Universal Physics Journal
|Author: Ethan Skyler
Publication Date: November 19, 2008
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.
(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.
(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.
(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.
(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.
In the drawing above, 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 correct to draw the average of
this myriad of internal forces as originating at the object's center of
(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 radial acceleration 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 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 in full agreement with Newton's LAW I, LAW III, the
Law of Mutual Forces, plus Rule 4b and Rule 7 of the
Rules for Force and Motion.
(7) In the Balancing A Broom Handle
Event we have established the following:
2) The magnitude of the torque force increases as the angle of the falling broom handle increases relative to vertical.
3) 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.
(8) Now that we have established that a gravitationally
caused torque force is the cause of the broom handle's clockwise radial
(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. Increase the force applied by your free hand
and the handle will begin to reduce its angle of lean.
Decrease this force and the handle will begin to increase its
angle of lean away from your person and away from
(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
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?
(10) This equal-rates-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
(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. But again there is a limit to how fast the van can
travel. When the van reaches its speed limit, its declining 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.
(12) Before working on a solution for the constant-acceleration
problem, let you
and I give some thought to the forces present for the short time that you
are able to keep the falling handle's angle of lean constant. From
our first event we know that the downward-directed force of Earth
gravitation that is being actively generated within every component of the
handle's matter is causing the application of a torque force within the
handle. This clockwise torque force is causing the radial and
centripetal acceleration of the handle's top while the axis of the
handle's rotation is located at the bottom where it is pivoting
against your inactive, non-accelerating hand. What do you think is
happening when you decide to cause an equal rate of acceleration for the
handle's bottom which up to now has been the handle's pivot point or axis
(13) If done correctly, the moment you begin accelerating the handle's bottom at a rate and direction equal to the acceleration occurring to the handle's top, all rotation of the handle will cease. As rotation comes to an end, one might think that the axis of this rotation can be abandoned. Yet this axis, located at the contact point between your hand and the broom handle, continues to be significant for it remains the site of the fulcrum or pivot point of the gravitational torque force as depicted on the drawing. Keep in mind that this clockwise gravitational torque force continues to be responsible for the forward-directed linear acceleration of the upper portion of the broom handle.
(14) What remains to be defined is the role of the action force of
your hand. It is clear that this force is responsible for the
forward-directed linear acceleration of the lower portion of the broom
handle. What is not clear is how this force appears to provide
balance for the clockwise gravitational torque force. For certain
this force is causing the linear acceleration of the fulcrum point for the
gravitational torque force. Yet one wonders if this force from your
hand is causing a counter-clockwise torque force on the broom handle and
if so, where does its fulcrum point lie? I think some sort of
counter-clockwise torque force has to be present to act as the balancing
force for the clockwise gravitational torque force in order for the
handle's angle of lean to continue to remain constant at 5 degrees during
the handle's acceleration.
(15) In our first experiment where the broom handle is falling
forward while being supported by your stationary hand, we are able to consider the
action of the gravitational torque force when it is not being balanced by
the action of an opposing torque force. Let's now consider an event
where the accelerating push of your hand is directed at right angle to one
end of the
handle's shaft while no opposing gravitational action force is present.
By isolating the accelerating push of your hand perhaps we can better
understand the role of this balancing force.
(16) First let us imagine we are weightless in deep space wearing protective
astronaut space suits. You position the broom handle "vertically" in
front of you as you lightly grip its base between your thumb and index
finger. When its position before you is generally the same as its position at
the start of the event back on Earth, you withdraw your index finger
releasing your grip on the handle and then using only your thumb, you
quickly push the base of the "vertical" handle away from you in
the forward, "horizontal" direction. I am positioned off
to your right as observer to this event.
(17) What I observe is the base of the broom
handle accelerating away from you as the force of your push initiates a
counter-clockwise rotation for the broom handle. The axis for this
sudden rotation appears to be about two-thirds of the way "up"
the handle. During the push by your thumb, the handle's lower
2/3rds rotates away from your waist while the handle's upper 1/3rd rotates
toward your head. Following your push in deep space, the handle
naturally and automatically takes on a motion where it rotates about its
center of matter at a constant rate as this new axis moves with a uniform
motion away from you into space.
(18) Given the presence of the handle's counter-clockwise rotation I think we can safely conclude that during the force of your push against the handle's bottom your push produces a torque force whose lever pivots at and drops "vertically" down from the fulcrum point which is approximately 2/3rds the way "up" from the handle's bottom.
(19) In pondering this deep space event while sitting at my desk, I
discovered that by placing a long pencil flat on the smooth surface of the
desk and then flicking the pencil's end with my index finger, a perfect
copy of the deep space event occurs. By placing my other index
finger on the far side and at various points along the pencil's length, I
am able to verify the 1/3rd - 2/3rds axis point during the pencil's
rotational acceleration. Here at my desk gravitation is at right
angle to the plane of this event so its role is a minimal one.
(20) Now I am imagining pushing or pulling horizontally and at right
angle on the end
of a yard stick that is being supported on a cushion of air while resting
on an active air hockey table. With the aid of an overhead movie
camera, accurate verification of the 1/3rd - 2/3rd axis point during the
push can be made. Then once the push or pull is over, verification
of the natural shift of the 1/3rd - 2/3rd axis point to the yard-stick's
1/2 - 1/2 center of matter will also be possible. I should think
overall this would make for a fine high-school science project.
Especially considering that the air hockey table will need "warming
up" prior to and "cooling down" following any such science
experiment. (I will appreciate hearing the results of every such
science project performed.)
(21) Let's return to our Earthly event where you are applying a
constant acceleration force to the handle's bottom in order to maintain a
constant 5 degree angle of lean. Now that we understand the role of
this counter-clockwise, external (contact) torque force, with its
fulcrum or pivot point 2/3rds of the way up the handle, let us consider
the complete event including gravitation's role. Keep in mind that
as the handle's angle of lean is changed, changes occur to each lever
length of these two action forces. As their respective lever lengths
change so changes the magnitude of their respective torque forces. I
promise interesting combinations of torque forces will occur here that we
can advance to Event 2 with its leaning and thereby cornering bicycle and
motorcycle, and Event 3 with its cornering automobile, pickup and Sport
(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.
(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.
(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.
(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 rate of acceleration
for the top of the broom handle needs to match that of the bottom.
Clearly no linear acceleration event will accomplish this task for long. That
leaves an event where acceleration takes the form of changes in direction
rather than changes in speed. Should you think changes in direction
doesn't represent "real" acceleration, you are in for a real surprise.
Climb aboard a playground turntable, begin balancing your broom handle and
hang on securely as I begin the turntable's rotation. As your
rotation increases in frequency, you will find that for the first time you
are able to maintain a steady angle of lean for the broom handle with it
top leaning generally in the direction of the turntable's axis of
rotation. The axis represents the horizontal direction of the centripetal
acceleration/Action force. The top is also naturally leaning a bit
into the "wind" to counter the entire handle's friction with air.
This air friction can be eliminated if the handle is being balanced inside
a tall, clear plastic box with no bottom. Then its lean will be
directed precisely toward the turntable's axis of rotation.
(26) Here, as long as I keep the turntable's rotation constant, you
are easily able to maintain a constant 5 degree inward-directed angle of
lean. Now the reality of centripetal acceleration become clear.
Viewing the leaning handle from your right side, I see the clockwise
torque of downward-directed non-acceleration/Action force of Earth
gravitation, with its axis located at your palm and its average located at
the handle's center of matter, is a match for the counter-clockwise torque
of the rotating turntable's inward-directed centripetal
acceleration/Action force with its axis again located at your palm and its
average located 2/3rds the way up the handle from your palm. Here
gravitational torque force is a match for accelerational torque force
brought to life through the action of constant centripetal acceleration
caused by a constant external centripetal acceleration/Action force that
in turn is causing the reactive generation of its own Newton LAW III
required-to-be-present internal acceleration/Reaction force.
(27 This concludes our "Balancing A Broom Handle" event. I
hope I did not lose your attention in this discussion regarding the action
forces, axis of rotation, center of matter, and accelerational forces
along with their directions. May I suggest you give Event 1 a quick
review before moving on to Event 2 where we will use our newly found
"broom handle" understandings as tools in exposing the same roles these
forces and accelerations play in our upcoming "Cornering a Bicycle" event.
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.
Click here to download a copy of Event 1: "Balancing A Broom Handle"