European Journal of Physics

PAPER Related content

- Effects of rolling friction on a spinning coin Dynamics of a spherical tippe top or disk Rod Cross

To cite this article: Rod Cross 2018 Eur. J. Phys. 39 035001 - Spinning eggs and ballerinas Rod Cross

- Experimenting with a spinning disk Rod Cross

View the article online for updates and enhancements. Recent citations

- A hemispherical tippe top Rod Cross

This content was downloaded from IP address 200.130.19.152 on 22/04/2021 at 19:35 European Journal of Physics Eur. J. Phys. 39 (2018) 035001 (10pp) https://doi.org/10.1088/1361-6404/aaa34f

Dynamics of a spherical tippe top

Rod Cross

School of Physics, University of Sydney, Sydney, Australia

E-mail: [email protected]

Received 9 November 2017, revised 7 December 2017 Accepted for publication 20 December 2017 Published 9 March 2018

Abstract Experimental and theoretical results are presented concerning the inversion of a spherical tippe top. It was found that the top rises quickly while it is sliding and then more slowly when it starts rolling, in a manner similar to that observed previously with a spinning egg. As the top rises it rotates about the horizontal Y axis, an effect that is closely analogous to rotation of the top about the vertical Z axis. Both effects can be described in terms of precession about the respective axes. Steady precession about the Z axis arises from the normal reaction force in the Z direction, while precession about the Y axis arises from the friction force in the Y direction.

Supplementary material for this article is available online

Keywords: precession, sliding friction, angular momentum

(Some figures may appear in colour only in the online journal)

1. Introduction

A tippe top is a well known physics toy that has been studied for many years [1–17]. The most familiar type is a hollow, truncated sphere with a short peg attached so that it can be spun between the thumb and index finger. The centre of mass of the top is offset from the geometric centre of the sphere. When placed gently on a horizontal surface, the top comes to rest with its spherical base in contact with the surface and with the peg pointing vertically upwards. When spun on the surface at sufficient speed, the centre of mass rises so that the top inverts and ends up spinning on its peg. It is generally accepted that inversion of a tippe top is due to the torque arising from sliding friction between the top and the horizontal surface. A relatively simple description of the effect of the friction force is given by Pliskin [3] and is supported by numerical calcu- lations presented by Cohen [9]. Featonby [17] devised a simple rule of thumb to help physics teachers explain the mystery of the tippe top to students.

0143-0807/18/035001+10$33.00 © 2018 European Physical Society Printed in the UK 1 Eur. J. Phys. 39 (2018) 035001 R Cross

Figure 1. Diagrams showing (a) an egg and (b) a spherical tippe top, each spinning at angular velocity, ω, about an axis of symmetry and precessing at angular velocity, Ω, around a vertical axis through the centre of mass, G. The small mass m inside the tippe top shifts the centre of mass away from the geometric centre of the sphere, C. The horizontal friction force, F, acts into the page in (a) and out of the page in (b) since Ω is initially much larger than ω. Precession about the Y axis causes θ to decrease for the egg and to increase for the tippe top. In both cases, G rises.

The rise of a spinning egg is a similar effect and is also due to sliding friction between the egg and the surface on which it slides. It was recently shown [18] that the horizontal friction force on a spinning egg results in precession about the horizontal axis, in the same way that the vertical reaction force is responsible for steady precession about the vertical axis. The precession frequencies about the vertical and horizontal axes are governed by equations of the same form, each describing the fact that the relevant torque about the centre of mass is equal to the rate of change of angular momentum. Consequently, the rise of a spinning egg can be explained simply as precession about the horizontal axis. If an egg is not spun fast enough, it will not rise to a completely vertical position. In the latter case, sliding friction gives way to rolling friction before the egg rises to a vertical position, so the torque about the horizontal axis decreases to zero and precession about the horizontal axis ceases. The transition from sliding to rolling occurs sooner if the egg slides at low speed, in the same way that a golf or billiard ball projected without spin on a horizontal surface starts rolling after an initial sliding stage. A question therefore arises as to whether the inversion of a tippe top can also be explained in terms of precession about the horizontal axis. To investigate the problem, measurements were undertaken using a spherical tippe top [13, 14] constructed from a hollow sphere with a small mass attached to the inside surface to shift the centre of mass away from the centre of the sphere. The results were compared with theoretical calculations, showing that the inversion is indeed due to precession about the horizontal axis. Differences between a spinning egg and a spherical tippe top are compared in figure 1. Both rotate at angular velocity Ω about the vertical Z axis, and both rotate at angular velocity ω about an axis of symmetry. In both cases, Ω is initially much larger than ω, and in both

2 Eur. J. Phys. 39 (2018) 035001 R Cross cases the vertical spin axis passes through the centre of mass, G,ifΩ is relatively large. As a result, the contact point of the egg slides out of the page, generating a horizontal friction force, F, in the positive Y direction (into the page). The contact point of the tippe top slides into the page, generating a horizontal friction force in the negative Y direction (out of the page). The torque due to F acting about the Z axis results in a decrease in Ω with time. The torque due to F acting about the X axis is in the positive X direction for the egg, resulting in precession about the Y axis. The torque due to F acting about the X axis is in the negative X direction for the tippe top, resulting in precession about the Y axis in the opposite direction to that for the egg. As a result, ddq t is negative for the egg and the egg rises. For the tippe top, ddq t is positive and the centre of mass rises (as it does for the egg).

2. Theoretical model

Theoretical descriptions of the behaviour of spinning eggs and tippe tops have been presented in most of the listed references. We summarise here the main results of interest. The geometry is shown in figure 1, where the XYZ reference frame is assumed to rotate about the Z axis at angular velocity Ω. The principal moments of inertia about axes through G are taken as I1, I1 and I3. The angular velocity of the egg or the tippe top in figure 1 is given by dq w =++W+wqsin IJ()wqcos K , () 1 dt while the angular momentum is given by

LIJK=++LLLXY Z,2() 2 where LX =-Wsinqw()II33 1 cos q, LY = It1ddq and LZ =W()II1 sinqwq +33 cos , and where w3 =+Wwqcos . Euler’s angular momentum equation is given by ¶L +´=W LXP ´() NF +,3 () ¶t where XPis the vector distance from G (the centre-of-mass) to the contact point, N is the vertical reaction force at the contact point and F is the horizontal friction force at the contact point. In the XYZ coordinate system, XPP=-=()()XhNN, 0, , 0, 0, and F = (FX, FY,0), in which case equation (3) can be written in component form as

¶LX -WLFhYY = ,4() ¶t

¶LY +WLNXFhXPX =- - ,5() ¶t and

¶LZ = FXYP.6() ¶t The latter three equations indicate that the relevant torque components are each equal to the corresponding rate of change of angular momentum. The terms WLX and WLY correspond to the rate of change in the angular momentum components due to precession about the vertical axis. The partial derivatives indicate that the magnitude of each angular momentum component can also change with time.

3 Eur. J. Phys. 39 (2018) 035001 R Cross

If R is the radius of the sphere and if D is the distance from C to G then hRD=-cos q and XDP = sin q. The initial motion of a tippe top results in sliding of the contact point, in which case FY can be taken as mMg where μ is the coefficient of sliding friction and M is the mass of the top. However, a tippe top will commence to roll at a later stage if XrP W=-w where r = R sin q is the perpendicular distance from the contact point to the axis of sym- metry. Rolling will result if ω reverses sign during inversion, as it usually does. If the top starts rolling then μ will decrease to a small value and the inversion rate of the top will also decrease to a small value. If FY decreases to zero when the top starts rolling, then the top will ( ) precess at a constant rate given from equation 5 by W=-LMgXX P. From the above relations it is easy to show that steady precession results when w =-D W R in which case MgD W=2 ,7() ()II13-+cos q IDR 3 consistent with the experimental results presented below. If the initial spin of the tippe top is relatively low then the top will not completely invert. However, there is no rise at all if I13IDR<-1 since then the denominator of equation (7) is negative even when q = 0. Similarly, a rise to q = 180 is possible only if I13IDR<+1 . Provided that I13IDR>-1 , the centre of mass will rise until the condition given by equation (7) is satisfied, but if I13IDR>+1 then the top will not completely invert, regardless of how fast it is spun [10, 13, 16].

3. Experimental method

The experiment was performed using a spherical tippe top rather than a conventional tippe top with a peg since the surface remained spherical during the whole inversion. Contact of a peg with the surface adds to the complexity of the problem and has no effect on the initial inversion of the top. The top was constructed from a 31.0g, 58 mm diameter hollow stainless steel sphere, with a 12mm diameter hole cut in one side to insert a 10mm diameter, 4.08g steel ball on the diametrically opposite side. The ball was attached to the inner surface of the sphere with 3.7g of Blu-Tack, a pliable adhesive. As a result, the centre of mass was offset -5 2 by 5.0mm from the centre of the sphere. The calculated value of I1 was 2.05´ 10 kg m -5 2 and the calculated value of I3 was 1.76´ 10 kg m . Since I13I = 1.165 and 11.17+=DR 2, the top was marginally capable of complete inversion at high spin rates. The tippe top was set in motion on a smooth, horizontal table using a hand drill and a conical nylon brush to spin it at about 100 rad s−1 about the vertical axis while the added mass was located at the bottom of the sphere. The resulting motion was filmed at 300 frames s−1 with a Casio EX-F1 camera, using marks on the ball to determine its angular motion, viewing side-on in order to measure the angle of inclination of the tippe top. The video film was analysed with Tracker software to measure Ω, ω and θ as functions of time while the tippe top inverted. The angular velocity, ω,isdefined as the spin about the axis of symmetry observed in a coordinate system rotating about the vertical axis at angular velocity Ω. The video camera was not rotated but remained fixed. Consequently, the observed spin in the laboratory frame of reference is wLab =+Ww . The spin, ω, was determined by recording the position of a mark on the top only after the top rotated by one full precession cycle. In that manner, the fixed camera simulated a rotating camera recording only one frame each precession cycle.

4 Eur. J. Phys. 39 (2018) 035001 R Cross

Figure 2. Experimental results obtained with the tippe top showing Ω, ω and θ as functions of time.

4. Experimental results

A wide range of different results was not obtained in this experiment since the primary objective was to compare a typical result with theoretical expectations. The main interest was to compare the relative magnitudes of the three terms in equation (4), under conditions of experimental interest. A typical result obtained with the tippe top is shown in figure 2, and also in supplementary video TippeTop.mov available online at stacks.iop.org/EJP/39/ 035001/mmedia. The top was set spinning at W=92 rad s−1 and it inverted over about two seconds, but it did not completely invert. Complete inversion to q = 180 was observed only when the initial value of Ω was about 120 rad s−1 or larger. In figure 2, Ω decreased to 61 rad s−1, while ω reversed from an initial value of about 10 rad s−1 to about −9 rad s−1 over the two second inversion period. The data shown in figure 2 for ω and Ω were obtained by curve fitting the angular displacement data obtained from the video film. The data points for θ were obtained directly from the video film. The reversal in the sign of ω is somewhat unexpected and a slightly confusing effect. The effect is illustrated in figure 3, where the observed spin directions are shown from the supplementary video film. The top was spun in the opposite direction to that shown in figure 1. Early in time, Ω and ω were in the same clockwise direction observed from above the top. Late in time, Ω and ω were also in the same clockwise direction observed from above the top. However, positive ω is defined as the direction of rotation when viewing a fixed point along the axis of symmetry, for example when viewing through the hole in the sphere. Viewed from below, through the hole, the sphere rotated anti-clockwise. The practical sig- nificance of the reversal in ω is that the contact point can come to rest on the horizontal surface, and hence a tippe top can roll if Ω and ω are in opposite directions. The direction of rotation after inversion is not the direction that one might intuitively expect. If a tippe top is slowly rotated by hand and if hand rotation is continued in the same direction while the top is inverted slowly by hand, then the top will rotate in the opposite direction to that shown in figure 3(b). Hand rotation in this manner reverses the direction of the angular momentum, whereas the direction of the angular momentum of a tippe top remains substantially unchanged when the top inverts. Even physicists can get this wrong.

5 Eur. J. Phys. 39 (2018) 035001 R Cross

Figure 3. Observed directions of Ω and ω for the spherical tippe top.

Figure 4. Numerical solutions of equations (4)–(6) for the spherical tippe top, with m = 0.2.

Freeman [5] incorrectly states that the angular momentum of a tippe top reverses direction when it inverts. Numerical solutions of equations (4)–(6) are shown in figure 4 to compare with the experimental results in figure 2. A reasonably good fit was obtained by setting the coefficient of sliding friction to 0.2. A higher coefficient resulted in a faster rise time. The values of I1 -5 2 -5 2 and I3 were taken as the calculated values, 2.05´ 10 kg m and 1.76´ 10 kg m respectively. The initial value of Ω was taken as 92 rad s−1, consistent with the measured value. The initial value of ω was taken as 12 rad s−1, rather than the measured value (10 rad s−1) in order to reduce the effects of nutation. Nutation has no significant effect on the rise time or the rotation frequencies, but it introduces a fast oscillation in all variables, particularly the LX and LY components of the angular momentum.

6 Eur. J. Phys. 39 (2018) 035001 R Cross

Figure 5. Calculated sliding speed of the contact point for the numerical solution shown in figure 4.

The numerical solution indicated that the tippe top commenced sliding immediately and continued to slide until t = 1.85 s, at which point it commenced to roll. The rolling phase was simulated by changing the coefficient of friction from 0.2 to either 0.02 or −0.02 depending on the rolling direction, whenever the sliding speed dropped below 0.001ms−1. Subsequent − sliding was permitted if the sliding speed exceeded 0.001ms 1. The calculated sliding speed of the contact point is shown in figure 5, indicating that there was a short period after 1.85s where the the tippe top alternated between rolling and sliding before settling into a pure rolling phase. Given that the contact point slides at relatively high speed in the Y direction and at a lower speed in the −X direction, the friction force in the X direction was taken as FMgX = 0.02 in equation (6). The resulting solutions were not particularly sensitive to the assumed value of FX. The main effect of FX was to damp nutation. The experimental results are also consistent with equation (7). For steady precession at − − q = 150, equation (7) indicates that W=60.8 rad s 1 and w =-10.4 rad s 1, close to the observed values.

5. Discussion

The role of the friction force in inverting a tippe top can be understood by examining the magnitude of the terms in equations (4)–(6). For the numerical solution presented in figure 4, the LX, LY and LZ terms during the sliding phase are shown in figure 6. The largest angular momentum component is LZ, which is about 50–100 times larger than the LX and LY com- ponents. Since XP varies from about 2 to 5mm during the sliding phase, and since FMgY ==-m 0.076 N is assumed to remain constant, dLtZ d~- 0.00023 Nm on aver- age, as given by equation (6). The three terms in equation (4) during the sliding phase are shown in figure 7. The ¶LtX ¶ term is clearly much smaller than the other two terms, which are equal and opposite to within 1%. The basic equation of motion describing inversion of the tippe top is therefore given to a very good approximation by FhY =-W LY , indicating that the torque in the X

7 Eur. J. Phys. 39 (2018) 035001 R Cross

Figure 6. The three components of the angular momentum vector during the sliding phase for the numerical solution shown in figure 4.

Figure 7. The three terms in equation (4) during the sliding phase for the numerical solution shown in figure 4.

direction is equal to the rate of change of LY due to rotation of the top about the vertical axis. Since FY is negative while the top is sliding, and since h and Ω are both positive quantities, LY = It1ddq is positive, meaning that the centre of mass rises. Intuition might suggest that the torque FhY acting around the X axis would simply result in a decrease in LX with time, and that ¶LtX ¶ would therefore be equal to FhY . However, when the top is spinning rapidly, the main effect of the torque is to rotate the LY component of the angular momentum vector around the Z axis, with the result that ¶LtX ¶ is much smaller than WLY . The effect is analogous to the effect of the torque due to the normal reaction force. Instead of falling rapidly onto the horizontal surface, as one might expect, the top rotates around the Z axis, in which case equation (5) reduces to NXLPX=-W . That is, the torque in the Y direction is equal to the rate of change of LX due to rotation of the top around the Z axis. The friction force in the Y direction causes the top to rotate about the Y axis, while the normal reaction force in the Z direction sustains steady precession about the Z axis. Rotation

8 Eur. J. Phys. 39 (2018) 035001 R Cross about the Y axis is not sustained if FY decreases to zero when the top starts rolling or if the top completely inverts. As is the case for a spinning egg, rotation about the Y axis can therefore be described as precessional motion, for the same reason that rotation about the Z axis is always described as precessional motion, at least when the precession frequency is constant or remains approximately constant. The equations describing the two separate precessional effects are identical in form and have the same physical explanation. Braams [2] noted that inversion of a tippe top can be described to within an order of magnitude by the relation FhY =-W LY , but he did not describe the inversion as being due to precession about the Y axis. As with many other authors on this topic, Braams noted that the torque due to the friction force acts in the correct direction to invert the top. The close analogy with steady precession about the vertical axis was not mentioned. In terms of explaining the inversion of a tippe top to students, the simplest explanation is that rotation about the Y axis occurs for exactly the same reason that precession occurs about the vertical axis.

6. Conclusion

Experimental and theoretical results concerning the inversion of a tippe top are presented, showing that the top inverts as a result of precession about the horizontal Y axis. The sliding speed of the contact point decreases with time until the top begins to roll, at which point the friction force drops substantially, the top stops rising rapidly and the top starts to precess about the vertical axis in a steady manner. The rise of a tippe top can be attributed to precession about the Y axis, arising from the friction force in the Y direction, in the same way that steady precession about the Z axis arises from the normal reaction force in the Z direction. Both effects are governed by equations of identical form, each indicating that the relevant torque due to the friction or normal reaction force is equal to the corresponding rate of change in the angular momentum arising from rotation about the vertical axis.

ORCID iDs

Rod Cross https://orcid.org/0000-0001-9409-2791

References

[1] Braams C M 1952 On the influence of friction on the motion of a top Physica 18 503–14 [2] Braams C M 1954 The tippe top Am. J. Phys. 22 568 [3] Pliskin W A 1954 The tippe top (topsy-turvy top) Am. J. Phys. 22 28–32 [4] Del Campo A 1955 Tippe top (topsy-turvee top) continued Am. J. Phys. 23 544–5 [5] Freeman I M 1956 The tippe top again Am. J. Phys. 24 178 [6] Parkyn D G 1956 The inverting top Math. Gazette 40 260–5 [7] Johnson F F 1960 The tippy top Am. J. Phys. 28 406–7 [8] Freier G D 1967 The tippy-top Phys. Teach. 5 36–8 [9] Cohen R J 1977 The tippe top revisited Am. J. Phys. 45 12–7 [10] Leutwyler H 1994 Why some tops tip Eur. J. Phys. 15 59–61 [11] Gray C G and Nickel B G 2000 Constants of motion for nonslipping tippe tops and other tops with round pegs Am. J. Phys. 68 821–8 [12] Soodak H 2002 A geometric theory of rapidly spinning tops, tippe tops, and footballs Am. J. Phys. 70 815–28 [13] Ciocci M C, Malengier B, Langerock B and Grimonprez B 2012 Towards a prototype spherical tippe top J. Appl. Math. 2012 268537

9 Eur. J. Phys. 39 (2018) 035001 R Cross

[14] Cross R 2013 Spherical tippe tops Phys. Teach. 51 144–5 [15] Cross R 2013 The rise and fall of spinning tops Am. J. Phys. 81 280–9 [16] Ueda T, Sasaki K and Watanabe S 2005 Motion of the tippe top: gyroscopic balance condition and stability SIAM J. Appl. Dyn. Syst. 4 1159–94 [17] Featonby D 2014 Can we understand the tippee top? Phys. Educ. 49 11–5 [18] Cross R 2017 Why does a spinning egg rise? Eur. J. Phys. 39 025002

10