The Dynamics of Hurricane Balls. Abstract

Transcription

1 The Dynamics of Hurricane Balls W. L. Andersen Department of Physical Sciences, Eastern New Mexico University, Portales, NM Steven Werner Engineering Dept, Lear Jet, Wichita, KS (Dated: September 18, 2014) Abstract We examine the theory of the Hurricane Balls toy. This toy consists of two steel balls, welded into a rigid structure which is sent spinning on a horizontal surface somewhat like a top. Unlike a top, at high frequency the symmetry axis approaches a limiting inclination that is not perpendicular to the surface. We calculate and experimentally verify the limiting inclinations for three toy geometries. 1

2 I. INTRODUCTION Recently an interesting physics toy under the name Hurricane Balls has been introduced to the marketplace. 1 This toy consists of two identical ball bearings welded together. After setting the apparatus spinning on one of the balls by hand a plastic pipe is used to blow at one side to increase the angular velocity. Unlike a coin, the Hurrican Balls apparatus can not be made to spin fully upright. The physics of this toy proves to be an instructive exercise in the theory of rigid body motion. Although moving under the influence of gravitational torque and rolling friction, at sufficiently high angular velocities the gravitational torque becomes neglible in comparison to the internal torques of the system so that Poinsot s theory of a freely rotating body becomes applicable. The interesting addition here is that the rolling friction enforces a constraint connecting the angular velocity about the axis of symmetry with the rotational velocity of the toy s form about the vertical axis. II. NOTATION AND HEURISTICS Before proceeding to the application of Poinsot s solution we describe a quick way to get the same results that raises an interesting question about rotating frames of reference. Although flawed, this argument gives some insight into the magnitude of torques involved and provides and opportunity to introduce notation. Figure 1 defines the angle θ used to describe the inclination of the rotating balls. The lower ball traces a circle on the ground centered on the point where the vertical line L passing through the center of mass intersects the ground. The component of the angular velocity in the direction of the axis of symmetry is ω 3. We take each ball to have mass M and radius R. In this heuristic derivation we put ourselves into a frame rotating about the axis perpendicular to the ground and passing through the center of mass of the toy. In this frame there appears two inertial centrifugal forces, each of magnitude MωT 2 R sin θ,acting on each ball in opposite directions. The torque due to this force couple is τ C = 2(MΩ 2 L sin θ)(l cos θ) (1) where Ω is the angular velocity about the vertical axis of the form of the balls as the toy rotates. We note that this is not the total angular velocity of the toy. 2

3 FIG. 1. In anticipation of later results the vertical axis is labeled L. The sense of ω 3 depends on the sense of Ω, the angular velocity of the symmetry axis about L. In addition to Eq. (1) there is torque due to the normal force of the ground supporting the weight 2Mg. This torque is τ N = 2(Mg)(R sin θ) (2) In order to treat the toy as a free object we must have τ C τ N. Neglecting the unknown angle factor in equation Eq. (1) we get Ω g R The Hurricane Balls used in our experiments have a radius of about 0.7 cm which implies a frequency much greater than 40 Hz. In this heuristic derivation we set the rate of change of the angular momentum about the symmetry axis, 4/5MR 2 ω 3,equal to its horizontal component times Ω, (3) L = Ω 4 5 MR2 ω 3 (4) Examination of the geometry in Figure 1 shows that the no-slip condition implies Ω = ω 3. This constraint is the essential difference between the free body case and Hurricane Balls. If 3

4 we set equation Eq. (4) equal to Eq. (1) and impose the no-slip condition then the inclination angle θ is cos θ = 2 5 (5) In this treatment we put ourselves in the rotating (nonspinning) frame of reference in order to get the centrifugal couple which accounts for the rate of change of the angular momentum. However, in that frame the spin axis does not rotate and therefore no net torque should exist! In the next section we avoid non-inertial frames of reference. III. NONCALCULUS TREATMENT This section follows the treatment of coin spolling in reference 2. We consider the case of two solid balls touching in this and the next section. The other cases follow identical treatments with different rotational inertial tensors. We first note that the frequency requirement for approximate free body motion is better characterized by the condition that the gravitational torque accounts for the precession of the angular motion vector L in the limit that θ be small. We then have L θrmg or ωθl ωθir 2 ω θrmg (6) which leads once again to Eq. (3). As in the previous section, we assume the geometry of Fig. 1. We must also make the a posteriori assumption that the total angular momentum is perpendicular to the surface defining the constraint, Ω = ω 3. We adopt an inertial frame rather than the rotating frame of the previous section. The total angular velocity, ω, of the toy is ω = Ω + ω 3 (7) The angular momentum of the toy is L = I ω where the rotational inertia tensor is represented in the diagonalized body axis basis by 4

5 I = MR (8) Calculation is simplified if Ω is expressed as the sum of projections along the principle axes. To this end we use the geometry of Fig. 2 to write Ω = Ω sin θ and Ω 3 = Ω cos θ. The vector relation is Ω = Ω + Ω 3 (9) Fig. 3 shows the angular velocity and angular momentum components associated with the symmetry axis and the axis instantaneously perpendicular to the symmetry axis and in the plane defined by the symmetry axis and the vertical. Ω 3 θ Ω Ω FIG. 2. The angular velocity Ω is projected onto principle directions. Since the total angular momentum is vertical the horizontal components of I (Ω + ω 3 ) = I (Ω + Ω 3 + ω 3 ) must add to zero. Referring to the geometry of Fig. 3 we have I 3 Ω sin θ + I 3 Ω cos θ sin θ I 1 Ω sin θ cos θ = 0 (10) Substituting for the rotational inertia components and isolating θ we get Eq. (5). IV. POINSOT S THEORY The main limitation of the above treatments is the a posteriori assumption that the total angular momentum is perpendicular to the surface defining the constraint. In order to 5

6 I Ω 3 Ω I Ω I ω 3 θ I Ω cosθsinθ 3 I Ω sinθcosθ 1 I 3 Ω sinθ FIG. 3. Angular velocities and angular momenta. completely understand the toy we start with Euler s equations of motion I 1 ω 1 + (I 3 I 2 )ω 3 ω 2 = 0; (11) I 2 ω 2 + (I 1 I 3 )ω 1 ω 3 = 0; (12) I 3 ω 3 = 0 (13) Poinsot s solution yields the geometric result shown in Fig. 4. The cone centered on ω 3 (traditionally the body cone) rolls without slipping on the cone centered on L (the space cone). The line of intersection is the total angular velocity vector of the toy. The angle of inclination is θ = α s + α b. If we define the oblateness β = (I 3 I 1 )/I 1 then it can be shown that the angular velocity ω rotates about the 3-body axis axis at the angular frequency βω 3. It can be further shown 3 that the space and body cones are related by cos α s = 1 + β cos 2 α b 1 + (2β + β2 ) cos 2 α b. (14) The frictional rolling constraint Ω = ω 3 amounts to the requirement α s = α b. Setting α α s = α b in Eq. (14) determines the cosine to be cos α = Considering that θ = 2α this is identical to Eq. (5) (15)

7 L ω ω 3 α s α b FIG. 4. The Poinsot construction. The body cone (centered on ω 3 ) rolls without slipping on the space cone (centered on L). V. EXPERIMENTAL The inclination angle θ was measured for three different geometries of custom manufactured Hurricane Balls: two solid balls (R = cm) touching two solid balls (R = cm) connected by a rod of length L = 1.21 cm. This set of Hurricane Balls is shown in Fig. 5 two hollow balls touching, interior and exterior radii of 1.60, 1.91 cm Theory predicts different limiting angles for each geometry. The measurements were accomplished by allowing the spinning balls to contact strike paper leaving marks from which the angle θ was determined. Fig. 5 shows Hurricane Balls with strike paper. Predicted and measured θ are shown in table I. Agreement of theory and experiment seems satisfactory though there is some issue with the hollow balls. This could be due to a manufactoring anomaly. Each hollow sphere is made of two hemispheres welded together. The thickness might not be uniform due to the weld. The hollow balls do appear to have extra mass from the weld as detected by sticking a tooth pick probe in the hole. 7

8 FIG. 5. Strike paper and Hurricane Balls. TABLE I. Limiting inclination at high frequency for three geometries. Estimated uncertainty in measured θ is about 0.5. The two radii indicated for the shell are the inner and outer radii. ball type R(cm) L(cm) θ theory θ exp solid solid shell 1.60, VI. CONCLUSION At high frequencies, Hurricane Balls provide an easily realized example of the Poinsot theory of freely rotating symmetrical bodies. The rolling constraint couples precession and symmetry axis rotation to produce a limiting inclination which can be measured to test the theory. William.Andersen@mailaps.org; permanent address: Station 33, Eastern New Mexico University, Portales, NM steve.werner@cox.net 8

9 1 available at < demonstration video at < youtube.com/watch?v=cvq8lapb498>. 2 W. L. Andersen, Noncalculus Treatment of Steady-State Rolling of a Thin Disk on a Horizontal Surface, Phys. Teach. 45 (7), Keith R. Symon, Mechanics, 3rd edition (Addison-Wesley, Reading, 1971),