Ball on the EdgePaul Beeken

Citation: The Physics Teacher 42, 366 (2004); doi: 10.1119/1.1790346 View online: http://dx.doi.org/10.1119/1.1790346 View Table of Contents: http://scitation.aip.org/content/aapt/journal/tpt/42/6?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Measuring the Flight Speed of Fire Bombers from Photos: An InClass Exercise in Introductory Kinematics Phys. Teach. 48, 106 (2010); 10.1119/1.3293657 Why Learn Physics? Phys. Teach. 47, 478 (2009); 10.1119/1.3225519 The Moment of Inertia of a Tennis Ball Phys. Teach. 43, 503 (2005); 10.1119/1.2120375 Television, football, and physics: Experiments in kinematics Phys. Teach. 43, 393 (2005); 10.1119/1.2033534 Ball Over the Edge Phys. Teach. 42, 516 (2004); 10.1119/1.1828715

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366 DOI: 10.1119/1.1790346 THE PHYSICS TEACHER ◆ Vol. 42, September 2004

While performing a simple kinematics ex-periment that involved rolling a ball offthe edge of a table, many of my students

observed a systematic error. Specifically, the horizon-tal component of the ball’s velocity when it hit thefloor was measured to be consistently higher than theinitial velocity it had at the edge of the table. As weincreased the initial speed of the ball, effect disap-peared. The repeatable nature of the observationdemonstrated that what we were seeing was real. Pre-sented here is a discussion of the problem using ananalysis accessible to students taking an algebra-basedphysics class.

The Effect When a ball rolls off a track or table, the action of

rolling over the edge can give a “kick” to the horizon-tal component of the velocity of the ball. Considerthe situation illustrated in Fig. 1. The ball has mass m,radius R, moment of inertia I, and initial velocity vi.Its total kinetic energy can be written as (rememberthat the angular speed � is equal to vi/R if there is noslipping as the ball rolls):

�1

2�mvi

2 + �1

2�I�2 . (1)

As it progresses over the edge, its center of mass(c.m.) drops, and it gains additional energy. Afterfalling a distance �h, its kinetic energy is

mg�h + �1

2�mvi

2 + �1

2�I�2 . (2)

There is a well-defined relationship between theheight through which the ball falls (�h) and the angle� defined in Fig. 1:

�h = R – R cos �. (3)

In modeling this situation, we recognize that theangle � is the most convenient dynamic variable, andso we cast our solutions in terms of �. Equating theenergy of the ball [Eq. (2)] as it rolls over the edgewith the total energy rotating about an axis throughthe contact point:

mg�h + �1

2�mvi

2 + �1

2�I�2 = �

1

2� Iedge ��

R

S��

2, (4)

where S is the speed of the ball’s center of mass as itmoves in an arc around the edge of the table. At thispoint it is important to emphasize the “no-slip”

Ball on the EdgePaul Beeken, Byram Hills High School, Armonk, NY

Fig. 1. Illustration of important vectors operating onthe ball going over the edge of the table. Note that S,the c.m. speed, changes as the ball rolls over the edge.

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THE PHYSICS TEACHER ◆ Vol. 42, September 2004 367

assumption for the ball (vi = R�). Substituting for�h using Eq. (3) and assuming a solid sphere so thatI = 2/5mR2 and Iedge = 7/5mR2, we find after simpli-fication:

S2 = �1

7

0�g(R – R cos �) + vi

2. (5)

Now let Sfbe the value of S at the instant that con-

tact with the table is broken. The horizontal compo-nent of its velocity is thus given by

Sfx = Sf cos �. (6)

The next step is to determine the value of � at theinstant the ball leaves the edge of the table. The ballleaves the table when the normal force acting on theball by the table drops to zero. Referring to Fig. 1and applying Newton’s second law gives

FG cos � – FN = m �S

R

2

�. (7)

This relation shows how the normal force decreasesas the speed of the ball increases. When the normalforce reaches zero, the ball leaves the edge. Settingmg = FG we obtain

mg cos � = m �S

Rf2

�. (8)

Combining Eq. (5) with Eq. (8), we find

cos � = �10g

1

R

7

+

gR

7vi2

�. (9)

Substituting into Eq. (5), we obtain

Sf = ��1�0�gR�

17�+� 7�vi�2

��. (10)

Finally, substituting this expression into Eq. (6) givesus the expression for the horizontal velocity as theball leaves the edge:

Sfx = �g

1

R� ��10gR

17+ 7vi

2��3/2

: vi2 < gR. (11)

Right away we can see some limiting values. As theinitial velocity approaches zero, there is a non-zero xcomponent of the velocity of the ball as it hits theground just from the kick of rolling off the edge. Atthe other extreme we can see that when the initial

speed is high enough, the angle � will approach zero,meaning that the ball “skips over the edge” without anincrease in horizontal velocity.

Using the limiting conditions outlined in the previ-ous paragraph, we can find what the minimum mea-sured horizontal speed might be for a ball rolling overthe edge. Letting vi → 0 in Eq. (9), we find that � →

53.9�. In the same limit Eq. (11) becomes

Sfx = ��1

1

0

7��

3/2�gR�, (12)

which means that the minimum speed should scaleas the square root of the radius of the ball. By thesame token, we can determine the minimum speedfor which we would no longer see an effect. When vi = �gR�, the ball is moving so fast that the finalhorizontal component is the same as the initialvelocity.

As long as vi < �gR�, we can use Eq. (11) to com-pare the horizontal component of the ball’s velocityafter it leaves the table with its initial speed vi. The re-sults of this comparison for a solid sphere having a

Fig. 2. Plot of the final x component of the velocity of theball versus the initial x component of the velocity. Thegreen line represents the analytical solution derived in thispaper, while the blue points come from a numerical inte-gration of the equations of motion. The dashed line is thenaïve expectation of the initial x component of velocitybeing the same as the final x component of the velocity(shown for reference). The points with error bars are mea-surements made from rolling a 1-in diameter steel ball overthe edge of an aluminum track.

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368 THE PHYSICS TEACHER ◆ Vol. 42, September 2004

1-in diameter are shown in Fig. 2.As a check we can also set up the differential equa-

tion that describes the equation of motion for a sphereconstrained to roll over the edge of the table subject tothe same limiting speed described above. This speeddetermines when the ball actually leaves the edge atthe point where the centripetal, gravitational, andnormal forces all balance. The solution hasn’t a closedform, so it must be integrated numerically. The valuesare identical to those derived algebraically.

Finally, an experiment was set up that closelymatched the one done by the students who observedthe effect in the first place. We used a 1-in diametersteel ball bearing rolling down an aluminum track andmeasured the speed before it left the edge using a PASCO sonic meter.1 The speed after leaving theedge was calculated from the position it hit the floorand its time of fall. These values are shown in Fig. 2,along with the theoretical curve.

While the analysis in this paper assumes a solidsphere, obviously the same analysis would work withother round objects as well. By substituting the mo-ment of inertia for a hollow sphere or a cylinder, onecan derive expressions for the kick these objects re-ceive.

ConclusionWe don’t always get what we expect. The systemat-

ic “error” encountered by students in rolling a ball offthe edge of a table led to a closer examination of thedetails of the experimental setup. A careful mechani-cal analysis readily accessible to the students allowedus to formulate a theory that successfully accountedfor the discrepancy seen in their data.

AcknowledgmentThe author would like to gratefully acknowledge theassistance of Lindsay Yao in acquiring the experimen-tal data, and the author’s entire Physics A class fornoticing the discrepancy in their 2D kinematics data.

Reference1. PASCO scientific, 10101 Foothills Blvd., Roseville, CA

95747-7100; http://www.pasco.com.

PACS codes: 01.50P, 46.02C, 46.05B

Paul Beeken received his Ph.D. from Columbia Universityand was a research staff member at California Institute ofTechnology and IBM Watson Research Center beforereturning to teaching. He currently teaches physics atByram Hills High School and at PACE University.

Byram Hills High School, 12 Tripp Lane, Armonk, NY10504; PACE University, White Plains, NY;pBeeken@byramhills.org

The Great Polymath“A quick quiz: Through meticulous observations with a 20-meter-long telescope that vibrated in

the slightest breeze, this 17th century scientist was the first to describe the shadow that Saturn’s

ring cast on the planet and to make detailed maps of the moon’s craters. He was an accom-

plished surveyor and architect who helped rebuild London after the Great Fire of 1666 and an

avid inventor whose creations include the balance spring watch and the compound microscope.

This ‘Leonardo da Vinci of England’ was also a maverick thinker who was one of the first to

articulate the concept of extinction and who suggested evolution two centuries before Charles

Darwin. Who was this polymath? If you guessed Robert Hooke (1635-1703), you know your his-

tory better than many of your peers…

“Earning Newton’s enmity proved to be disastrous for Hooke. ‘None of the thousands of instru-

ments and models he constructed or the fossil specimens he collected survived Newton’s presi-

dency of the Royal Society,’ notes Drake. As a further insult, Hooke’s only known portrait was

reputedly destroyed after his death by Newton supporters. The Royal Society of Chartered

Surveyors held a competition to create a new portrait; … It has taken three centuries—and a

hardy band of revisionist scholars—to coax Hooke from the shadows." 1

Observations are being held to commemorate the 300th anniversary of Hooke’s death.

1. Richard Stone,"Championing a 17th century underdog," Science 301, 152 (July 2003).

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