relatively small, allowing the rings to rotate many times around each rod before coming to a stop.The results indicate that each contact point on the ring slides during the brief period that the contact points remain in contact with the stationary rod since the contact points contact the rod with a non-zero tangential velocity component.The sliding friction force reverses direction and changes magnitude during each  revolution of the ring, with the result that the average friction force during each complete cycle is relatively small.

Figure 1 .
Figure 1.Geometry of a ring rotating around a horizontal rod of radius r.
the thickness of the ring is much smaller than R. If the ring rolls without slipping then the contact point remains at rest provided that v G = RΩ = (R + r)ω.In that case, equations (1)-(3) can be combined to show that

Figure 2 .
Figure 2. Experimental results showing (a) versus t and (b) Ω/ω versus t for the 28 mm ring on the 2 mm rod, just before the ring lost contact with the rod.

Figure 3 .
Figure 3. Experimental results showing the trajectory of two different points marked on the bracelet as they approach and leave the copper tube, at two different times during a given rotation cycle.The data points were digitised at equal time intervals of 6.67 ms.

Figure 4 .
Figure 4. Numerical solutions of equations (1)-(3) showing (a) Ω, ω and Ω/ω versus t and (b) v P and F versus t for the 28 mm ring on the 2 mm rod.

Figure 5 .
Figure 5. Qualitative features of the calculations in figure4, indicating the positions of the ring when v P and F reverse sign.The ring starts rolling without slipping when it arrives at its lowest point at θ = 0, then slides forward from θ = 60°to θ = 236°, and then slides backwards from θ = 236°to θ = 360°.

Figure 6 .
Figure 6.Numerical solutions in figure 4 plotted to show F/N and v P versus θ.The ring is at its lowest position when θ = 0.