High School Math-Physics SMILE Meeting
05 February 2002
Notes Prepared by Porter Johnson

Fred Farnell (Lane Tech HS, Physics) Newton's Second Law; Momentum
passed around the new book The Grip of Gravity[The Quest to Understand Laws of Motion and Gravitation] by Prabhakar Gondalekar [Cambridge 2001]  ISBN 0-521-80316-0.  According to this book [p 94], Leonard Euler, [circa 1750] was the first person to write Newton's Second Law in the modern form, F = ma.  Newton is also described as having a "flawed character" in Gonalekar's book [p 114], as well as elsewhere [http://www.worldscibooks.com/histsci/p299.html].  For details of Euler's life and his considerable accomplishments, see the website http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html.

Fred next illustrated the concept of momentum by dropping and catching two different masses [100 grams and 2 kilograms] from a fixed height about 2 meters. He took the precaution of putting heavy gloves on his hands, while catching the dropped masses. He caught the lighter weight while holding his hands still, whereas he began moving his hands just before catching the heavier weight.  He was able to stop the heavier weight by letting it fall a somewhat greater distance, so that the average force on his hands would remain modest in size.  Because the heavier mass acquires more momentum than the lighter one when falling through the same distance, it takes either a substantially greater force, or else a significantly greater time, to stop it.  A simple, direct way to show the relation between force, momentum, and impulse.  Thanks, Fred!
Earl and Porter
pointed out that a good fielder catches a baseball while moving the glove away from the ball to absorb the blow and to keep the ball from popping out of the glove, and then promptly puts the other hand over the glove to grab the ball. Arlyn Van Ek uses a blanket to catch a raw egg, and then breaks a water balloon by throwing it against the wall, to illustrate the effect of the "stopping time" in a collision.  Watch out for those falling masses and flying eggs!

Larry Alofs (Kenwood HS, Physics) Pendula
set up a pendulum with a cylindrical bob that was supported by two strings [bifillar] for  better control, which passed on its swing through a photo-gate timer.  The cylindrical bob blocked out the light signal during the time of its passage, which is given in terms of its diameter, L = 0.028 meters, and its velocity v as T = L / v.  [Note:  a cylindrical bob is used because its cross section remains fixed, even if its orientation changes slightly.]  The bob was released at a height h = 0.20 meters above its lowest point, where the photo-gate was located.  Thus, the bob theoretically would have speed v = Ö (2 g h) = Ö (2 ´ 9.8 ´ 0.2) m/sec = 1.98 m/sec when passing through the gate. And the time T is predicted to be T = L / v = 0.028 / 1.98 sec = 0.014 sec.. The measured time was 0.013 sec, indicating reasonable agreement.

Larry then set the photo-gate to measure the period of the pendulum.  He first measured the period for a small angle (about 5°), and found it to be T = 1.416 sec.  Then, he carefully held the bob so that the strings made a large angle of about  q0 = 57°  (around 1 radian) to the vertical direction, and released the pendulum.  He made several measurements of the period, obtaining 1.484 sec, 1.493 sec, and 1.501 sec, respectively, corresponding to an average To = 1.493 sec. Then he found the following formula [allegedly accurate for the period T at large angles q0, and expressed in this notation] in a somewhat older ***CRC Handbook of Chemistry  and Physics, under the category "simple pendulum":

To / T = 1 + (1/4) sin2(q0/2) = 1.0574
But, according to our observations, 
To / T = 1.493 / 1.416 = 1.0543
Not bad agreement between observation and formula, Larry!
*** The 40th Edition [1958-1959] of the CRC Handbook contains the following statement [p 3113, in the section on Definitions and Formulas]:  "If the period is T0 [in our notation] for maximum angle q0 [in our notation], the time of vibration in an infinitely small arc  is approximately To / T = 1 + (1/4) sin2(q0/2)  [in our notation]. But what does that statement actually mean?? 
Porter Johnson (Physics, IIT) pointed out that the correct expression for the period for the large amplitude simple pendulum of length L and maximum displacement q0 from equilibrium is expressed in terms of  K(k), the complete elliptic integral of the first kind.  Here are some details
T = [4Ö(L/g) ] ´ K(k)   ,   with k = sin(q0/2),

The function K(k) is defined as the (elliptic) integral:

K(k) = ò0p/2 dj /Ö (1 - k2 sin2j )

One may make either of the following expansions of the elliptic integral [See the web article Large Amplitude Period of a Physical Pendulum http://webphysics.davidson.edu/alumni/BeKinneman/pendulum/report.htmas well as the Java Applet http://hyperphysics.phy-astr.gsu.edu/hbase/pendl.html :

K(k) = p/2 ´ [ 1 + (1/2)2 sin2(q0/2) + (1 · 3 /2 · 4 )2 sin4(q0/2) + ... ]


K(k) = p/2 ´ [ 1 + (1 / 16) q02 + (11 / 3072 ) q04 + ... ]

Both expansions are viable if q0 is sufficiently small, but the first expansion is neither more accurate, nor more rapidly convergent, than the second one. Let us compare them

q0 2/p ´ K( sinq0  1 + (1/4) sin2(q0/2)   1 + (1 / 16) q02
 0° 1.0000 1.0000 1.0000
15° 1.0043 1.0043 1.0043
30° 1.0174 1.0167 1.0173
45° 1.0400 1.0366 1.0385
(1 radian)
1.0663 1.0574 1.0625
60° 1.0731 1.0625 1.0685
75° 1.1190 1.0926 1.1070
90° 1.1803 1.1250 1.1542
105° 1.2622 1.1573 1.2099
120° 1.3728 1.1875 1.2741
150° 1.7622 1.2333 1.4284
180° ¥ 1.25 1.6168

If anything, the first expression is less accurate than the second, since the term of fourth order in q0  has the wrong sign in that expression, although neither one is correct at large q0.

Ann Brandon (Joliet West, Physics) Pendula, Continued
described an class exercise to study the dependence of a period T of a small amplitude pendulum upon its length L.  She tied cords to round metal washers and cut them to produce pendula of lengths from, say, 10 to 150 cm.  Each student took a pendulum, suspended it from a paper clip taped to the lab bench, and measured its period, averaging over ten complete oscillations.  She took each pendulum, taped it directly to the board along a ruled horizontal line at a point corresponding to the measured T. Thereby, she constructed a graph of length versus time, using each pendulum to mark its own length.  The graph was definitely not linear.  Then, she drew another ruled line indicating time-squared, and placed each pendulum at the appropriate point on that graph.  This time, the graph was straight.  Here are typical data:

 L( m)   T(sec)   T2 (sec2
0.10 0.6 0.4
0.25 1.0 1.0
0.50 1.4 2.0
0.75 1.7 3.0
1.00 2.0 4.0
1.25 2.2 5.0
1.50 2.4 6.0
[Do these numbers look contrived, perchance?]

The formula T = 2 p Ö (L/g) may also be written as L = g / 4 p2 T 2, so that the slope of the graph of L versus T2 is g / 4 p2 » 4 m/sec2. Very interesting, Ann!

Bill Blunk (Joliet Central, Physics) Garbage
has given up trying to make garbage attractive, so he showed a method to make your garbage repulsive to everybody else.  Specifically, he used rabbit fur [the remains of Poor Thumper, who gave his/her all to science], a hard plastic rod [for electrostatics experiments], and plastic foam packing material in sheet form.  He formed a ring out of the packing material, and used the Poor Thumper [rabbit fur] to charge both the rod and the ring.  Then he threw the ring into the air, and it "floated" in the air above the rod.   Since the ring and the rod contained charges of the same sign, courtesy of Poor Thumper, the ring was held aloft by the repulsive electric force between them.  The ring could conveniently [or inconveniently] be dropped on a nearby person's head by taking the rod away. [Note for animal activists:  artificial fur works pretty well also!]  If you don't have a rod, it will work very well with an inflated and electrically charged rubber balloon, as Bill showed us.  You made a repulsive display attractive to us, Bill!  Thanks!

Roy Coleman (Morgan Park HS, Physics) Various:

Let's hope the second law of thermodynamics isn't repealed also.  Very fine, Roy!

Earl Zwicker (IIT, Physics) String Blow Pipe
showed each of us a plastic "pipe" with a string loop attached to one end, which he had obtained years ago from a fast food outlet.

The pipe pretty much had the shape of a conventional tobacco pipe, with stem and bowl.  But the bowl had an opening at its bottom end, as well as its top.  The continuous loop of string passed through the holes, entering at the bottom and leaving at the top.  When Earl blew air into the pipe stem, the air came out of the top of the bowl, but not the bottom (because the bowl was designed that way).  The string --- which was very light and fuzzy --- was carried along in the air stream.  It took the shape of an elongated loop, leaving the bowl at its top and turning around to re-enter at its bottom.  Simple enough! But wait!

Earl pointed out that, near the very top of the loop there was a small "depression", or "valley".  How come?  This question was thrown out for our consideration.

It may be related to the similar indentation that occurs on a broad, flat belt in an old-fashioned machine shop, in which the power from one master motor is transferred by the belt to a long shaft.  Pulleys along the shaft are then used with other belts to drive machines located at various positions to the shaft.

Interesting! What do you think?

Notes taken by Porter Johnson