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N T E R D I S C I P L I N A R Y
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I V E L Y
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P P L I C A T I O N S
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R O J E C T
The
Hopping Hoop
Interdisciplinary Lively Application Project
Title: The Hopping Hoop
Authors:
Tim Pritchett
Department of Physics, United States Military Academy,
West Point, New York 10996
Stan Wagon
Department of Mathematics and Computer Science,
Macalester College, St. Paul, Minnesota 55105
Editor:
David C. Arney
Mathematics Classifications: Calculus
Disciplinary Classifications: Physics, Calculus
Prerequisite Skills:
1. Basic Calculus
2. Differential Equations
Physical Concepts Examined:
1. Motion
2. Friction
Materials: None
Computing Requirements: Mathematica
Project Intermath 1-43. ©Copyright 2001 by COMAP, Inc. All rights reserved.
Permission to make digital or hard copies of part or all of this work for personal or
classroom use is granted without fee provided that copies are not made or distributed
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COMAP must be honored. To copy otherwise, to republish, to post on servers, or to
redistribute to lists requires prior permission from COMAP.
2 Project Intermath
Contents
1.
2.
3.
4.
5.
6.
7.
8.
Introduction
Problem Statement and Assumptions
Overview of Solution Procedure
Constrained Motion: Rolling Without Slipping
Some Basic Physics
Removing One Constraint: Rolling With Slipping
The Hop
Postscript: A Final Suprise
1. Introduction
Tape a heavy battery or two to a hula hoop and roll the hoop along the floor.
Or, attach a weight to the inside of the rim of a bicycle wheel and give the wheel a roll.
If you do it right, you’ll discover, as did Littlewood [1, p. 37] (see also [3, 6, 7]), that the
hoop/wheel will hop into the air at a certain point. Why does this happen? In this project,
we’ll create a mathematical model of a weighted hoop that exhibits this unexpected
hopping behavior. In the process, we’ll learn a fair amount of physics. We’ll also
develop techniques that are extremely useful in determining the evolution in time of
systems subject to contraints. This is a very important dividend, since constrained
systems are absolutely everywhere; they are commonplace not only in science and
engineering, but in business and finance as well.
Figure 1. A stroboscopic photo (created with the help of Dan Schwalbe)
that shows the surprising small hop that occurs after about 90 ° of
rotation.
The Hopping Hoop 3
The mechanical system that is the subject of this project is illustrated in figure
2: a perfectly rigid circular hoop of mass (1 — l)M and radius a with an object of mass
l M (the weight) rigidly attached to its rim. Here, M is the total mass of the combined
system consisting of the hoop and the object. The parameter l is the ratio of the
attached mass to the total; of particular interest will be the limit l Ø 1, in which the
mass of the hoop itself is negligible relative to that of the attached object.
y
lM
q
x
Figure 2. The system: a hoop with an additional weight attached to its
rim. Together, the hoop and the weight have mass M; the mass of the
hoop alone is (1 — l)M while that of the attached weight is l M.
This project is laid out as a series of exercises. Some of these exercises explore
aspects of the problem that will be of particular interest to students of physics and
engineering, but are not 100% necessary to complete the project. Such exercises,
designated by an asterisk, are optional (though all students are strongly encouraged to
attempt them). One such exercise follows.
4 Project Intermath
* Exercise 1. Center of Mass
Consider a system of N particles having masses m1 , m2 , …, mN , and position
÷”
”
vectors r1 , ”r2 , …, ”rN . The position R of the center of mass of the system is defined by
÷ ” ⁄Ni=1 mi ”ri
R = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
N mÅÅÅÅÅ
⁄i=1 i
and it represents — pun very much intended! — a “weighted average” of the positions
of the N particles comprising the system. Show that for the system we are considering
here, namely, the hoop of mass H1 - lL M with the additional attached mass l M, the
center of mass lies along the line connecting the center of the hoop with the attached
mass a distance l a from the center of the hoop.
2. Problem Statement and Assumptions
To simplify the problem somewhat, we will make two assumptions:
1. We assume that both the hoop and the surface upon which it rolls are perfectly
rigid; in other words, neither the hoop nor the surface suffers deformation of any kind
as the former rolls along the latter. This is admittedly an idealization — the bottom of a
real hoop is slightly flattened while the supporting surface shows a slight depression in
the vicinity of the point of contact — but it is an extremely reasonable one.
2. We assume that the hoop rolls without slipping, at least initially. It turns out that
this assumption is a little tricky to implement in an actual experiment, at least if you’re
interested in seeing a big hop, which requires fairly high speeds. (One way to insure
that the hoop initially rolls without slipping in an actual experimental trial is to
position the attached weight so that it is not directly above the center of the hoop, and
allow it to accelerate from rest entirely under the influence of gravity. This will not,
however, result in a very impressive hop.)
The problem is then to predict the subsequent motion of the hoop.
The Hopping Hoop 5
3. Overview of the Solution Procedure
The motion of the hoop consists of three distinct phases: an initial phase in
which the hoop rolls entirely without slipping; an intermediate phase in which the
hoop slips as it rolls, but has not yet lost contact with the supporting surface; and a
final phase in which the hoop has lost contact with the supporting surface and is
airborne. Figure 3 shows what we are aiming for: a simulation that yields a graphic
display of the hoop and shows the experimentally observed hop.
Figure 3. The hopping hoop. The hop is not large, but its existence is very
surprising. The trajectory of the center of mass is indicated by the
downward sloping curve that appears in the vicinity of the large dots.
Exercise 1 formally defines the center of mass. We show, in exercise 2, that the
trajectory of the center of mass during the initial rolling-without-slipping phase must
be a cycloid. After a brief review of the relevant physics, we proceed to write down, in
exercises 3 and 4, the three differential equations of motion for the weighted hoop.
During the rolling-without-slipping phase, these three equations can be written
(exercises 5, 6, and 8) as a single equation, which we then solve (exercises 9 and 10).
This solution is only valid, however, so long as the hoop does not slip, so the next four
exercises, 11 through 14, are devoted to determining the critical angle at which slipping
begins and the initial phase of the motion terminates.
We next turn out attention to the intermediate phase of the motion, in which
the hoop slips but maintains contact with the supporting surface. We describe in some
detail a procedure for solving the differential equations of motion during this phase
and, in exercise 15, ask the student to implement this procedure numerically.
Exercise 16 treats the hop itself, the final phase of the motion in which the hoop
is actually airborne. The hop ends when the hoop regains contact with the supporting
surface, and we determine the time of impact in exercise 17.
6 Project Intermath
Exercise 18 assembles the previously obtained solutions for the three phases of
the motion into a single simulation.
4. Constrained Motion: Rolling Without Slipping
The most general motion of a rigid body in 3-space consists of a translation of
the center of mass, combined with a rotation of the body about an axis containing the
center of mass. Accordingly, we shall specify the motion of the hoop by the Cartesian
coordinates HX@tD, Y@tDL of the center of mass of the system — indicated by the small dot
in figure 2 — and by the angle q@tD through which the hoop has turned.
Of course, X@tD, Y@tD, and q@tD are not the only variables we could choose to
specify the configuration of the weighted hoop. Indeed, instead of the horizontal and
vertical coordinates of the center of mass of the system, it might seem more natural to
choose the horizontal and vertical coordinates of the center of the hoop, quantities that
we will denote by x@tD and y@tD, respectively. However, as we shall see presently, the
dynamics of the system will serve to make the choice of coordinates X@tD, Y@tD, and q@tD
particularly convenient.
Because of our assumptions, the three coordinates X@tD, Y@tD, and q@tD are not
independent, but are subject to various constraints:
1. Since the hoop is assumed to be a rigid object that suffers no deformation as it rolls,
the center of mass is always a fixed distance from the center of the hoop: It is always
true that HX@tD - x@tD L2 + HY@tD - y@tD L2 = l2 a2 , or equivalently,
X@tD - x@tD = a l sin@qHtLD
(1)
Y@tD - y@tD = a l cos@qHtLD
2. Since the supporting surface is also assumed to be rigid, the vertical height of the
center of the hoop must satisfy
y@tD = a
(2)
so long as the hoop and the surface are in contact.
3. As long as there is no slipping, the horizontal position of the center of the hoop and
the angular coordinate q[t] are related by
The Hopping Hoop 7
x@tD = aHq@tD - q0 L
where
(3)
q0 = q@0D
These three constraints together imply that as long as the hoop rolls without
slipping along the supporting surface, the center of mass of the system moves along a
curtate cycloid (“curtate” means “shortened”), also known as a trochoid [4]. The
coordinates of the center of mass are then related to the angular coordinate q[t] by
X@qD = aHq - q0 + l sin@qDL
(4)
Y@qD = aH1 + l cos@qDL
2
1
0
0
π
2π
3π
4π
Figure 4. The trajectory of the center of mass of an asymmetrically
weighted hoop of radius 1 with l = 0.8 is a curtate cycloid.
2
1
0
0
π
2
π
3π
2
2π
5π
2
3π
Figure 5. A more detailed depiction of the cycloidal curve that is the
trajectory of the center of mass. Also shown are the positions at equal
time intervals of the additional weight attached to the rim of the wheel, as
well as a spoke connecting the weight to the center. The motion is that of
an asymmetrically weighted hoop of unit radius with l = 0.75 rolling
without slipping under the influence of gravity with an initial angular
speed of 1 radian/second.
8 Project Intermath
Exercise 2. Cycloid Constraint
Show how the three constraints (1), (2), and (3) combine to give the equations
(4), which constrain the center of mass of the weighted hoop to move on a curtate
cycloid.
Aside for the Advanced Student: Configuration Space
NOTE: The material in this section may be useful if this ILAP module is used in
conjunction with an advanced course in Langrangian/Hamiltonian mechanics or
dynamical systems. This material is entirely supplementary.
Constraints have a geometric interpretation. As we have already mentioned,
the configuration of our system is specified by a triplet of (generalized) coordinates; we
have chosen HX, Y, qL, though this is by no means the only choice. In the absence of
constraints, X, Y, and q can assume all possible real values, so the set of all possible
configurations of the system, termed the configuration space of the system, is isomorphic
to Ñ3 . The constraints restrict the possible configurations of the system to lie on some
surface of lower dimension; in the case of the hoop, the configuration space of the
system subject to constraints (1), (2), and (3) — which are equivalent to the cycloid
constraint, equations (4), as you demonstrated in the preceding exercise — is the onedimensional curve shown in figure 6.
The Hopping Hoop 9
Y
0
1
0
X
5
10
10
5
θ
0
Figure 6. Configuration space for the hoop when the cycloid constraint is
satisfied.
The complete state of a system is specified by the values not only of the
generalized coordinates, but of the generalized velocities as well; a complete
°
°
°
description of the hoop at an instant in time requires the six values HX, X, Y, Y, q, qL, or,
equivalently, HX, PX , Y, PY , q, Pq L, where PX , PY , and Pq are the momenta canonically
conjugate to the coordinates X, Y, and q, respectively. The set of all possible states of a
classical system is called the phase space of the system.
5. Some Basic Physics
Forces Acting on the Hoop
The motion of any object is determined by the forces exerted on it by the
outside world (external forces). In the case of our weighted hoop, there are two such
external forces: the gravitational attraction of the earth for the hoop and the attached
object, and the contact force that the supporting surface exerts on the hoop.
10 Project Intermath
The gravitational force behaves as if the entire mass of the system were
concentrated at the position of the center of mass. This force is directed straight down
and has magnitude M g, where M is the combined mass of the hoop and the attached
object and g = 9.8 m ê s2 is the constant acceleration imparted by the earth’s gravity to
every object near the earth’s surface.
The force of contact exerted by the supporting surface on the hoop may be
resolved into components normal and tangential to the surface, and these components
are conventionally treated as if they were separate forces. The component normal to
the surface is commonly known as the normal force, and the component tangential to
the surface is friction.
q
Mg
N
F
Figure 7.
Free body diagram of the asymmetrically weighted hoop
showing the external forces acting upon it.
For later convenience, we define f = F ê M and n = N ê M. These quantities are
the forces per unit mass (accelerations) arising respectively from the the x- and ycomponents of the force of contact between the hoop and the supporting surface. (For
brevity, we will frequently refer to f and n as “forces” even though they are actually
forces per unit mass.) Unlike g, f and n are not constant, but vary as the hoop rolls.
Friction, the tangential component of the force of contact between two surfaces,
is directed so as to oppose the relative motion of the surfaces, or, in cases where there
The Hopping Hoop 11
is no relative motion, the tendency for such motion. In the present case, we assume that
the hoop rolls without slipping, and thus there is no relative motion between the hoop
and the supporting surface. If, however, the hoop were able to slip freely in the
situation depicted in figure 7, then it would rotate clockwise. The friction force in
figure 7 is thus directed to the right to oppose this tendency for relative motion.
Indeed, it is precisely the action of the friction force that prevents the hoop from
sliding; friction is the force responsible for maintaining the “rolling without slipping”
constraint expressed mathematically by (3).
In much the same way, the normal component of the contact force (the
“normal force”) maintains the constraint that the center of the hoop be no closer to the
supporting surface than the hoop radius — y@tD ¥ a — where equality holds when the
hoop and the supporting surface are in contact. We shall have more to say about forces
of constraint later.
Newton’s Second Law
Newton’s second law expresses mathematically how the interactions of a body
with its environment (manifested by the external forces exerted on the body)
determine the body’s subsequent motion (expressed by the acceleration of the body’s
÷”
÷–”
center of mass): ⁄ Fexternal = M R
Exercise 3. Center of Mass Equations of Motion
Review the previous subsection’s discussion of the forces acting on the hoop,
and apply Newton’s second law to write down the following pair of second-order
differential equations for X@tD and Y@tD, the horizontal and vertical coordinates of the
center of mass of the system.
–
X@tD = f
(5)
–
Y@tD = n - g
(6)
We have already observed that an arbitrary motion of a rigid body may be
decomposed into a translation of the body’s center of mass combined with a rotation of
the body about an axis containing the center of mass. If the direction of the rotation
axis does not change with time, the rotational form of Newton’s second law is given by
–
⁄ texternal = I q, where q specifies the angle of rotation. The sum on the left-hand side is
over the components parallel to the rotation axis of all external torques about the center
of mass; I is the moment of inertia of the body about the center of mass.
12 Project Intermath
Exercise 4. Equation of Rotational Motion
Consider torques about the center of mass, and apply the rotational form of
Newton’s second law to obtain the following differential equation for q[t], the angle
through which the hoop has rotated.
–
M
q@tD = ÅÅÅÅ
ÅÅ H nHX@tD - x@tDL - f Y@tD L
I
(7)
* Exercise 5. Derivation of the Equation of
Energy Conservation
Consider the differential equations of motion that you obtained in the previous
°
°
two exercises. Multiply both sides of equation (5) by M X@tD, of equation (6) by M Y@tD,
°
and of equation (7) by I q@tD. Eliminate the unknown forces n and f from the resulting
three equations using the constraint equations (1), (2), (3), and (4). In this way, obtain a
single differential equation for the three coordinates. Integrate this equation with
respect to time and so obtain the equation of energy conservation for the weighted
hoop.
° 2 ° 2
° 2
ÅÅÅÅ12 MIX@tD + Y@tD M + ÅÅÅÅ12 I q@tD + M g Y@tD = constant
(8)
Exercise 6. Energy Conservation Equation in
Terms of q Only
Using the cycloid constraint (4), eliminate the coordinates X@tD and Y@tD in favor
of q[t] from the equation of energy conservation (8) and so obtain:
° 2
a g M H1 + l cos@qHtLDL + ÅÅÅÅ12 @I + a2 M H1 + l2 + 2 l cos@qHtLD LD q HtL = constant
(9)
The Hopping Hoop 13
* Exercise 7. Moment of Inertia of the
Asymmetrically Weighted Hoop
The quantity I appearing in (8) and (9) is the moment of inertia of the system
consisting of the hoop plus the attached object about the axis that passes through the
system center of mass and is perpendicular to the plane of figure 7. Take the attached
object to be a mathematical point and show that in such a case I is given by
I = M a2 H1 - l2 L. (Hint: You will need to use the parallel axis theorem, which states that
the moment of inertia of a rigid body about any axis is equal to the moment of inertia
about a parallel axis passing through the center of mass plus the product of the mass of
the body and the square of the distance between the two axes.)
Of course, in the real world the attached object will not be a mathematical
point, but will be of nonvanishing physical dimension, and thus will possess a
nonvanishing moment of inertia about its centroid. We will take
I = M a2 H1 - l2 + l ¶L
(10)
where ¶ is a dimensionless parameter (generally quite small) related to the physical
dimensions of the attached object.
Initial Conditions
The coordinate system shown in figure 2 is chosen so that at t = 0 the center of
the hoop is situated at (0, a). In general, three parameters are required to specify the
initial state of the weighted hoop: one to specify the initial orientation of the hoop, and
two more to completely specify its initial motion. One very natural choice would be to
specify the angle q0 = q@0D; the initial translational speed x° 0 of the center of the hoop;
°
°
and the initial angular speed of rotation q0 = q@0D. In the present case, however, where
we assume that the hoop rolls without slipping, the latter two parameters are not
°
independent, but are related by x° 0 = a q0 . We will thus specify the initial state of the
°
weighted hoop by giving q0 and q0 .
14 Project Intermath
Exercise 8. Mechanical Energy
The constant appearing on the right-hand side of equations (8) and (9) is the
total mechanical energy E of the weighted hoop, and it may be evaluated from the
initial conditions. Take the attached object to be initially located directly above the
point of contact (q0 = 0) and show that E = M a g @H1 + lL H1 + cL + ÅÅÅÅ12 ¶ l cD
°2
where we have introduced the dimensionless quantity c = a q0 ê g.
°
Substitute this result into (9) and solve the resulting equation for q@tD to obtain
the following first-order differential equation describing the time evolution of the
weighted hoop constrained to roll entirely
without slipping.
°
g
cH1+ l+ ¶ lê2L+ lH1- Cos@qHtLD
########
#######
q@tD = "#####
ÅÅÅÅa "################################
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅL
1 + ¶ lê2+ l Cos@qHtLD
(11)
Exercise 9. Motion on the Cycloid: Numerical
Solution for a Typical Case
Use a numerical differential equation solver (such as the Mathematica function
NDSolve) to obtain a numerical solution to (11) subject to the initial condition q@0D = 0
for 0 § t § 3 (sec). Use the following values for the parameters: g = 9.8 Hmeters ê sec2 L,
a = 1 (meter), l = 0.95, ¶ = 0.01, c = 0.1. To help you visualize your solution, create a
plot similar to figure 5 (e.g., with the Mathematica function ParametricPlot.)
Exercise 10. Motion on the Cycloid: Analytic
Solution for a Special Case
Take l = 1 (corresponding to a hoop whose mass is negligible relative to that of
the attached object) and ¶ = 0 (effectively treating the attached object as a point mass).
For this special case, obtain an analytic solution to the differential equation (11) subject
to the initial condition q@0D = 0.
The Hopping Hoop 15
Forces of Constraint
We were fortunate to be able to eliminate the forces f and n from the equations
of motion. These forces are not known a priori; rather, they are determined by the
effects they produce: by the constraints they maintain. In fact, one may even view
equations (5) and (6) as defining the forces f and n in the case that the cycloid constraint
(4) holds.
* Exercise 11. Explicit Expressions for the Forces
of Constraint
Use the cycloid constraint to obtain the following expressions for the forces of
constraint n and f as a function of x = cos@qD.
1
f @xD = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Ig l
H2+¶ l+2 l xL2
è!!!!!!!!!!!!2
1 - x H2 + l H-2 + ¶ + 6 xL -
c H2 + H2 + ¶L lL H1 + ¶ l + l xL + l2 H-2 ¶ - 2 x + 3 ¶ x + 4 x 2 LL
(12)
1
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
n@xD = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
H2+¶ l+2 l xL2
H4 g + 4 g ¶ l - 2 g l2 - 2 c g l2 + g ¶2 l2 - 2 g l3 - 2 c g l3 - g ¶ l3 - c g ¶ l3 +
8 g l x - 4 c g l x - 4 g l2 x - 4 c g l2 x + 4 g ¶ l2 x - 4 c g ¶ l2 x - 2 g ¶ l3 x -
(13)
2 c g ¶ l3 x - c g ¶2 l3 x + 10 g l2 x 2 - 2 c g l2 x 2 2 g l3 x 2 - 2 c g l3 x 2 + 3 g ¶ l3 x 2 - c g ¶ l3 x2 + 4 g l3 x3 L
The fact that n and f are absent from the equation of energy conservation (8) is
no accident, but reflects a general property of forces of constraint. Forces that maintain
constraints necessarily act perpendicular to the configuration space of the system, and
consequently do no work.
16 Project Intermath
6. Removing One Constraint: Rolling With
Slipping
If the weighted hoop is to become airborne, then its center must acquire a
vertical velocity. For this to occur, one of the conditions that constrain the center of
mass to move along a cycloid must fail to be satisfied: the hoop must slip. The next
exercise examines why.
Exercise 12. Why the Hoop Must Slip
Use the rigid hoop condition (1) to obtain an expression for y° @tD, the vertical
component of the velocity of the center of the hoop. Interpret the two terms in the
resulting expression. What values must these terms have (a) if the hoop rolls without
slipping, and (b) if the hoop is to become airborne?
Having established that there must be some slippage of the hoop relative to the
supporting surface just prior to the hoop’s becoming airborne, we now examine at
what point this slippage occurs.
Equations of Motion for Rolling with Slipping
Once the hoop has started to slip, its motion is no longer described by the
single equation (11). Rather, the slipping hoop is described by the three equations of
motion (5), (6), and (7), subject to the constraints (1) and (2). The friction force is now
given by f = -mk n, where the minus sign indicates that the force is directed to the left,
a fact that follows from a consideration of the physical particulars of the situation and
the observation that n@tD ¥ 0. The preceding definition of kinetic friction and the rigid
hoop constraint (3) allow us to recast the equations of motion in the form:
–
Y@tD = n - g
(14)
–
X@tD = -mk n
(15)
–
M
q@tD = ÅÅÅÅ
ÅÅ a @l sin @qHtLD + mk H1 + l cos@qHtLDLD n
(16)
I
The normal force n@tD must be sufficiently large to ensure that the height of the
center of the hoop above the supporting surface is never less than the radius of the
The Hopping Hoop 17
hoop, i.e., the point of contact never lies below the x-axis. This constraint is used to
dynamically determine the value of n.
Solution Procedure: Rolling With Slipping
To obtain the motion of the system during the phase in which the hoop rolls
with slipping but has not yet lost contact with the supporting surface, one must solve
simultaneously the differential equations (14) and (16), subject to the constraint that
y@tD ¥ a. This constraint must be enforced explicitly at each time step of the integration:
one checks at each stage whether y@tD < a, and if this is the case, one adjusts n to give
y@tD = a for that step, then, using the new value of n, one recomputes X and q from (14)
and (16), respectively. The integration continues until y@tD is strictly greater than a.
The height of the center of the hoop is y@tD = Y@tD - a l cos @qHtLD.
–
–
If one differentiates twice with respect to time and substitutes for Y@tD and q@tD
from equations (6) and (16) respectively, one obtains
° 2
M
y– @tD = n - g + a lIsin @qHtLD ÅÅÅÅ
ÅÅ a nH l sin @qHtLD + mk H1 + l cos@qHtLD L L + cos@qHtLD q@tD M
I
(17)
In the solution procedure just described, n is updated once every Dt using the
constraint y@tD ¥ a. For the duration of any single time step Dt, n is assumed to be
constant. For simplicity, we will take not just n, but also y– @tD to be constant over the
interval Dt. This implies that y° @t + DtD = y° @tD + y– @tD Dt and
y@t + DtD = y@tD + y° @tD Dt + ÅÅÅÅ12 y– @tD HDtL2
= y@tD + ÅÅÅÅ12 Dt Hy° @tD + y° @t + DtD L
One may view the above expressions for y° @t + DtD and y@t + DtD as arising from a
slightly modified version of the improved Euler method for numerical solution of
differential equations [2]. (The method is “improved” because the term second-order in
Dt in the expression for y@t + DtD represents a simple but significant improvement over
the basic Euler method.)
If y@t + DtD fails to be greater than or equal to a, then one sets y@t + DtD = a and
solves the above equation for n.
°
2 Aa-y@tD-Dt y° @tD- ÅÅ1ÅÅ D t2 I-g+a l cos@qHtLD q@tD2 ME
2
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
n = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
M a2
Dt2 A1+ ÅÅÅÅÅÅÅÅIÅÅÅÅÅÅÅÅ l sin@qHtLD Hl sin@qHtLD+H1+l cos@qHtLDL mk LE
The procedure may be represented by the following pseudocode:
18 Project Intermath
while (y[t] ≤ a) {
if (y[t+∆t] < a) recompute n to give y[t+∆t] = a;
use new n to compute X[t+∆t], Y[t+∆t], θ[t+∆t],
X [t+∆t], Y [t+∆t], θ[t+∆t];
t = t + ∆t;
} end while;
A remark is in order about a matter that arises frequently in numerical work:
the comparison of floating-point quantities. As we have said, the loop terminates when
the height y@tD of the center of the hoop is strictly greater than a. In order to determine
whether or not this is the case, the computer must compare two floating-point values,
and these are known to only a finite precision. To insure that the loop is not terminated
prematurely, the termination condition is tested only to a certain tolerance d, chosen so
as not to exceed the precision of the machine; the while loop ends when y@tD - a > d.
Exercise 13. Numerical Solution: Rolling WITH
Slipping
Use the programming language of your choice to implement the solution
procedure just described. Take mk = 0.7 and use your program to calculate the motion
of the hoop from the onset of slipping until time tHop when the hoop loses contact with
the supporting surface. To verify that the hoop is indeed slipping, print the values of
°
a q and of the horizontal velocity x° of the center of the hoop at selected times during the
integration. For your use in later exercises, note the values at t = tHop of X, Y, and q and
° °
°
of their time derivatives X, Y, and q.
Aside: Checking for Numerical Correctness
Whenever one computes an aproximation to a solution, one must try to
ascertain whether the solution is approximately correct. In the present case, the
numerical procedure necessitated the introduction of two parameters: the time step Dt
and the tolerance d. It is straightforward to assess the effect of the finite step size Dt on
the results of the computation: One would expect that for any given value of d, as
Dt Ø 0 the numerical solution ought to “converge” to a limiting "numerical function"
(i.e., table of values) and the takeoff time tHop , which marks the endpoint of the
numerical integration, ought to approach some limiting value. One can verify that this
The Hopping Hoop 19
is in fact the case. For instance, let us take d ê a = 10-4 .
Dt HsecL tHop HsecL
nsteps
0.002
0.0005
1.01507
0.99757
14
21
0.0001
0.00005
0.992471
0.991621
54
91
Interpreting the impact of the tolerance d on the final result demands a certain
amount of physical insight. Since floating-point quantities can be computed only to
finite precision, we have no choice but to implement the termination condition for the
numerical integration, namely, that y@tD be strictly greater than a, as y@tD - a > d; in
effect, we must consider the hoop to be in contact with the supporting surface until the
point of contact clears the surface by at least d. Since the parameter d effectively defines
when the hoop is considered to be airborne, the amount of time tHop - tSlip during
which the hoop rolls while slipping prior to takeoff depends on d, and so too does the
height of the hop. (By the “height of the hop” we mean the maximum height above the
supporting surface attained by the center of the hoop; it is max ( y@tD ) on the interval
0 § t § tImpact.) Roughly speaking, this is because the longer the rolling-with-slipping
phase prior to takeoff, the greater the takeoff speed that the hoop can build up; more
°
precisely: the greater the amount by which » y° » can exceed » Y ». Numerical simulations
confirm that the height of the hop decreases with d. We obtained the following results
for Dt = 0.002:
dêa
tHop nsteps hop height
0.01 1.39
0.001 1.049
0.0001 1.015
76
31
14
0.097
0.0011
0.00012
The observed decrease in the height of the hop with d might lead one to expect
in the limit d Ø 0 to observe no hop at all. However, the d Ø 0 limit is not only in
conflict with physical reality, it violates even the assumptions of this admittedly
idealized problem! The point is simply this: Even if we were able to perform our
numerical computations to infinite precision, physical reality will still not let us make d
arbitrarily small. It is, after all, microscopic imperfections in the hoop and the
supporting surface that give rise to friction in the first place, and the scale of these
20 Project Intermath
imperfections provides a physical lower bound to d. In the present case, we began with
the assumption that hoop and surface are sufficiently rough that the former would
initially roll without slipping over the latter. This precludes d Ø 0.
7. The Hop
When the hoop loses contact with the supporting surface, then it is clear that
all forces of contact must vanish: n@tD = f @tD = 0.
Exercise 14. Airborne Hoop
Assuming that the hoop is flying through the air, write down the equations of
motion for the center of mass coordinates X@tD and Y@tD and for the angular coordinate
q@tD. Given that the time at which the hoop loses contact with the supporting surface is
tHop, and that the values at that instant of the three coordinates and their time
°
°
°
derivatives are XHop , YHop , qHop , XHop, YHop, and qHop , solve the equations of motion to
obtain expressions for X@tD, Y@tD and q@tD valid while the hoop is in the air.
The airborne hoop simply rotates about its center of mass at a constant angular
speed equal to its angular speed at the instant the contact forces went to zero, while the
center of mass itself falls freely under the influence of gravity, describing the
characteristic parabolic trajectory. This combined free rotation/free fall continues until
the height of the center of the hoop decreases once again to a, at which time n@tD
acquires the positive value required to keep the point of contact between hoop and
ground from moving below ground level.
Exercise 15. Time of Impact
Compute the time tImpact at which the hopping hoop regains contact with the
supporting surface.
The Hopping Hoop 21
Exercise 16. Putting It All Together
Assemble your results from exercises 9, 11, 14, 13, and 14 into a complete
solution of the motion of the hoop, and use the resulting combined solution to create a
plot similar to the dots in figure 3 showing the motion of the hoop from time t = 0 until
tImpact.
8. Postscript: A Final Surprise
In computing the motion of the asymmetrically weighted hoop, we took q0 = 0;
that is, we assumed that the attached object is initially located at the top of the hoop,
directly above the point of contact. Would the behavior of the hoop have been
substantially different if the attached object had started out at the bottom of the hoop, i.e.,
with q0 = p? The answer is a resounding “yes”! We challenge the ambitious among you
to examine the behavior of the hoop when the total mechanical energy is the same as in
the case which we considered in this module, but with the attached object initially at
the bottom instead of the top.
Some care is required to modify equations (11–13) to the case where q0 = p (or
more generally, to the case where q0 assumes an arbitrary value), but if you do so
correctly you will find that the hoop begins to slip at a critical angle qSlip = 210.7 °, at
°
which point the angular speed of rotation is qSlip = 10.12 radians/second. You may then
proceed as in exercise 13 to compute a numerical solution for the motion of the hoop
during the phase in which it rolls with slipping along the supporting surface, and as in
exercise 14 to obtain a solution for the motion of the hoop during the phase in which it
is airborne.
The resulting motion, illustrated in figure 8 and readily observed
experimentally, is truly surprising. Note that for the experiment one can start with the
weight at the top and ignore the small hop that can occur on the way down. At some
point in the upward motion of the weight there should be a very big hop.
The electronic version of [5] contains a complete Mathematica package for
simulating the hopping hoop with a variety of initial conditions.
22 Project Intermath
3.5
3
2.5
2
1.5
1
0.5
0
-1
0
1
2
3
4
Figure 8. Motion of the hoop for q0 = p. The mechanical energy of the
system is identical to that for the case considered throughout this module.
The hoop is airborne for most of the motion depicted here; observe that
the center of mass (black line) follows the parabolic trajectory
characteristic of an object falling freely under the influence of gravity.
The Hopping Hoop 23
References
[1] B. Bollabás, editor, Littlewood’s Miscellany, Cambridge Univ. Pr., London, 1986.
[2] Martin Braun, Differential Equations and their Applications, 2nd ed., Springer, New
York, 1975, pp. 104-105.
[3] James P. Butler, Hopping hoops don’t hop, American Mathematical Monthly, 106
(1999) 565-568.
[4] Shokichi Iyanaga and Yukiyasi Kowada, Encyclopedic Dictionary of Mathematics, MIT
Press, Cambridge, Mass., 1977, p. 322.
[5] Tim Pritchett, A mechanical system with constraints, Mathematica in Education and
Research Vol 8:3-4 (1999) 20-27.
[6] Tim Pritchett, The hopping hoop revisited, American Mathematical Monthly 106
(1999) 609-617.
[7] T. F. Tokieda, The hopping hoop, American Mathematical Monthly 104 (1997) 152-154.
24 Project Intermath
Notes to the Instructor
This Interdisciplinary Lively Applications Project is probably best suited for use in
a course in mathematical modeling, and it was written with this in mind. We make no
assumptions regarding the physics background of the student audience. Those
students who have taken the first term of a typical calculus-based general physics
course will find themselves in familiar territory. But such a course is by no means
prerequisite to this module, since all the necessary physics is fully explained in the
module itself; in this sense, the module is self-contained. An instructor who wishes to
deemphasize the physics in favor of the mathematics can easily do so simply limiting
the number of optional (starred) exercises the students are required to complete.
Indeed, certain of the optional exercises (e.g., exercise 11) simply require the student to
work through the algebra leading to a given result. Such exercises can probably be
skipped with impunity, the stated results being accepted on faith.
The module can be implemented in several ways. It could, for instance, form
the basis for a semester project by a group of two or three students who would work
through it in its entirety. Students with an experimental bent may even wish to
construct a model of the hoop and use it to obtain qualitative experimental
confirmation of the hop. It is very exciting to see that physical reality matches the
model, and building a working model is not difficult: use a hula hoop with holes
drilled in the inside to accept four metal spokes, since it is essential that the hoop stay
rigid. Students who have an interest in both mathematics and physics might well enjoy
this experimental aspect. Alternately, all or a portion of the module could be used as a
class project spanning multiple class meetings, with selected exercises assigned as
homework.
This ILAP project could also serve well in a physics course in classical
mechanics, either with individual exercises assigned as homework, or as a semester
project by a small group.
Prior familiarity with a computer package (such as Mathematica, MathCad, or
Maple) offering routines for root-finding, numerical solution of ODEs, and symbolic
manipulation will free the students to concentrate on the mathematics and physics of
this project, and such familiarity, though not absolutely essential, is very desirable. On
the other hand, in order to complete exercise 15, it is essential that students have some
experience programming a computer in some language.
Absent that knowledge, the computer code necessary to compute the motion of
the hoop as it rolls with slipping along the supporting surface will have to be supplied
by the instructor. Perhaps the main point of this project is that an undergraduate can
handle a simulation where the frictional component is somewhat more complicated
The Hopping Hoop 25
than the familiar air resistance models. Creating such a simulation can be a valuable
exercise in the art of model-building, and in learning the capabilities (and limits and
quirks) of modern software.
Sample Solutions to Exercises
Exercise 1. Center of Mass
In this case, the system consists of two masses: the hoop with mass H1 - lL M
and the attached object with mass l M. We can express the position vectors in any
coordinate system we choose, so let’s choose a plane polar coordinate system whose
origin lies at the center of the hoop. In this system, the position vector of the hoop is the
null vector: ”rh = 0, and that of the attached object is given by ”rm = a r̀. (Here, r̀ is the unit
vector defined at every point in the plane to point radially away from the origin.
Although we do not need it here, we remind the student that the second unit vector in
`
plane polar coordinates, q, points in the direction of increasing polar angle q; this is
usually taken to be in the sense of counterclockwise rotation about the origin. For a
`
more complete discussion of r̀ and q, see, e.g., Daniel Kleppner and Robert J.
Kalenkow, An Introduction to Mechanics, McGraw-Hill, 1973.) The position vector of the
center of mass is then
÷ ” H1 - lL M ÿ 0 + a M l r̀
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = l a r̀
R = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
M H1 - lL + M l
÷”
Since R ∂ r̀, the center of mass of the system lies along the line connecting the
center of the hoop (the origin) with the position of the attached object.
We observe that for l Ø 1, i.e., when the attached object is much more massive
than the hoop, the center of mass coincides with the position of the object. Conversely,
for l Ø 0, i.e., when the hoop is much more massive than the attached object, the center
of mass lies at the origin (the center of the hoop).
Exercise 2. Cycloid Constraint
The rigid hoop constraint (1) relates the coordinates of the center of the hoop to
the coordinates of the system center of mass.
X@tD - x@tD = a l sin q@tD
Y@tD - y@tD = a l cos q@tD
Substituting for x@tD from the “no slipping constraint,” equation (3),
x@tD = a Hq@tD - q0 L,
26 Project Intermath
yields the given result for X@tD : X@qD = aHq - q0 + l sin@qDL, while substituting for y@tD
from the “normal force constraint,” equation (2), y@tD = a, yields the given result for
Y@tD : Y@qD = aH1 + l cos@qDL.
Exercise 3. Center of Mass Equations of Motion
Referring to the free body diagram, figure 7, we see that the net force acting to
–
the right is simply F, the tangential component of the contact force, so F = M X, or
–
–
X = F ê M = f . Similarly, the net upward force is N - M g, so N - M g = M Y, or
–
Y = N ê M - g = n - g.
Exercise 4. Equation of Rotational Motion
Of three forces shown in the free body diagram, figure 7, only two exert a
nonvanishing torque about the center of mass. (The weight M g, which acts at the
center of mass, can exert no torque about an axis through the center of mass because
the torque arm is zero.) The torque resulting from the normal component of the contact
force is N H X@tD - x@tD L, and this torque acts to increase q. The torque arising from the
tangential component of the contact force is given by -F Y@tD, and this torgue acts to
decrease the angle q, which is the reason for the minus sign. Summing these two
–
torques, we obtain, F Y@tD - N HX@tD - x@tD L = I q, or, with F = M f and N = M n,
–
q = ÅÅÅÅ1I 8M f Y@tD - M n HX@tD - x@tD L <, which is (7).
Exercise 5. Derivation of the Equation of Energy
Conservation
Taking our cue from the problem statement, we multiply both sides of (5) by
°
°
°
M X@tD, both sides of (6) by M Y@tD, and both sides of (7) by I q@tD. This gives
–
°
°
M X@tD X@tD = M f X@tD
–
°
°
M Y@tD Y@tD = MHn - gL Y@tD
–
°
°
I q@tD q@tD = M 8 nHX@tD - x@tD L - f Y@tD < q@tD
The left hand side of each of these equations can be written as a total derivative:
d M ° 2
°
ÅÅÅÅÅÅÅÅÅ 9 ÅÅÅÅÅÅÅÅÅ X@tD = = M f X@tD
dt 2
d M ° 2
°
ÅÅÅÅÅÅÅÅÅ 9 ÅÅÅÅÅÅÅÅÅ Y@tD = = M Hn - gL Y@tD
(18)
dt 2
d I ° 2
ÅÅÅÅÅÅÅÅÅ 9 ÅÅÅÅÅ q@tD = = M 8 nHX@tD - x@tDL - f Y@tD <
dt 2
Using the chain rule to differentiate (4) with respect to time, we get
The Hopping Hoop 27
°
°
°
°
X@tD = X£ @q@tDD q@tD = H1 + l cos@qD L q@tD = Y q@tD
°
°
°
°
Y@tD = Y£ @q@tDD q@tD = -lsin@qD q@tD = -HX - xL q@tD
(19)
We now add the three equations (18) and use (19) to eliminate the cancelling
pairs of terms involving n and f . The result is:
d M ° 2
d I ° 2
d M ° 2
°
ÅÅÅÅÅÅÅÅÅ 9 ÅÅÅÅÅÅÅÅÅ X@tD = + ÅÅÅÅÅÅÅÅÅ 9 ÅÅÅÅÅÅÅÅÅ Y@tD = + ÅÅÅÅÅÅÅÅÅ 9 ÅÅÅÅÅ q@tD = = -M g Y@tD
dt 2
dt 2
dt 2
or,
d M ° 2
M ° 2
I ° 2
ÅÅÅÅÅÅÅÅÅ 9 ÅÅÅÅÅÅÅÅÅ X@tD + ÅÅÅÅÅÅÅÅÅ Y@tD + ÅÅÅÅÅ q@tD + M g Y@tD = = 0
dt 2
2
2
Integration yields the energy conservation equation (8).
Exercise 6. Energy Conservation Equation in
Terms of q Only
Differentiating equations (4) with respect to time, we obtain
°
°
X@tD = a H1 + l cos@qHtLD L q@tD
°
°
Y@tD = -a l sin@qHtLD q@tD
In terms of q, the translational kinetic energy of the center of mass is thus given by
° 2 ° 2
° 2
ÅÅÅÅ12 M IX@tD + Y@tD M = ÅÅÅÅ12 M a2 H1 + l2 + 2 l cos@qHtLD L q@tD
and the gravitational potential energy is
M g Y@tD = M g aH1 + l cos@qHtLD L.
Substituting for these terms in (8), one obtains the required result (9).
Exercise 7. Moment of Inertia of the
Asymmetrically Weighted Hoop
The hoop has mass MHoop = H1 - lL M, assumed to be concentrated entirely in
the rim, a distance a from the center. Thus, the moment of inertia of the hoop about its
center is MHoop a2 , or H1 - lL M a2 . The parallel axis theorem states that the moment of
inertia of a rigid body about any axis is equal to the moment of inertia I0 about a
parallel axis passing through the center of mass of the body plus the product of the
mass m of the body and the square of the distance R between the two axes:
I = I0 + m R2 . Now, the center of mass of the hoop alone coincides with the geometric
center of the hoop, and, as you showed in Exercise 1, the center of mass of the system
consisting of the hoop and the attached object lies a distance R = l a from the center of the
28 Project Intermath
hoop. Thus, the moment of inertia of the hoop alone about an axis passing through the
center of mass of the hoop-object system is given by
IHoop = MHoop a2 + MHoop R2
= H1 - lL M a2 + H1 - lL MHl aL2
= H1 - lL M a2 H1 + l2 L
The attached object has mass l M and is situated a distance H1 - lL a from the
center of mass of the hoop-object system. If the object is a point mass, then the moment
of inertia of the attached object alone about an axis passing through the center of mass of the
hoop-object system is IObject = l MH1 - lL2 a2 . We add these two results to obtain the total
moment of inertia of the hoop-object system about an axis through its center of mass:
I = IHoop + IObject
= H1 - lL M a2 H1 + l2 L + l MH1 - lL2 a2
= M a2 H1 + l2 - l - l3 + l - 2 l2 + l3 L
= M a2 H1 - l2 L
Exercise 8. Mechanical Energy
°
°
Setting q@tD = 0 and q@tD = q0 in (9) gives immediately
° 2
constant = E = a M I g H1 + lL + a H1 + H1 + ÅÅÅÅ2¶ L lL q0 M
°2
One then substitutes g c for a q0 , and so obtains the result sought:
E = M a g HH1 + lL H1 + cL + 1 ê 2 ¶ l cL
°
Thus, for the weighted hoop rolling without slipping with q@0D = 0 and q@0D = è!!!!!!!!
g c ê !a!! ,
the equation of energy conservation (9) takes the form
° 2
a g M H1 + l Cos@q@tDDL + ÅÅÅÅ12 HI + a2 MH1 + l2 + 2 l Cos@qHtLD LL q@tD =
M a g @H1 + lL H1 + cL + ÅÅÅÅ12 ¶ l cD
°
Solving for q@tD, one obtains a single differential equation for the motion of the
hoop, equation (11).
Exercise 9. Motion on the Cycloid: Numerical
Solution for a Typical Case
When Mathematica solves a differential equation numerically it returns an
InterpolatingFunction object, which can be treated just as an ordinary function.
Using the Mathematica function NDSolve, we obtain the solution q1 in the form of such
an interpolating function
The Hopping Hoop 29
g = 9.8; a = 1; λ = 0.95; ∂ = 0.01; c = 0.1; t1 = 3.0;
θ1 = θ ê. NDSolveA
∂λ
c H1 + λ + L + λ H1 − Cos@θ@tDDL
g
2
''''''''''''''''''''''''''''''''
''''''''
'''' , θ@0D == 0=,
9θ @tD == $%%%%%%
&''''''''''''''''''''''''''''''''
∂λ
a
1 + + λ Cos@θ@tDD
2
θ, 8t, 0, t1 <EP1T
InterpolatingFunction@880., 3.<<, <>D
We can then use the function q1 to compute the positions of the center of mass, the
center of the hoop, and the attached object.
rCM@t_D := a 8θ1 @tD + λ Sin@θ1 @tDD, 1 + λ Cos@θ1 @tDD<
rCtrHoop@t_D := rCM@tD − λ a 8Sin@θ1 @tDD, Cos@θ1 @tDD<
rMass@t_D := rCtrHoop@tD + a 8Sin@θ1 @tDD, Cos@θ1 @tDD<
The following Mathematica code generates the plot.
ntSteps = 30;
∆t = t1 ê ntSteps;
ptsCM = Map@rCM, Range@0, t1 , ∆tDD;
ptsCtrHoop = Map@rCtrHoop , Range@0, t1 , ∆tDD;
ptsMass = Map@rMass , Range@0, t1 , ∆tDD;
lines = 8
[email protected],
Map@Line, Transpose@8ptsCtrHoop, ptsMass<DD<,
8AbsolutePointSize@4D, Map@Point, ptsMassD<,
8AbsolutePointSize@2D, Map@Point, ptsCtrHoopD<<;
ParametricPlot@Evaluate@rCM@ttDD, 8tt, 0, t1 <,
Epilog → lines,
AspectRatio → Automatic,
PlotRange → 8−0.1, 2.2<, Axes → None, Frame −> NoneD;
30 Project Intermath
Exercise 10. Motion on the Cycloid: Analytic
Solution for a Special Case
In the case of a massless hoop (l = 1) with attached point mass (¶ = 0), the
equation of motion (9) reduces to
"################
°
c + sin2########
A ÅÅ2qÅÅ E###
g
g
2 c + 1- cos@q@tDD
dq
"#####
########
#Å##Å# , or ÅÅÅÅ
q@tD = "#####
ÅÅÅÅa "################
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅ
Å
Å
=
ÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅ
q
1 + cos@q@tDD
dt
a
cosA ÅÅ2ÅÅ E
where we have made use of the trigonometric identities 1 + cos@qD = 2 cos2 @q ê 2D and
1 - cos@qD = 2 sin2 @q ê 2D. The above expression is separable, so we can straightforwardly
integrate. We will use Mathematica to perform these computations, although they can
equally well be done by hand.
Clear@g, a, cD
thetaIntegral = ‡
θ@tD
0
Cos@θ ê 2D
!!!!!!!!!
θ
è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
c + H Sin@θ ê 2D L2
i
j
z
θ@tD
θ@tD %2%%% y
è!!!
j
z
$%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%
−2 j
Log@
c
D
−
LogASinA
E
+
c
+
SinA
E Ez
j
z
j
z
j
z
2
2
k
{
The other side of the equation is of course just è!!!!!!!!
g ê a t.
i
y
g t
j thetaIntegral
z
z
$%%%%%%
j
z
integratedEqn = j
==
êê
Simplify
j
z
j
z
2
a 2
k
{
y
θ@tD
θ@tD 2
1 i
j g
z
j
z
LogASinA E + $%%%%%%%%%%%%%%%%
c + SinA%%%%%%%%%%%%%%%%
E%%%% E == j
t + Log@cDz
$%%%%%%
j
z
2
2
2
a
k
{
eqnsimpler = Simplify@Map@Exp, integratedEqnDD
1
θ@tD
θ@tD 2
SinA E + $%%%%%%%%%%%%%%%%
c + SinA%%%%%%%%%%%%%%%%
E%%%% == 2
2
2
g#
I"#####
a t+Log@cDM
θ@tD
θ@tD
sinThetaOver2 = SinA E ê. SolveAeqnsimpler, SinA EEP1T
2
2
g
g
1 "######
"######
1 è!!!
c − 2 a t J−1 + a t N
2
simplerSinThetaOver2 =
PowerExpand@Simplify@ExpToTrig@sinThetaOver2DDD
The Hopping Hoop 31
è!!!
g t
è!!!
c SinhA E
è!!!
2 a
θ@tD
θ@tD ê. SolveASinA E == simplerSinThetaOver2, θ@tDEP1T
2
è!!!
g t
è!!!
2 ArcSinA c SinhA EE
è!!!
2 a
The above expression for q@tD is valid as long as the hoop remains in contact
with the ground and rolls without slipping; in other words, as long as the attached
object is constrained to move along a cycloid.
Exercise 11. Explicit Expressions for the Forces of
Constraint
We wish to find the forces of constraint f and n required to keep the
asymmetrically weighted hoop rolling without slipping, i.e., the forces required to keep
the the center of mass moving along the cycloidal trajectory (4).
X@t_D := a Hθ@tD + λ Sin@θ@tDDL
Y@t_D := a H1 + λ Cos@θ@tDDL
°
We’ve already determined that under these conditions, q@tD must satisfy (11).
Clear@g, a, λ, ∂, cD;
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!
g è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
"#####
2 c + 2 λ + 2 c λ + c ∂ λ − 2 λ Cos@θ@tDD
a
θ = è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!
!!!!!!!!
2 + ∂ λ + 2 λ Cos@θ@tDD
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!
g è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
"#####
2 c + 2 λ + 2 c λ + c ∂ λ − 2 λ Cos@θ@tDD
a
è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!
!!!!!!!!!
2 + ∂ λ + 2 λ Cos@θ@tDD
–
d °
We will need an expression for the second derivative q@tD = ÅÅÅÅ
Å q@tD, so we differentiate
dt
one more time.
–
θ = Simplify@D@θ, tD ê. θ @tD → θD
H1 + cL g λ H2 + H2 + ∂L λL Sin@θ@tDD
a H2 + ∂ λ + 2 λ Cos@θ@tDDL2
2
d
From (5) we see that the required force of constraint f @tD is just given by ÅÅÅÅ
ÅÅÅÅ X@tD
dt2
forceOfConstraint = D@X@tD, 8t, 2<D
32 Project Intermath
a Hθ @tD + λ H−Sin@θ@tDD θ @tD2 + Cos@θ@tDD θ @tDLL
We substitute to eliminate the time derivatives of q and obtain an expression in terms
of x = cos@qD (//. abbreviates ReplaceRepeated, which causes the substitutions to
be made until there is no further change: twice in this case).
forceOfConstraint = Simplify@
–
forceOfConstraint êê. 8θ @tD → θ, θ @tD → θ, θ@tD → ArcCos@ξD<D
1
H2 + ∂ λ + 2 λ ξL2
è!!!!!!!!!!!!!
Ig λ 1 − ξ2 H2 + λ H−2 + ∂ + 6 ξL − c H2 + H2 + ∂L λL H1 + ∂ λ + λ ξL +
λ2 H−2 ∂ − 2 ξ + 3 ∂ ξ + 4 ξ2 LLM
2
d
In the same way, (6) guarantees that n@tD is just g + ÅÅÅÅ
ÅÅÅÅ Y@tD
dt2
normalForce = g + D@Y@tD, 8t, 2<D
g + a λ H−Cos@θ@tDD θ @tD2 − Sin@θ@tDD θ @tDL
normalForce =
–
Together@normalForce ê. θ @tD → θ ê. θ @tD → θ ê. θ@tD → ArcCos@ξDD
1
H2 + ∂ λ + 2 λ ξL2
H4 g + 4 g ∂ λ − 2 g λ2 − 2 c g λ2 + g ∂2 λ2 − 2 g λ3 − 2 c g λ3 − g ∂ λ3 −
c g ∂ λ3 + 8 g λ ξ − 4 c g λ ξ − 4 g λ2 ξ − 4 c g λ2 ξ + 4 g ∂ λ2 ξ −
4 c g ∂ λ2 ξ − 2 g ∂ λ3 ξ − 2 c g ∂ λ3 ξ − c g ∂2 λ3 ξ + 10 g λ2 ξ2 −
2 c g λ2 ξ2 − 2 g λ3 ξ2 − 2 c g λ3 ξ2 + 3 g ∂ λ3 ξ2 − c g ∂ λ3 ξ2 + 4 g λ3 ξ3 L
Exercise 12. Why the Hoop Must Slip
Rearranging (1), we obtain the following expression for the vertical height of
the center of the hoop
y@tD = Y@tD - a l cos @qHtLD
Differentiation with respect to time yields
°
°
y° @tD = Y@tD + a l sin @qHtLD q@tD
°
The first term, Y@tD, is the speed at which the center of mass falls under the
°
influence of gravity. The second term, a l sin @qHtLD q@tD, is the speed at which the center
of the hoop rises as a result of the rotation of the hoop about the center of mass. If the
hoop is to become airborne, then the magnitude of the latter term must exceed that of
The Hopping Hoop 33
the former. As long as the hoop rolls without slipping, however, then these two
quantities are equal in magnitude, as can be seen by differentiating the second equation
of the cycloid constraint (4).
Exercise 13. Onset of Slipping for a Massless
Hoop
With l = 1 (massless hoop) and ¶ = 0 (attached object is a point mass), the
expression (12) for the frictional force f required to keep the center of mass of the
weighted hoop moving along the cycloidal trajectory required by the constraints
becomes
g
"#########2###
1-x H4 x+4 x2 -4 c H1+xLL
g Hx-cL
"#########2###
1-x
f @xD = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ,
H2+2 xL2
x+1
while the corresponding expression (13) for the normal force reduces to
4 g x3 +8 g x2 +4 g x-4 c g-8 c g x-4 c g x2
n@xD = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = g Hx - cL,
H2 x+2L2
°2
where c = a q0 ê g. Alternately, since x = cos@qD,
f @qD = gHcos@qD - cL tan@q ê 2D
n@qD = gHcos@qD - cL
Slipping occurs at the minimum angle q for which » f @qD » = ms » n@qD ». Since 0 § c < 1,
this minimum angle will necessarily occur in a regime where both f and n are nonnegative and the absolute value signs may be dropped: f @qD = ms n@qD. One may see by
inspection that this equation has two possible solutions:
tan@q ê 2D = ms
and
cos@qD - c = 0
Thus
qSlip = Min(arccos@cD, 2 arctan@ms D)
Exercise 14. Onset of Slipping for a Typical Case
The hoop begins to slip at the minimum angle for which
H f @cos qD L2 = Hms n@cos qD L2 , where the forces of constraint f and n are given respectively
by (12) and (13). So, assuming the computations of exercise 11, we seek the roots of the
following equation.
forceOfConstraint2 == Hµs normalForceL2
34 Project Intermath
1
H2 + ∂ λ + 2 λ ξL4
Hg2 λ2 H1 − ξ2 L H2 + λ H−2 + ∂ + 6 ξL − c H2 + H2 + ∂L λL H1 + ∂ λ + λ ξL +
1
λ2 H−2 ∂ − 2 ξ + 3 ∂ ξ + 4 ξ2 LL ^ 2L == H2 + ∂ λ + 2 λ ξL4
HH4 g + 4 g ∂ λ − 2 g λ2 − 2 c g λ2 + g ∂2 λ2 − 2 g λ3 − 2 c g λ3 − g ∂ λ3 −
c g ∂ λ3 + 8 g λ ξ − 4 c g λ ξ − 4 g λ2 ξ − 4 c g λ2 ξ + 4 g ∂ λ2 ξ − 4 c g ∂
λ2 ξ − 2 g ∂ λ3 ξ − 2 c g ∂ λ3 ξ − c g ∂2 λ3 ξ + 10 g λ2 ξ2 − 2 c g λ2 ξ2 −
2 g λ3 ξ2 − 2 c g λ3 ξ2 + 3 g ∂ λ3 ξ2 − c g ∂ λ3 ξ2 + 4 g λ3 ξ3 L ^ 2 µ2s L
The denominators coincide, so we need look at numerators only.
g = 9.8; a = 1; λ = 0.95; ∂ = 0.01; c = 0.1; µs = 0.8;
slipEqn =
Numerator@forceOfConstraintD2 == Hµs Numerator@normalForceDL2
86.6761 H1 − ξ2 L H2 − 0.39095 H1.0095 + 0.95 ξL +
0.95 H−1.99 + 6 ξL + 0.9025 H−0.02 − 1.97 ξ + 4 ξ2 LL ^ 2 ==
0.64 H1.53795 + 31.9737 ξ + 68.4348 ξ2 + 33.6091 ξ3 L ^ 2
The relevant root is
θSlip = Min@ArcCos@Cases@ξ ê. NSolve@slipEqn, ξD, _RealDDD;
θSlip ê Degree
88.5037
So the hoop begins to slip at q = 88.5037 degrees. We use the solution q1 obtained in
exercise 9 to determine the corresponding time tSlip . First we generate a plot to show
that the desired time is near 1 second.
Plot@8θSlip, θ1 @tD<, 8t, 0, 3<D;
The Hopping Hoop 35
7
6
5
4
3
2
1
0.5
1
1.5
2
2.5
3
tSlip = t ê. FindRoot@θ1 @tD == θSlip, 8t, 1<DP1T
0.987071
cycloid@θ_D := a 8θ + λ Sin@θD, 1 + λ Cos@θD<
8XSlip, YSlip< = cycloid@θSlipD
8XSlip, YSlip< = D@cycloid@θ1 @tDD, tD ê. t → tSlip
θSlip = θ ê. θ@tD → θSlip
82.49436, 1.02481<
83.34709, −3.10171<
3.26607
Exercise 15. Numerical Solution: Rolling WITH
Slipping
µk = 0.7;
We employ a numerical integration scheme with fixed time step Dt.
∆t = 0.002;
(One could devise a more sophisticated scheme utilizing an adaptive step size, but this
is by far the simplest approach.) Our solution will consist of a table of values of the
three coordinates X, Y, and q computed at the discrete times tSlip , tSlip + Dt, tSlip + 2 Dt,
and so on. As a result, it is convenient to parameterize the evolution of the system in
terms not of the time t, but rather in terms of the integer variable i, which counts the
number of time steps completed. We define new variables relevant to this slipping
36 Project Intermath
phase: x[i] and y[i] for the horizontal and vertical coordinates of the center of mass,
vx[i] and vy[i] for the horizontal and vertical components of the center of mass velocity,
and th[i] and thDot[i] for the rotation angle q and its derivative. The values of this
quantities at the onset of slipping (i = 0) are given by
8x@0D, y@0D< = 8XSlip, YSlip<;
8vx@0D, vy@0D< = 8XSlip, YSlip<;
8th@0D, thDot@0D< = 8θSlip, θSlip<;
We define the height, vertical velocity, and vertical acceleration of the center of the
hoop
yctr@i_D := y@iD − a λ Cos@th@iDD
vyctr@i_D := vy@iD + a λ Sin@th@iDD thDot@iD
n
i
ayctr@i_D := n − g + a λ j
jSin@th@iDD a H1 − λ2 + λ ∂L
k
y
Hλ Sin@th@iDD + µk H1 + λ Cos@th@iDDLL + Cos@th@iDD thDot@iD2 z
z
{
where the expression for the acceleration is simply equation (17).
Finally, to be able to start the integration procedure, we require the value of the normal
force at the onset of slipping, as well as the value of the vertical acceleration of the
center of mass one time step prior to the onset of slipping. Both quantities may be
obtained from equation (13). We assume that the quantity is stored in normalForce
from the work in exercise 11.
normalForce
1.53795 + 31.9737 ξ + 68.4348 ξ2 + 33.6091 ξ3
H2.0095 + 1.9 ξL2
nCyc@ξ_D := Evaluate@normalForceD
One begins each step in the integration by checking explicitly to determine whether the
constraint is violated: yctr@iD < a. If it is, the normal force n to increased to give
yctr@iD = a for that time step, and, using the revised value of n, the new values x[i + 1],
y[i + 1], vx[i + 1], vy[i + 1] , th[i + 1], and thDot[i + 1] are computed for the next time step.
The procedure continues until yctr@iD is strictly greater than a. Of course, since all of
these comparisons involve floating point values, they can be done only to a given
tolerance
1
tolerance = ;
102
Thus, the termination condition actually takes the form
yctr[i]-a ¥ tolerance
The Hopping Hoop 37
At each time step, we compute the values x[i + 1], y[i + 1], vx[i + 1], vy@i + 1], th[i + 1],
and thDot[i + 1] for the next time step by assuming that the corresponding accelerations
are constant over a time step. For x and y, whose accelerations depend only on n
(which we update only once every time step), this is strictly true, and it is a very good
approximation for th.
The integration is accomplished in a For loop:
n = nCyc@Cos@θSlipDD;
ay@−1D = nCyc@Cos@θ1 @tSlip − ∆tDDD − g;
ForAi = 0, yctr@iD − a ≤ tolerance, i ++,
H* Compute normal force required to prevent yctr@i+1D < a. *L
IfAyctr@iD + vyctr@iD ∆t + ayctr@iD ∆t2 ê 2 < a, n = MaxA0,
1
i i
−j
j2 j
j−a − a λ Cos@th@iDD + ∆t2 H−g + a λ Cos@th@iDD thDot@iD2 L +
2
k k
i 2i
yy
∆t Ha λ Sin@th@iDD thDot@iD + vy@iDL + y@iDz
zz
zìj
j∆t j
j1 + λ
{{ k
k
Sin@th@iDD
yy
Hµk H1 + λ Cos@th@iDDL + λ Sin@th@iDDLz
zz
zEE;
2
H1 + ∂ λ − λ L
{{
H* Recalculate current accelerations to reflect the change in n. *L
ax@iD = −µk n;
ay@iD = n − g;
thDblDot@iD =
n
Hλ Sin@th@iDD + µk H1 + λ Cos@th@iDDLL;
a H1 − λ2 + λ ∂L
H* Compute new values of coordinates and their derivatives. *L
vx@i + 1D = vx@iD + ax@iD ∗ ∆t;
vy@i + 1D = vy@iD + ay@iD ∗ ∆t;
thDot@i + 1D = thDot@iD + thDblDot@iD ∗ ∆t;
x@i + 1D = x@iD + Hvx@iD + vx@i + 1DL ∗ ∆t ê 2;
y@i + 1D = y@iD + Hvy@iD + vy@i + 1DL ∗ ∆t ê 2;
th@i + 1D = th@iD + HthDot@iD + thDot@i + 1DL ∗ ∆t ê 2;
H* Estimate new value of n. *L
n = Max@0, n + Hay@iD − ay@i − 1DLD E
For future reference, we save the final values.
H* End For loop *L
38 Project Intermath
nsteps = i
tHop = tSlip + nsteps ∆t;
8XHop, YHop< = 8x@nstepsD, y@nstepsD<;
8XHop, YHop< = 8vx@nstepsD, vy@nstepsD<;
8θHop, θHop< = 8th@nstepsD, thDot@nstepsD<;
76
°
Finally, we generate the required table showing t, a q, and vxctr.
vxctr@i_D := vx@iD − a λ Cos@th@iDD thDot@iD
TableFormATable@
8tSlip + j ∗ ∆t, a ∗ thDot@jD, vxctr@jD<,
8j, 0, nsteps, 5<D,
TableHeadings −> 9None, 9"Time", "a
\ .\
θ ", "vx of hoop ctr"==E
.
Time
0.987071
0.997071
1.00707
1.01707
1.02707
1.03707
1.04707
1.05707
1.06707
1.07707
1.08707
1.09707
1.10707
1.11707
1.12707
1.13707
a θ
3.26607
3.36306
3.46729
3.5784
3.69599
3.81964
3.94886
4.08313
4.22189
4.36451
4.51032
4.65861
4.80861
4.9595
5.20052
8.57936
vx of hoop ctr
3.26607
3.36515
3.47351
3.59222
3.7223
3.86473
4.02037
4.18999
4.37417
4.57334
4.78768
5.01708
5.26115
5.51913
5.82928
7.72878
The Hopping Hoop 39
Exercise 16. Airborne Hoop
Setting n@tD = f @tD = 0 in (5), (6), and (7), we see that for t ¥ tHop , the equations of
motion reduce to
–
X@tD = 0
–
Y@tD = -g
–
q@tD = 0
For the given the initial conditions at t = tHop,
X@tHop D = XHop
Y@tHop D = YHop
q@tHop D = qHop
°
°
X@tHop D = XHop
°
°
Y@tHop D = YHop
°
°
q@tHop D = qHop
the equations of motion have the solution
°
X@tD = XHop + XHop Ht - tHop L
°
Y@tD = YHop + YHop Ht - tHop L - ÅÅÅÅ12 g Ht - tHop L2
°
q@tD = qHop + qHop Ht - tHop L
Exercise 17. Time of Impact
In order to see what’s going on, we start by plotting the height of the center of
the hoop.
yCH@tt_D :=
1
YHop + YHop Htt − tHopL − g Htt − tHopL2 − a λ Cos@θHop + θHop Htt − tHopLD
2
Plot@yCH@ttD, 8tt, tHop, tHop + 0.1<,
AxesLabel → 8"t", "yCtr \\\ Hoop"<D;
yCtr Hoop
1.1
1.08
1.06
1.04
1.02
1.14
1.16
1.18
0.98
1.22
1.24
t
0.96
Now we find tImpact.
tImpact = tt ê. FindRoot@yCH@ttD == a, 8tt, tHop + 0.5, tHop + 1<DP1T
40 Project Intermath
1.22917
Exercise 18. Putting It All Together
We combine the results of exercises 9, 13, and 14 into a single comprehensive
solution . . .
Clear@θSoln, rCMSolnD
rCMSoln@tt_ ê; 0 ≤ tt < tSlipD = rCM@ttD;
θSoln@tt_ ê; 0 ≤ tt < tSlipD = θ1 @ttD;
rCMSoln@tt_ ê; tSlip ≤ tt < tHopD = 8
Interpolation@
Table@8j ∆t, 8x@jD, vx@jD<<, 8j, 0, nsteps<DD@tt − tSlipD,
Interpolation@Table@8j ∆t, 8y@jD, vy@jD<<, 8j, 0, nsteps<DD@
tt − tSlipD<;
θSoln@tt_ ê; tSlip ≤ tt < tHopD = Interpolation@
Table@8j ∆t, 8th@jD, thDot@jD<<, 8j, 0, nsteps<DD@tt − tSlipD;
rCMSoln@tt_ ê; tt >= tHopD := 8XHop, YHop< +
8XHop, YHop< Htt − tHopL + 80, −g< ê 2 Htt − tHopL2
θSoln@tt_ ê; tt > tHopD := θHop + θHop Htt − tHopL
. . . and use it to create a plot.
The Hopping Hoop 41
rCtrHoop@t_D := rCMSoln@tD − λ a 8Sin@θSoln@tDD, Cos@θSoln@tDD<
rMass@t_D := rCtrHoop@tD + a 8Sin@θSoln@tDD, Cos@θSoln@tDD<
ntSteps = 40;
deltat = tImpact ê ntSteps;
ptsCM = Map@rCMSoln, Range@0, tImpact, deltatDD;
ptsCtrHoop = Map@rCtrHoop, Range@0, tImpact, deltatDD;
ptsMass = Map@rMass, Range@0, tImpact, deltatDD;
lines = [email protected],
Map@Line, Transpose@8ptsCtrHoop, ptsMass<DD<,
8AbsolutePointSize@4D, Map@Point, ptsMassD<,
8AbsolutePointSize@2D, Map@Point, ptsCtrHoopD<<;
ParametricPlot@rCMSoln@ttD, 8tt, 0, tImpact<,
Compiled → False, Epilog → lines, AspectRatio → Automatic,
PlotRange → 8−0.2, 2.2<, Frame → True, Axes → 8True, None<D;
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
42 Project Intermath
About the Authors
Stan Wagon studied at McGill (BSc) and Dartmouth (PhD), and is now a
professor at Macalester College. He has written many articles and books about
mathematics, and has received three prizes for his writing: The L. R. Ford award, the
Trevor Evans Award, and the Chauvenet Prize. His books include The Banach-Tarski
Paradox, Mathematica in Action, and Which Way Did the Bicycle Go?. He recently
appeared in Ripley's Believe-It-Or-Not for the square wheel bicycle that he built, and
which is on display at Macalester. Every July he teaches a Mathematica course in the
mountains of Colorado, and that is how the hopping hoop project got its start: he
purchased a hula hoop and he and Tim Pritchett tried to see the bounce. The hoop was
too flexible! Other interests include rock-climbing, wilderness skiing, mountain
climbing, and snow sculpture. In May 2000, he and three others skied to over 19000
feet on Mt. Logan during a 17-day expedition. In January 2000, his snow sculpture
team took second place at an international snow sculpting competition (images at
http://www.stanwagon.com).
Tim Pritchard graduated with distinction from the Integrated Science Program at
Northwestern University. He subsequently attended the Georg-August-Universität
The Hopping Hoop 43
Göttingen, where he received a vordiplom in Mathematics, and the University of
California Berkeley, from which he holds MA and Ph.D. degrees in Physics. Tim is
currently a professor at the U.S. Military Academy at West Point, where he divides his
time between teaching physics to cadets and pursuing his own research interests in
nonlinear optics. An avid cyclist, Tim recently spent a month touring northern Italy on
his (round-wheeled!) bicycle.