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 XPis 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.