Rolling and spinning coin: A level gyroscopic processional motion

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1 18 TEACHING MATHEMATICS AND ITS APPLICATIONS Volume 2, No. 1, 21 Rolling and spinning coin: A level gyroscopic processional motion Steve Humble Submitted December 1998; accepted September 2 Abstract This paper presents an example of precessional motion in mathematical terms available to students studying Further Pure and/or Further Mechanics A level mathematics. It shows how a real life example can be easily demonstrated. The mathematical solution helps students to see the applications of their Pure mathematics studies and gives a practical method to calculate the coe cient of friction of any surface. Introduction I was sitting playing with my 9-month-old son. He has these doughnut shaped plastic rings. I rolled them to him and he would try to catch them. Young children never want you to stop playing, so as you can imagine I performed this experiment many times. Then for a change I started to spin them towards him. Giving them angular momentum in a near vertical axis, and velocity in the horizontal. They slid across the carpet in a spiral path. After performing this experiment a few times, I remembered the rolling case, and a question came to me; Is the horizontal motion of a rolling coin and a spinning coin the same a spiral, and if so do these motions have the same equations? Out of this question I created a worksheet for my students. I have used it successfully with a number of groups of students studying Pure 1 and/or Mechanics 2 A level. The following is a mixture of the worksheet that I give to the students and teaching support ideas. Rolling coin To solve the problem of a precessing rolling coin, you need a co-ordinate system which lends itself to circular movements. Intrinsic co-ordinates does this well. An overview of intrinsic co-ordinates The variable s is defined as the distance along the curve from some fixed point on the curve. The angle is the angle made by the tangent to the curve at some variable point with the positive direction of the x-axis, see Figure 1. ß The Institute of Mathematics and its Applications 21

2 TEACHING MATHEMATICS AND ITS APPLICATIONS Volume 2, No. 1, Fig 1. Intrinsic co-ordinate system Fig 2. Angle of tilt of the coin At some variable point on the curve the rate of change of s with respect to is called the radius of curvature ¼ ds=d. Intrinsic co-ordinates are covered in detail in the London A Level Mathematics course. For further information on this coordinate frame please see my references 1;2. A coin rolled at a slight angle to the vertical, see Figure 2, precesses like a gyroscope or a spinning top, with its axis free to move. So the precessional motion is combined with linear motion, see Figure 3, to produce a curved path. The frictional force is F ¼ Mv 2 =, this is necessary if the coin is to follow a curved path. From Figure 2 you can see the change in angular momentum (L) is due to the moment about ~o. This moment acts to change the direction of the angular momentum of the coin, but not its magnitude. It is mathematically clear why this is so if you think about the vector cross product. Taking moments about ~o : FR cos Mgd: This is the change in angular momentum due to the moment about ~o. The tangent to the path makes an angle of with the x-axis. So the change in the angular momentum in the x direction, as a result of a small rotation d in time dt is dl cos ¼ L cos d : ; FR cos Mgd ¼ dl cos ¼ d dt dt L cos Assuming that the coin is tilted only slightly we can use small angle approximations cos 1 and sin ¼ d=r.

3 2 TEACHING MATHEMATICS AND ITS APPLICATIONS Volume 2, No. 1, 21 Newton (!): Ma ¼ Mg Assuming constant acceleration, v ¼ v gt with a acceleration M mass v velocity v initial velocity! angular speed coefficient of fiction N Normal reaction Fig 3. Linear motion of the coin (a) The angular momentum (L) of the rolling coin and the horizontal component of L. (b) Vector triangle showing the change in the angular momentum in the x-y plane. Fig 4. The angular momentum L, of a disc with moment of inertia of MR 2 =2 about the rotation axis passing through the centre of mass is L ¼ MR 2!=2. So using the above the moment about ~o we obtain Mgd Mv2 R ¼ d MR 2! dt 2 ð1þ d =dt is the angular speed of precession, so from Figure 1 this gives v ¼ d =dt. Due to the fact that the coin is rolling we can also say that v ¼ R!. Using (1) with the rolling relationship you can show that ¼ 3Rv2 2gd ð2þ Then by differentiating (2) w.r.t. t obtain

4 TEACHING MATHEMATICS AND ITS APPLICATIONS Volume 2, No. 1, d dt ¼ 3R d d dt with v ¼ v gt from Figure 3. Let k ¼ 3R=d to simplify the algebra, then integrate equation (3) to obtain the radius of curvature at any point on the path as ¼ e k with ¼ kv2 2g From these equations it can be seen that as the coin slows down, the rate of precession increases and the radius of curvature decreases. I have extended this theory into the following areas, which make interesting investigations for the students. These are given below ð3þ The equation of the path Need to use complex numbers i ¼ p 1, i 2 ¼ 1 and e i ¼ cos þ i sin. So the path is ð t ð ð x þ iy ¼ ve i dt ¼ e i d ¼ e ði kþ d Given that v dt ¼ ds ¼ d from intrinsic co-ordinate system. Solving this integral gives the following equations of the path x ¼ ðk ke k cos þ e k sin Þ=ð1 þ k 2 Þ y ¼ ð1 e k cos ke k sin Þ=ð1 þ k 2 Þ Using a Graphics calculator students can see that these equations will produce a spiral curve. Since you can show that v ¼ d dt ¼ 2gd 3Rv ¼ 2 Ln v k v Which is the intrinsic angle of the path in terms of the velocity. Link between linear and precessional distance ð travelled The equation of the length of the path is s ¼ d You can show this equals s ¼ ð1 e k Þ k The total length can be shown to be s ¼ v2 as!1 ð2gþ This total length can also be obtained via v ¼ v gt.

5 22 TEACHING MATHEMATICS AND ITS APPLICATIONS Volume 2, No. 1, 21 Experiment By finding the angle between the start direction and the final position of the rolling coin you can determine the coefficient of friction and the initial velocity. You can show that as!1 then x ¼ k ð1 þ k 2 Þ and since the range D ¼ p ðx 2 þ y 2 Þ y ¼ ð1 þ k 2 Þ ; D ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and Angle : tan ¼ 1 ð1 þ k 2 Þ k Fig 5. Students will encounter a number of problems with this experiment. One of these is due to approximating the constant d, see Figure 2. A method of improving the accuracy in d is to use a slanted starting wedge. Another area of difficulty is with the type of coin used. I have found that a 1p coin rolls well due to its rounded edge. Other coins tend to roll in an upright position due to their flat edges. The large plastic doughnut rings I talked about in the introduction also work very well. Another alternative is to make a large cardboard disc and stick some blu-tack on one side at its centre. As you can see the experiment has a number of regions where errors may occur. So I do not tend to place too much emphasis on the experiment, using it mainly to promote interest in the problem and further research. This will then drift nicely into homework. Some numbers for this problem seem to occur frequently. For example, with a 1p coin values such as D ¼ 15 cm and ¼ 78 on a paper surface. Once the student has found these he can then go on to find values for the coefficient of friction :7 and the initial velocity v :2 ms 1. If you get the Physics department involved, you will find that they have all sorts of equipment that can track the coins path. I must admit to trying this, but I think that it is straying from the point, which is to create a simple model and to solve it using mathematics. Since the basic equipment needed is only a protractor, ruler and 1p coin, it is the sort of experiment that you can do anywhere. In a way this is why technical equipment subtracts from its primitive appeal.

6 TEACHING MATHEMATICS AND ITS APPLICATIONS Volume 2, No. 1, As you can see this work sheet solves only part of my question. It also produces a method to find the coefficient of friction of any surface, and the initial linear and angular velocities. What about a spinning coin? The main differences between the rolling and spinning coin theory are as follows: i) The angular momentum of a spinning coin is perpendicular to the that of a rolling coin. So the component for the spinning coin is L sin and not L cos. ii) The moment of inertia is MR 2 =4. iii) v ¼ ðd =dtþ 6¼ R! this is a major difference when it comes to producing a solution and it means that you have to use numerical methods. A further assumption that! ¼! ct, where c is a small constant and! is the initial angular speed is also necessary. Using Euler s method, which is in the London A level Mathematics syllabus. The equations of motion for the spinning coin in terms of time can be solved. x nþ1 ¼ x n þ hðv cos Þ y nþ1 ¼ y n þ hðv sin Þ with ¼ gd Rðg þðcd=4þþ ln ðv þð! d=4þþ v þð! d=4þ ðgþðcd=4þþt! ¼! ct; v ¼ v gt Program written in PowerBASIC to show path of a spinning coin SCREEN 12 WINDOW (-1,-1) - (1,1) ***Initialise *** X= Y= VO=2 WO=5 ***Coefficient of friction*** U=.1 ***Loop with h=d=.1*** FOR T= TO 25 STEP.1 X=X+.1*(VO-1*U*T)*COS((LOG(VO+(WO/2))-LOG(VO+(WO/2)- 1*U*T))/(1*U)) Y=Y+.1*(VO-1*U*T)*SIN((LOG(VO+(WO/2))-LOG(VO+(WO/2)- 1*U*T))/(1*U)) PSET(X,Y) NEXT T This shows the motion to be a spiral for a spinning coin.

7 24 TEACHING MATHEMATICS AND ITS APPLICATIONS Volume 2, No. 1, 21 Conclusion I have used this idea with students after teaching intrinsic co-ordinates in either P4 or M4. They have found only a few of the concepts difficult and by using the experiment we have been able to work through these problems. The rolling and spinning coin problem demonstrates to students how useful this co-ordinate frame can be to certain types of mechanics problems. Since the solution involves so many areas of the A level syllabus it has also become a useful revision tool. The areas of the A level syllabus used in the solution are co-ordinate systems (intrinsic), curvature, complex numbers, differentiation, integration, vectors, numerical methods, differential equations, moments, angular momentum, and centripetal force. References 1. Pure Mathematics 4 by Mannell & Kenwood, Heinemann. 2. Mechanics 4 by Hebborn & Littlewood, Heinemann. Steve Humble is Head of Mathematics at Bellerbys College. After graduating with BSc, BA and PGCE he has worked in a number of Private and State Schools. He enjoys trying to work out mathematically why things work. Address for correspondence: Bellerbys College, 44 Cromwell Road, Hove, East Sussex BN3 3ER, UK. shumble@bsg.ac.uk