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Universal Physics Journal
Event 1: Balancing A Broom Handle
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Author: Ethan Skyler
Publication Date: To be announced...
Revision Date: February 20, 2008 |
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Purpose
The first in a series of three articles,
Balancing a Broom Handle is an investigation designed to sharpen our
understanding of the roles played by acceleration/Action and
acceleration/Reaction forces during balancing events. Since these same forces, including the same geometry of
application, are equally present during more common and somewhat more
complex bicycle, motorcycle and automotive cornering events, Balancing a
Broom Handle is perhaps the least complex event within which we can begin
this study. |
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Event 1
Obtaining a wooden handle of the type screwed
into the wide head of a shop boom is a prerequisite to these
considerations. I will wait right here for you to accomplished this
task....
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(2) In a level area, clear of moving or stationary obstacles, hold
one hand out, palm up, at waist height. Place one end of the broom
handle forward of your palm on the inside of the first joints of your first
two fingers. Using your other hand, or other means if necessary,
adjust the handle's shaft until it is vertical. As you release the
shaft be prepared to adjust the position of your supporting hand to
prevent the broom handle from falling away from vertical. While at
first it may be difficult for you to shift your hand in the correct
direction to the required degree to maintain the handle's vertical
balance, in a few minutes of effort you will likely be able to acquire
this skill. Performing this exercise inside an open room with a low
ceiling will help, for you can raise your supporting hand thereby trapping
the top of the handle against the ceiling when things start to get out of
control. Keep practicing until you are able to make and recover from
minor adjustments in the position of the top of the handle.
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(3) After maintaining the
top of the handle in the vertical position for a few moments, I want you
to pull your hand back just a bit so that the top of the handle begins to
accelerate away from you in the forward direction. For the next few
tries, do not attempt to prevent the handle's acceleration and rotational fall to the floor. Repeat this event several times, noting the
acceleration and rotation of the handle prior to its leaving your
hand. Also note the reduction in the force of the handle's weight
against your open hand just prior to its departure. The handle does
not drag off the edge of your hand with constant down-force but instead at
the last moment lifts off your hand with little to no weight-force at all.
This no-weight-force departure from your hand is an important clue to the
forces and geometric rotation present in this event.
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(4) For a closer look at the falling broom handle's no-weight-force
departure from its support, try the same experiment as before only this
time begin by balancing the handle on the arm of a chair instead of on
your open hand. During the handle's fall, after its release from
vertical, you can move in for a good look at the moment of the rotating
handle's lift-off from the chair's arm.
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(5) Next let us look at the forces and
geometrical motion present during this relaxed event where the broom handle is
allowed to fall from vertical while being supported by your non-moving hand. Expand the drawing to the right with a mouse
click and observe the five still frames "a" through "e" of the broom handle falling toward
the floor. Notice in the middle of each broom handle I have inserted a C/M
or center of matter icon. Note how the five downward-directed
gravitational force vectors are drawn with their tails originating at the
center of these five C/M icons. In every event where the action force
of gravitation is involved, it is always correct to draw the average of
this myriad of internal forces as originating at the object's center of
matter.
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(6) As the event progresses from left to right, see how the
downward-directed gravitation (Grav.) vectors become more and more
misaligned with the upward-directed support force from your hand which, by
the way, is also gravitationally based. Below each still frame,
there is drawn an imaginary lever that represents this vector
misalignment. The increasing length of this lever represents
the increasing mechanical advantage the constant downward-directed action
force of gravitation has upon generating the increasing torque force that
is being applied to the broom handle as the handle's angle of lean departs
from vertical at an increasing rate. This torque force is
responsible for causing clockwise rotation of the broom handle with the
axis of this rotation located at the point where the broom handle is in
contact with your hand. Once the handle is no longer in contact with
your hand, as represented by still frame (e), the axis of the handle's
rotation naturally shifts to its center of matter. From this point
on, the handle's downward-directed internal force of gravitation (Grav.) is
almost exclusively an acceleration/Action force, on average centered at the C/M icon and
responsible for both the handle's downward-directed acceleration and the
reactive generation of the handle's supporting, internal
acceleration/Reaction matter force. Here this pair of mutual forces are both internal and both equally present
within the same object, the rotating and falling broom handle. Make
no mistake, there exists no fabled "net force" here.
Instead, within each component of the handle's matter there exists both a net
internal acceleration/Action force of gravitation supported by a net
internal acceleration/Reaction force of matter in full agreement with Newton's LAW I, LAW III, the Universal
Law of Mutual Forces, plus Rule 4b and Rule 7 of the Universal
Rules for Force and Motion.
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(7) In the Balancing a Broom Handle
Event we have established the following:
a) Misalignment between the downward-directed
action force of gravitation (Grav.) and the opposing upward-directed action
force provided by your hand causes the application of a torque force to
the falling broom handle.
b) The magnitude of the torque
force increases as the angle of the falling broom handle increases
relative to vertical.
c) When the broom handle is
supported by your hand, the axis of the torque force is located at the point of
contact between your hand and the broom handle.
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(8) Now that we have established that a gravitationally
caused torque force is the cause of the broom handle's clockwise rotation
(from the perspective of the drawing) as it falls to the ground, let us
figure a way to stop the handle's
fall at about 5 degrees away from vertical. What do you think
has to be done to halt the handle's rotational fall at this point?
Grabbing the handle at its C/M with your free hand is one possible way. That will work fine
since it causes a balancing counterclockwise torque to be applied to the broom handle.
The vector of this balancing force will be represented by an arrow drawn
from the handle's C/M icon horizontally to the left on my drawing which is straight
back toward your person while being opposite to the direction of the
handle's lean. The handle's fall will now come to a halt at about
5 degrees away from vertical.
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(9) Is there another way to halt the handle's angle of
lean when it reaches 5 degrees? Say you are limited to finding a workable method
using
only horizontal changes in motion of the hand that is in contact with the
base of the wooden handle. Consider that when the broom is
balanced prior to any fall, the rate of acceleration of the handle's top
is a match with the rate of acceleration of the handle's
bottom. They are both equal to zero if you will allow me to ignore
the minor orbital and rotational accelerations of Earth. Now consider that when balance is lost,
the handle's top begins to accelerate away from you while the bottom
remains relatively stationary. Here it should be no surprise that the position
of the handle's top begins changing relative to the position of the
handle's bottom, since the two ends are experiencing different rates of
acceleration. Are you reaching the same conclusion as am I in that
"freezing" the handle's angle of lean at 5 degrees when the
handle's top is experiencing a certain rate of generally forward-directed acceleration
is logically possible only when the handle's bottom is experiencing the
same rate and direction of acceleration?
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(10) This equal-rates-and-directions-of-acceleration conclusion means that first
you balance the broom handle and then withdraw your supporting hand a bit
to initiate the misalignment of vertical action forces that results in the
gravitationally-based torque force causing radial acceleration of the handle's top away from your
position. Then as the handle's top approaches 5 degrees of lean,
you quickly begin accelerating the handle's bottom using the application
of a forward-directed force from your supporting hand. If done
correctly, the handle's angle of lean will "freeze" at a
constant angle. Here the forward-directed horizontal acceleration
rates of each end are a perfect
match. The handle is now balanced at a
constant 5 +/- degrees of lean, at least for as long a time as you are
able to maintain the required rate of acceleration of the handle's lower
end. As you speed up, friction with air will begin providing an
increasing support for a portion of handle's forward-directed
gravitational torque force.
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(11) But in reality, you can only maintain the required rate of
linear acceleration for but a short time for there is a limit to how fast
you can run. If you decide to perform this experiment while inside the
enclosed cargo box of a large delivery van, it will be possible to
"freeze" the falling handle at 5 degrees of lean for a longer
period of time. Here friction with air will not be a factor. But again there is a limit to how fast the van can
travel. When the van reaches its speed limit, its hopefully constant rate of
acceleration will be reduced to zero and the leaning broom handle will
complete its fall to the deck. Clearly in order to keep the falling
handle's angle of lean constant, a way of maintaining a constant rate of
acceleration of the handle's bottom needs to be found.
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(12) Fortunately, acceleration comes in two types and so far we are
considering only one type, linear acceleration, which is
characterized by changes in speed while a constant direction of travel is
being maintained. Let us now shift to considering centripetal
acceleration, characterized by changes in direction while a constant
speed of travel is being maintained. Surely this new constant-speed
requirement will solve the speed limit dilemma we soon encountered in our
linear acceleration events. But where best to set up our centripetal
acceleration event? And will friction between the broom handle and
the surrounding air be a problem as well? |
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(13) A human centrifuge often found at an amusement park is
one possibility. You know the type where the tub revolves so quickly
causing the outer vertical wall to apply such a strong
centripetal-directed acceleration/Action force against the participants's
bodies, supported by equally strong centrifugal-directed
acceleration/Reaction forces being reactively generated within the
participant's matter, that by simple friction they remain pinned to the
wall even after the supporting floor is dropped away! An exciting
event for sure, and always poorly described in the many physics texts in
my possession.
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(14) There does exists a simpler, slower and more easily controlled
event where we can hope to easily and accurately maintain a constant rate
of acceleration of the broom handle's base so that a 5 degree angle of
lean can be maintained. Remember the foot-powered child's
merry-go-round at the nearby playground? Taking our broom handle
there, you climb aboard and face inward toward the axis of rotation while
leaning back against the outer handrail. I'll handle the constant
rate of rotation for as long as I am able. |
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(15) We'll start this event with you balancing the broom handle
while standing on the outer rim of the stationary merry-go-round.
Then I begin causing the disk to rotate about its axis. Once up to a
comfortable and constant speed of rotation, you notice that to maintain
the handle's balance, the top has moved inward in response to the
inward-directed centripetal acceleration/Action force from your
accelerating hand. If you had not adjusted the handle's balance inward
from vertical, it would have soon flipped over backward or outward since
the top and bottom of the handle would not be experiencing the same rate
of acceleration.
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(16) You also notice when balanced on the rotating disk, the
handle's top has not only moved inward but forward in the direction of
travel as well. Clearly
this forward-directed component is in response to the
backward-directed forces of friction with the surrounding still air
through which the tilting handle is moving. Better results will be
obtained if air friction is absent. Knowing this would be an issue,
I brought along a tall clear plastic box, without a bottom, to be mounted
to the merry-go-round handrail. This way the handle can be held up
inside the box where the air remains generally still during our next test. |
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(17)
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(18)
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(19)
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(20)
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(21)
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(22) Next we will consider the range of these opposing torque
forces. When you are successful in balancing the broom handle while
holding your position, there is no misalignment of forces so the
gravitational lever is at zero length. Here no gravitational torque
force exists. Then when the handle is horizontal with one end
supported by your hand, the gravitational lever is longest meaning the
gravitational torque force is at its greatest value.
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(23) Meanwhile, when the handle is vertically balanced, the
"push" lever is longest which means that the torque force
developed by any push you choose to make will be at its highest
value. But when the handle is horizontal, with one end supported by
your hand, the push lever is at zero length so no matter how great the
force of your push, no torque force will occur.
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(24) To recap their torque force capabilities, the
downward-directed gravitational force will be least effective when the handle is vertical
and most effective when the handle is horizontal. Conversely, your
forward-directed push force will be most effective when the handle is vertical and least
effective when the handle is horizontal. It should now be clear to
you that only when conditions are perfect will these two torque-producing
action forces provide perfect balance for each other. At all other
times their imbalanced torques will cause an acceleration resulting in changes in the broom handle's
angle of lean.
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(25) Now let us have a look at solving the earlier problem of how to
forcefully cause a constant rate of horizontal acceleration for the
bottom of the handle so that a 5 degree forward-directed angle of lean can
be maintained for considerably more than a few seconds of time. The
key to the solution is to first recognize that the top of the broom
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References
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Author's Commentary
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Copyright Notice
Event 1"" (C) Copyright 2003 - 2006 Ethan Skyler. All
rights reserved. No
portion of Event 1, minus the exceptions noted below, may be copied by any means without the author's
written permission and even then only if the author's copyright notice is
permanently affixed to each approved copy. To request written permission
visit Contacts. The author grants each visitor to The Universal Physics Journal the right to make one copy of Event 1 for his or her own
personal archive as long as the author's copyright notice is permanently
affixed to the archive copy.
The Author's Commentary to Event 1 is
hereby granted by the author, Ethan Skyler, into the realm of the Public
Domain. As such it may be freely copied, in full, or in part, by any
means so long as a reference to the author and this web site address
of http://www.UniversalPhysics.org/
are included with each copy.
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