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A jumping cylinder on an inclined plane

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Raúl W Gómez at Universidad Nacional Autónoma de México
Raúl W Gómez
J. J. Hernández-Gómez
J. J. Hernández-Gómez
J. J. Hernández-Gómez
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Vivianne Marquina at Universidad Nacional Autónoma de México
Vivianne Marquina
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Abstract and Figures

The problem of a cylinder of mass m and radius r, with its centre of mass out of the cylinder's axis, rolling on an inclined plane that makes an angle α with respect to the horizontal, is analysed. The equation of motion is partially solved to obtain the site where the cylinder loses contact with the inclined plane (jumps). Several simplifications are made: the analysed system consists of an homogeneous disc with a one-dimensional straight line mass parallel to the disc axis at a distance y < r of the centre of the cylinder. To compare our results with experimental data, we use a styrofoam cylinder to which a long brass rod is embedded parallel to the disc axis at a distance y < r from it, so the centre of mass lies at a distance d from the centre of the cylinder. Then the disc rolls without slipping on a long wooden ramp inclined at 15 • , 30 • and 45 • with respect to the horizontal. To determine the jumping site, the movements are recorded with a high-speed video camera (Casio EX ZR100) at 240 and 480 frames per second. The experimental results agree well with the theoretical predictions. S Online supplementary data available from stacks.iop.org/EJP/33/1359/mmedia (Some figures may appear in colour only in the online journal)
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A jumping cylinder on an inclined plane
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2012 Eur. J. Phys. 33 1359
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IOP PUBLISHING EUROPEAN JOURNAL OF PHYSICS
Eur. J. Phys. 33 (2012) 1359–1365 doi:10.1088/0143-0807/33/5/1359
A jumping cylinder on an inclined plane
RWG
´
omez,JJHern
´
andez-G ´
omez and V Marquina
Facultad de Ciencias, Universidad Nacional Aut´
onoma de M´
exico, DF 04510, Mexico
E-mail: rgomez@unam.mx,jorge_hdz@ciencias.unam.mx and marquina@unam.mx
Received 28 April 2012, in final form 28 June 2012
Published 25 July 2012
Online at stacks.iop.org/EJP/33/1359
Abstract
The problem of a cylinder of mass mand radius r, with its centre of mass
out of the cylinder’s axis, rolling on an inclined plane that makes an angle α
with respect to the horizontal, is analysed. The equation of motion is partially
solved to obtain the site where the cylinder loses contact with the inclined
plane (jumps). Several simplifications are made: the analysed system consists
of an homogeneous disc with a one-dimensional straight line mass parallel to
the disc axis at a distance y<rof the centre of the cylinder. To compare our
results with experimental data, we use a styrofoam cylinder to which a long
brass rod is embedded parallel to the disc axis at a distance y<rfrom it, so
the centre of mass lies at a distance dfrom the centre of the cylinder. Then the
disc rolls without slipping on a long wooden ramp inclined at 15◦,30
◦and 45◦
with respect to the horizontal. To determine the jumping site, the movements
are recorded with a high-speed video camera (Casio EX ZR100) at 240 and
480 frames per second. The experimental results agree well with the theoretical
predictions.
SOnline supplementary data available from stacks.iop.org/EJP/33/1359/mmedia
(Some figures may appear in colour only in the online journal)
1. Introduction
The motion of a homogeneous cylinder rolling without slipping on a horizontal surface and
on an inclined plane is a typical problem in most mechanics textbooks, but only a few address
the same problem when the centre of mass of the cylinder is out of its axis; actually, we
only know of two examples where this case is put forward as an end-of-the-chapter exercise
[1,2]. Moreover, we have only found a few references that deal with similar movements
for symmetrical inhomogeneous bodies [3–9]. In [3], the Lagrangian formulation is used to
describe the motion of two cylinders moving in a horizontal plane. References [4–7]usea
cumbersome torque analysis in the solution of the movement. None of them establishes the
jumping site of an inhomogeneous cylinder. The purpose of this paper is to put forward a
common type of movement forgotten in most undergraduate and postgraduate textbooks.
0143-0807/12/051359+07$33.00 c
2012 IOP Publishing Ltd Printed in the UK & the USA 1359
1360 RWG
´
omez
et al
Figure 1. The rolling cylinder at an arbitrary position.
2. Theoretical solution
We analyse a simplified model of the inhomogeneous cylinder, in which we take the centre of
mass to be out of the geometrical centre of the cylinder. The mechanical system considered,
depicted in figure 1, consists of a homogeneous disc of mass m1, with a line mass m2located
at some distance from the centre of the cylinder. The physical quantities of interest in this
development are (figure 1) as follows.
•αis the inclined plane angle with respect to the horizontal;
•ris the cylinder radius;
•M=m1+m2is the total mass of the system;
•dis the distance from the centre of the cylinder to the centre of mass;
•θis the rotation angle;
•Pis the instantaneous axis of rotation;
•is the distance from Pto the centre of mass;
•ICM =1
2m1r2+m1M
m2d2is the moment of inertia of the cylinder with respect to a
perpendicular axis passing through the centre of mass;
•IP=ICM +M2is the moment of inertia with respect to P.
The simplest way to partially solve the equation of motion is through energy conservation.
After the cylinder has rolled a distance x=rθalong the inclined plane, its geometrical centre
has descended a distance h1=rθsin αand the centre of mass has descended a distance
h2=d−dcos θcos α, so the change Uin the potential energy is
U=−Mg[rθsin α+d(1−cos θ)cos α],(1)
where gis the value of the acceleration of gravity.
If the initial conditions are such that the cylinder is at rest and the centre of mass lies on a
line perpendicular to the inclined plane, then the change Kin the kinetic energy of the disc,
after it has rotated an angle θ,is
K=1
2IP˙
θ2,(2)
where IPis the cylinder’s moment of inertia with respect to the instantaneous axis of
rotation P.
Taking a reference frame in which the x-axis is along the inclined plane, the y-axis is
perpendicular to the same plane and the z-axis is perpendicular to the x–yplane, the following
relation holds between the different quantities involved in the problem (figure 1)
=r+d=ıdsin θ+
j
(r+dcos θ), (3)
A jumping cylinder on an inclined plane 1361
0
π
2
π
3
π
2
2
π
5π
2
3
π
7
π
2
4
π
9
π
2
5
π
11
π
2
6
π
13
π
2
7
π
15
π
2
8
π
θ
rads
Θ2
g
cos Α
dCos Θ
15°, r 10cm
Figure 2. Plot of ˙
θ2/gversus cos α/dcos θfor α=15◦. The first intersection is at θ=18.01 rad,
so the cylinder jumps at 180.1cm.
where is the vector from the centre of mass to the instantaneous axis of rotation Pand θis
the rotation angle. The magnitude of is
=r2+d2+2rd cos θ. (4)
Combining equations (1) and (2) one obtains
˙
θ2=2Mg d(1−cos θ)cos α+rθsin α
ICM +M(r2+d2+2rd cos θ).(5)
The condition the cylinder must satisfy in order to lose contact with the inclined plane is that
the normal to the inclined plane component of the centrifugal force,
FC=−Mω×(ω×d)=M(idsin θ+jdcos θ)˙
θ2,(6)
equals the normal component of the cylinder total weight, that is,
Md ˙
θ2cos θ=Mgcos α, (7)
from which the following expression can be obtained, using equation (5):
2Md(1−cos θ)cos α+rθsin α
ICM +M(r2+d2+2rd cos θ)=cos α
dcos θ.(8)
A simple way to arrive at a solution of this transcendental equation is plotting both sides
and looking for the first intersection of the resulting curves. The results for 15◦,30
◦and 45◦are
shown in figures 2,3and 4, respectively, where the dashed lines correspond to the right-hand
side of equation (6), while the continuous one corresponds to its left-hand side.
It is interesting to note the oscillation of the kinetic energy (proportional to the left-hand
side of equation (8)) due to the centrifugal force effect produced at different positions of the
centre of mass of the cylinder.
1362 RWG
´
omez
et al
0π
2
π3π
2
2π5π
2
3π7π
2
4π
θrads
Θ2
g
cos Α
dCos Θ
30°, r 10cm
Figure 3. Plot of ˙
θ2/gversus cos α/dcos θfor α=30◦. The first intersection is at θ=11.46 rad,
so the cylinder jumps at 114.6cm.
0π
2
π3π
2
2π5π
2
3π7π
2
4π
θrads
Θ2
g
cos Α
dCos Θ
45°, r 10cm
Figure 4. Plot of ˙
θ2/gversus cos α/dcos θfor α=45◦. The first intersection is at θ=5.34 rad,
so the cylinder jumps at 53.4cm.
A jumping cylinder on an inclined plane 1363
Figure 5. Photogram at the jumping site for an inclined plane at α=15◦.
3. Experimental results
To compare our theoretical results with experimental data, we use a styrofoam cylinder of radius
r=10.00 ±0.05 cm, height h=5.55 ±0.05 cm and mass m1=34.20 ±0.05 g, to which a
9.50±0.01 mm diameter and 5.10 ±0.01 cm long brass road of mass m2=32.10 ±0.05 g was
embedded parallel to the disc axis at a distance y=5.50±0.05 cm from it, so the centre of mass
lies at a distance d=2.65±0.09 cm from the centre of the cylinder. Then the disc rolls on a long
wooden ramp inclined at 15◦,30
◦and 45◦with respect to the horizontal, by means of squares
with such corresponding angles. To determine the jumping site, the motion was recorded with
a high-speed video camera (Casio EX ZR100) at 240 and 480 frames per second (fps), with
corresponding resolutions of 432 ×320 and 224 ×160 pixels. For the measurements, we used
the higher resolution (240 fps) videos (available from stacks.iop.org/EJP/33/1359/mmedia),
which leads to an uncertainty in the position of the jumping site of the cylinder of about 0.5 cm.
The acceleration of gravity in Mexico City is g=9.78 ±1ms
−2.
The main sources of error in our experiment were the initial position of the cylinder,
which was taken in such a way that its centre of mass lay normal to the inclined plane, and the
site where the cylinder actually jumps.
The initial position of the cylinder was set manually using as a guide a square fixed
perpendicularly to the inclined plane at the end where the motion started (see figures 5–7). The
centre of mass lay on a line from the cylinder’s point of contact to the centre of the embedded
rod. We estimate an error of about 3◦in the initial position. The jumping site is difficult
to determine in the video (available from stacks.iop.org/EJP/33/1359/mmedia), especially in
the 45◦case, because the jump takes place before the disc has completed a whole turn and
its speed is relatively small, so the jump is minute. We estimate a maximum error of about
5.0 cm in the measurements of the jumping site. Taking this into account, no error analysis was
attempted; instead, we report our experimental results in the jumping site with a maximum
error of ±5.0cm.
Figures 5–7are photograms, taken from the videos of the jumping sites (available from
stacks.iop.org/EJP/33/1359/mmedia), and in table 1we compare the theoretical solution with
the experimental results.
1364 RWG
´
omez
et al
Figure 6. Photogram at the jumping site for an inclined plane at α=30◦.
Figure 7. Photogram at the jumping site for an inclined plane at α=45◦.
Tab l e 1. Comparison of the experimental results with the theoretical predictions.
Inclined plane angle α(deg) 15 30 45
Theoretical jumping angle θτ(rad) 18.01 11.46 5.34
Theoretical jumping position (cm) 180.1 114.653.4
Experimental jumping position (cm) 185.0±5.0 115.0±5.055.0±5.0
Of special interest (to be seen in the videos) are: the 15◦case, where the speeding up and
slowing down of the cylinder are clearly seen, and the spectacular second and third jumps in
the 45◦case. The videos can be seen on YouTube 1.
Along with the paper, we present an animated plot of the graphics in figures 2–4, in which
we show the evolution of ˙
θ2/gversus cos α/dcos θwith respect to the inclination angle αof
1http://www.youtube.com/watch?v=iuemR3XtSSE.
A jumping cylinder on an inclined plane 1365
the plane, within the interval from 0 to π/2. From the animation, it is clear that when α−→ 0,
the jump would never take place, while when α−→ π/2 the jumping site positions tend to
θ=0.
4. Conclusion
The first integral of the equation of motion of a cylinder whose centre of mass is not at its
geometrical centre, and rolls on an inclined plane, is obtained. From this solution, the site
where the cylinder should ‘jump’ is determined. An experimental setup that resembles the
assumptions made to obtain the theoretical solution is furnished. The experimental results
agree well with the theory. We believe that this is the first time that this problem has been
theoretically and experimentally addressed.
Acknowledgments
This work was partially supported by DGAPA-UNAM IN115612, Mexico.
References
[1] Landau L D and Lifshitz E M 1976 Mechanics (Course of Theoretical Physics vol 1)3rd edn (Oxford: Pergamon)
p 103
[2] Wittenburg J 2008 Dynamics of Multibody Systems (Berlin: Springer) problem 3.5
[3] Carnevalia A and Mayb R 2005 Rolling motion of non-axisymmetric cylinders Am.J.Phys.73 909–13
[4] Akulenko L D, Bolotnik N N, Kumakshev S A and Nesterov S V 2006 Control of motion of an inhomogeneous
cylinder with internal movable masses along a horizontal plane J. Appl. Math. Mech. 70 843–58
[5] Turner L and Turner A M 2010 Asymmetric rolling bodies and the phantom torque Am. J. Phys. 78 905–8
[6] Jensen J H 2011 Rules for rolling as a rotation about the instantaneous point of contact Eur. J. Phys. 32 389–97
[7] B Y-K Hu 2011 Rolling of asymmetric discs on an inclined plane Eur. J. Phys. 32 L51–4
[8] Narayanan P, Rostamian R, Tasch U, Lefcourt A M and Kim M S 2008 Rolling dynamics of an inhomogeneous
ball on an inclined planed track http://www.math.umbc.edu/∼rouben/pubs/rolling-ball.pdf
[9] Rutstam N 2010 Study of equations for Tippe Top and related rigid bodies PhD Thesis Link¨
oping University
http://liu.diva-portal.org/smash/get/diva2:359340/FULLTEXT01

Supplementary resource (1)

A Jumping Cylinder on an Incline
Preprint
April 2012
Raúl W Gómez · J J Hernández-Gómez · Vivianne Marquina
... In all the references we have found (with the exception of [21]), it was assumed that the necessary condition for the jump is when the normal force F y (acting perpendicularly from the ramp over the cylinder) vanishes at the instant of the hop. Let us remark that a different jump condition will be used in this work, and this condition turns out to be equivalent to that of [21]. ...
... In all the references we have found (with the exception of [21]), it was assumed that the necessary condition for the jump is when the normal force F y (acting perpendicularly from the ramp over the cylinder) vanishes at the instant of the hop. Let us remark that a different jump condition will be used in this work, and this condition turns out to be equivalent to that of [21]. As argued in reference [21], the equation that the angle θ and the angular velocityθ must satisfy at the position where the cylinder jumps are given by −g cos α + dθ 2 cos θ = 0. ...
... Let us remark that a different jump condition will be used in this work, and this condition turns out to be equivalent to that of [21]. As argued in reference [21], the equation that the angle θ and the angular velocityθ must satisfy at the position where the cylinder jumps are given by −g cos α + dθ 2 cos θ = 0. ...
Jump effect of an eccentric cylinder rolling on an inclined plane
Article
Full-text available
This work addresses the intriguing phenomenon of the jump effect exhibited by an eccentric cylinder rolling on an inclined plane. Our main objective is to determine the region of the parameter space where the cylinder can undergo a jump after pure rolling motion. In previous works, it was assumed that slipping always precedes a jump, leading to the belief that slip is necessary. In contrast, our study challenges this prevailing notion by demonstrating the existence of situations where a jump can occur after pure rolling motion. We present a detailed theoretical description of the jump effect, investigating the dependence of the jump behavior on the initial conditions, the forces involved, and the parameter values. Through rigorous analysis, we determine the restricted region in the parameter space that allows for jumps to occur without slipping. Our findings contribute to the ongoing debate surrounding the jump effect in eccentric cylinder dynamics.
View
... In all references we have found (with exception of [20]), it was assumed that the necessary condition for the jump is when the normal force F y (acting perpendicularly from the ramp over the cylinder) vanishes (F y = 0) at the instant of the hop. Let us remark that a different jump condition will be used in this work, and this condition turns out to be equivalent to that of [20] but derived from a different approach. ...
... In all references we have found (with exception of [20]), it was assumed that the necessary condition for the jump is when the normal force F y (acting perpendicularly from the ramp over the cylinder) vanishes (F y = 0) at the instant of the hop. Let us remark that a different jump condition will be used in this work, and this condition turns out to be equivalent to that of [20] but derived from a different approach. As argued in reference [20], the equation that the angle θ and the angular velocityθ must satisfy at the position where the cylinder jumps is given by −g cos α + dθ 2 cos θ = 0. ...
... Let us remark that a different jump condition will be used in this work, and this condition turns out to be equivalent to that of [20] but derived from a different approach. As argued in reference [20], the equation that the angle θ and the angular velocityθ must satisfy at the position where the cylinder jumps is given by −g cos α + dθ 2 cos θ = 0. ...
The jump effect of a general eccentric cylinder rolling on a ramp
Preprint
Full-text available
  • Mar 2023
Interesting phenomena happen when an eccentric rigid body is rolling on an inclined or horizontal plane. For example, a variety of motions between rolling and sliding is exhibited until suddenly a jump occurs. We provide a detailed theoretical description of the last effect for a general eccentric cylinder. Before the jump, when the cylinder moves along the ramp, we could assume a pure rolling motion, however it turns out that when the cylinder is reaching its jump position, both the normal and static frictional forces approach zero. Thus, it seems that there will no longer be sufficient force to maintain rolling without slip. In order to have a jump without slipping, we prove that the parameters that characterize the dynamic behavior of the cylinder must belong to some restricted region.
View
... Littlewood claimed that a massless hoop with an added point mass will hop, but skids first [9]. In the years since, numerous studies have investigated the oscillatory (rocking) motion of the disk, if the disk or hoop will hop, the necessary conditions for hop, and the angular position of hop [2,3,6,10,12,[14][15][16][17]. Additionally, recent studies have shown similar intriguing behavior (i.e., rocking, rolling, slipping, and hopping) in elliptic disks or cylinders whose center of mass rises and falls during general planar motion on a surface [4,5,7,8]. ...
... Thus, there will no longer be sufficient force to maintain roll without slip. The follow-up studies to Tokieda, in addition to later investigations by Ontorato et al and Gomez et al, suggest that a mass asymmetric disk has three qualitatively distinct regimes of dynamic behavior: roll without slip, roll with slip, and hop [2,6,9,12,13,15,17]. Furthermore, Butler, Pritchett, and Theron note that slip can manifest as either skidding or spinning. ...
... In all instances the model was able to capture the onset and direction of slip as well as the onset of hop. In simulation and experiment, slip preceded hop further verifying the findings of [6,12,13]. Additionally this paper is, to the best of the authors knowledge, the first investigation of the behavior of an eccentric disk in a Lagrangian framework. Subsequent works may wish to model the disk's projectile motion, impact, and restitution to predict subsequent hops. ...
View
... The pure rolling motion of a homogeneous cylinder is a typical example dealt with in most mechanics textbooks, but only few of them address the same problem when the center of mass of the cylinder lies out of its axis. Some recent papers discussed the dynamics of a not azimuthally symmetric body rolling on a horizontal 1,2,3 or inclined plane 4,5 showing how an in depth study of asymmetric rolling bodies can stimulate a more accurate comprehension of the physics of rolling. ...
... As a consequence, the asymmetric rolling spool (ARS) can lose grip and start slipping (for the same reason an ARS can jump when it is rolling down an inclined plane. 5 FIG The mechanical system (Figure 1), consists of a symmetric body of mass M 0 , and radius R with some extra masses M i located at distance r i from the center. From a theoretical point of view, the only simplifying assumption we make is to ignore possible vibrations and oscillations of the spool due to the displacement of the center of mass of the system in the "depth" dimension (referring to figure 2). ...
... Note that ( ) eff I ϑ in eq. (2) corresponds to ( ) P I ϑ as calculated by ref. [5] and [1] for different systems. ...
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We designed a very simple asymmetric rolling spool and we studied it experimentally, in the librational regime, by using a video camera and video analysis software. Comparison of experimental results and theoretical analysis allows addressing critical aspects related both to the characteristic of the oscillatory anharmonic motion of the system and to the role of friction forces in determining it. For that reasons the asymmetric rolling spool can be presented to undergraduate students as an interesting case study where to apply their physics knowledge to understand the behavior of real objects.
View
... Be it a hoop, disk, cylinder, or sphere rolling without slipping down an inclined plane, when treated as a planar problem, all the treatments are essentially the same. These treatments have been extended to include spheres that have a center of mass that is not at the geometric centroid, 1 asymmetric disks, 2-4 spinning balls, [5][6][7][8][9] cylinders, 10,11 and prisms. 12 Perhaps the problem most closely aligned with a rolling ellipse is the so-called "massless hoop" problem initially posited by Littlewood in 1953. ...
... when combined with Eq. (10). Note that we use the branch of the arccotangent function for which / ¼ Àp=2 when h ¼ 0. ...
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An ellipse on a non-slip inclined plane can rock, roll, or jump. Below a threshold energy, it rocks about its static equilibrium configuration. Above this energy, it rolls, and, if it rolls, it will eventually jump off the incline after some number of rolls. It is shown that jumping can only occur in certain configurations of the ellipse.
View
... In this work we analyzed a classical mechanical problem: a disc with a circular hole rotating and ascending on a inclined plane. There are several similar studies in the literature [2,3], then in this brief report the principal idea is to show two of the three modes of representation previously mentioned. First, we present the basic equations in both Lagrangian and Newtonian formalisms Figure 1. ...
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... Ciekawym aspektem związanym z ruchem niewyważonej bryły po równi pochyłej jest możliwość jej "wyskoku" z równi [11]. Wspomniane oderwanie ciała od równi zachodzi, gdy spełniony jest następujący warunek: ...
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View
... We develop the analysis of the harmonic and the anharmonic regimes of oscillation of the spool when an increasing number of extra masses is added and discuss how the normal force, which determines the value of the friction force, is not equal to the weight of the spool and depends on its motion. As a consequence, the system can lose grip and start slipping [2, 3] (for the same reason the spool can jump when it is rolling down an inclined plane [4,5]). ...
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  • Nov 2007
  • J FLUID MECH
We consider the dynamics of a hollow cylindrical shell that is filled with viscous fluid and another, nested solid cylinder, and allowed to roll down an inclined plane. A mathematical model is compared to simple experiments. Two types of behaviour are observed experimentally: on steeper slopes, the device accelerates; on shallower inclines, the cylinders rock and roll unsteadily downhill, with a speed that is constant on average. The theory also predicts runaway and unsteady rolling motions. For the rolling solutions, however, the inner cylinder cannot be suspended in the fluid by the motion of the outer cylinder, and instead falls inexorably toward the outer cylinder. Whilst ‘contact’ only occurs after an infinite time, the system slows progressively as the gap between the cylinders narrows, owing to heightened viscous dissipation. Such a deceleration is not observed in the experiments, suggesting that some mechanism limits the approach to contact. Coating the surface of the inner cylinder with sandpaper of different grades changes the rolling speed, consistent with the notion that surface roughness is responsible for limiting the acceleration.
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Rolling of asymmetric discs on an inclined plane
Article
Full-text available
  • Oct 2011
In recent papers, Turner and Turner (2010 Am. J. Phys. 78 905–7) and Jensen (2011 Eur. J. Phys. 32 389–97) analysed the motion of asymmetric rolling rigid bodies on a horizontal plane. These papers addressed the common misconception that the instantaneous point of contact of the rolling body with the plane can be used to evaluate the angular momentum L and the torque in the equation of motion . To obtain the correct equation of motion, the 'phantom torque' or various rules that depend on the motion of the point about which L and are evaluated were discussed. In this letter, I consider asymmetric discs rolling down an inclined plane and describe the most basic way of obtaining the correct equation of motion, that is, to choose the point about which L and are evaluated that is stationary in an inertial frame.
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ROLLING DYNAMICS OF AN INHOMOGENEOUS BALL ON AN INCLINED TRACK
Article
Experiments and computer simulations show that when a rigid ball with inhomoge- neous but symmetric density distribution rolls on an inclined track, it tends to adjust its orientation to rotate about its major axis of inertia. We present tentative arguments toward explaining that behavior and sketch ideas that may lead to rigorous analysis and better understanding of the phe- nomenon. 1. Introduction. In this article we report on experimental results and computer simulations of an interesting phenomenon observed in a simple rigid body motion. We consider a rigid spherical ball of radius R with an inhomogeneous mass distri- bution. We assume that its mass is distributed symmetrically about the ball's center, therefore the center of mass coincides with the geometric center. However, the three principal moments of inertia I1 I2 I3 (relative to the center) are not necessarily equal. We construct a \track" consisting of a pair of parallel slender rigid cylinders (rails) of radius r each, with their axes set at a distance of a apart. The track is inclined and makes an angle of relative to the ground; see Figure 1.1. The ball is placed on the track and allowed to roll down. We take a > 2r so that the rails do not overlap and 2R > a 2r so that the ball does not fall through the gap between the tracks. The contact between the ball and the rails is of Coulomb type, i.e., dry friction. The ball will slip at the contact points if the tangential contact force exceeds what the dry friction can support. The ball may lift o the tracks, losing contact with one or both rails, if the dynamics so dictates. The initial conditions of the motion may be quite generic and is not instrumental to the discussion that follows. In the simplest case, we start with the ball in contact with both rails, with zero linear and angular velocities, and its principal axes of moment of inertia oriented in a random way. As the ball rolls down the track, it picks up speed and spins progressively faster. In the early stages of the motion it maintains no-slip contact with both rails. However as its angular velocity increases, the inhomogeneous distribution of mass results in dynamic imbalance (due to unequal principal moments of inertia) and a sequence of events ensues. Depending on the situation, one or both contacts may begin to slip. It is also possible that dynamic forces may cause the ball to lift o
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Asymmetric rolling bodies and the phantom torque
Article
  • Sep 2010
The center of mass and the point of contact of two-dimensional rolling bodies are used almost interchangeably to understand the dynamics of rolling bodies in introductory physics courses. The use of the contact point has the virtue that one does not need to know the normal and frictional forces at the point of contact. However, the correct use of the point of contact requires a concomitant torque that we call a ``phantom torque.'' This torque vanishes for azimuthally symmetric bodies that are conventionally used as examples in introductory physics courses and, thus, goes unnoticed. We show that the phantom torque can be evaluated easily.
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Rolling motion of non-axisymmetric cylinders
Article
  • Oct 2005
Theoretical and experimental results are compared for the rolling motion of cylinders on a ramp. An asymmetric distribution of the mass makes the motion jerky and complex and is an interesting and simple example of Lagrangian mechanics.
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Rules for rolling as a rotation about the instantaneous point of contact
Article
  • Jan 2011
It is a widespread misunderstanding in introductory physics courses that the motion of rolling bodies in general can be calculated using the point of contact as a reference point when equating the rate of change of angular momentum to the torque. In this paper I discuss in general two correct rules to be used instead, in order to derive the equation of motion of rolling bodies, taking moments about the point of contact. I also clarify that the point of contact either can be reckoned the fixed point on the rolling body instantly at the point of contact or the geometrical point defined as the point of contact at any time. Altogether this gives four different ways of deriving the equation of motion with some of them being easier than others depending on the case under consideration. The four different methods are as an illustration applied to a case presented recently by Turner and Turner (2010 Am. J. Phys. 78 905–7).
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Control of motion of an inhomogeneous cylinder with internal movable masses along a horizontal plane
Article
  • Dec 2006
  • PMM-J APPL MATH MEC+
The controlled motion of a rigid inhomogeneous cylinder over a rough horizontal plane is considered. The control is provided by controlled motion of internal masses. Mathematical models are constructed that correspond to rolling without loss of contact or slippage. The conditions for the physical implementability of such a motion are derived. The case where the internal moving masses from a rigid flywheel the centre of inertia of which lies on the axis of the cylinder is investigated in detail. A near-time-optimal feedback control that enables the total energy to be changed in a required way is constructed on the basis of an asymptotic approach. The main operating modes are simulated, namely, swinging up of the cylinder to a large angular amplitude, rotation with a prescribed energy, deceleration of rolling to a complete stop, and oscillations and rotations in the neighbourhood of the separatrix.
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  • Jan 1976
  • L D Landau
  • E Lifshitz
Landau L D and Lifshitz E M 1976 Mechanics (Course of Theoretical Physics vol 1) 3rd edn (Oxford: Pergamon) p 103