Fred Farnell (Lane Tech HS, Physics) Newton's Second Law; Momentum
Fred 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
Larry 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 q_{0} = 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 T_{o} = 1.493 sec. Then he found the following formula [allegedly accurate for the period T at large angles q_{0}, and expressed in this notation] in a somewhat older ***CRC Handbook of Chemistry and Physics, under the category "simple pendulum":
*** 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 T_{0} [in our notation] for maximum angle q_{0 }[in our notation], the time of vibration in an infinitely small arc is approximately T_{o} / T = 1 + (1/4) sin^{2}(q_{0}/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 q_{0} from equilibrium is expressed in terms of K(k), the complete elliptic integral of the first kind. Here are some details
The function K(k) is defined as the (elliptic) integral:
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 :
or
K(k) = p/2 ´ [ 1 + (1 / 16) q_{0}^{2} + (11 / 3072 ) q_{0}^{4} + ... ]
Both expansions are viable if q_{0} is sufficiently small, but the first expansion is neither more accurate, nor more rapidly convergent, than the second one. Let us compare them
q_{0} | 2/p ´ K( sinq_{0}) | 1 + (1/4) sin^{2}(q_{0}/2) | 1 + (1 / 16) q_{0}^{2} |
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) 57.3° |
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 q_{0 } has the wrong sign in that expression, although neither one is correct at large q_{0}.
Ann
Brandon (Joliet West, Physics) Pendula, Continued
Ann 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) | T^{2} (sec^{2}) |
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 |
The formula T = 2 p Ö (L/g) may also be written as L = g / 4 p^{2} T ^{2}, so that the slope of the graph of L versus T^{2} is g / 4 p^{2} » 4 m/sec^{2}. Very interesting, Ann!
Bill Blunk (Joliet Central, Physics) Garbage
Bill 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:
Earl Zwicker (IIT, Physics) String Blow Pipe
Earl 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