WHY SHOULD MY PADDLE HAVE HOLES ANYWAY?
Good old Isaac Newton. Newton's Second Law states: A
particle acted upon by an unbalanced force F experiences an acceleration a that has the same direction as the
force and a magnitude that is directly proportional to the force. If F is applied to a particle of mass m,
this law may be expressed mathematically as
F = ma
Let's first investigate the path of travel that a
paddle brings about from high in the air, down onto the culprit's bare
buttocks. As the paddle descends through the air, the path of travel will
actually correlate more closely to curvilinear motion as opposed to linear kinematics
directional travel. We would therefore have to introduce some basic principles
defining general curvilinear motion in order to better understand what is
really happening. However, these concepts are far beyond the scope of this
brief paper and will hence, not be discussed. Yet what we can do, is to analyze
the board's properties at any one instance in time, and because of the centripetal
forces acting, our paddle with always possess a path of travel tangent to the
circle of real travel. The paddle will have two real accelerations, normal
acceleration an whose vector is perpendicular to the tangential
acceleration vector & heading toward the center of the circle defining the
path of travel, and tangential acceleration at which has the more
commonly thought of magnitude for its vector, a path tangential to the circular
motion, appearing to ‘head straight forward,' the same acceleration which your
automobile measures acceleration. This is the acceleration which we will
concern our studies with, at and apply in Newton's Second Law.
Newton's First Law states the following: A particle
originally at rest, or moving in a straight line with constant velocity, will
remain in this state provided that the particle is not subjected to an
unbalanced force. Hence, a paddle will keep falling until the buttocks will
yield the predominantly unbalanced force that will abruptly stop the paddle as
the two objects SMACK
together. Yet, Another force resisting the paddle's free fall and ever more
highly powered flight, is the same quality that suspends multi-ton aircrafts
high in the sky, air friction.
Because the resisting wind friction acts equally at
each and every infinitesimal point, thus forming a uniform pressure
distribution, if we decrease the area, say by drilling some holes, we then are
decreasing the area for which the wind can no longer cause as much resistance,
because at these points (holes) the wind will simply ‘fly right through them.'
Conversely however, now let's take into account
Newton's Second Law. Remember, we are analyzing instantaneous points in time
and thus are concerned with linear tangential paths of travel. This second law
can also be stated another way: the unbalanced force acting on the particle is
proportional to the time rate of change of the particle's linear momentum.
Momentum M, equals the
product of mass m and velocity V
Furthermore, the rate of change of the particle's
linear momentum will be due to a changing velocity dV. However, dV is simply measuring a change in
velocity occurring over some small period of time. In fact, this is the very
definition of acceleration. Therefore, an increased acceleration will correlate
with an increase change in linear momentum.
An equation inherently means that if one side of it
is reduced, then in order to remain a true statement of equality, the other
side must also be reduced by equal magnitude. By drilling holes in the paddle,
we have already established that you are decreasing the surface area,
unfortunately however, you are also decreasing the mass of the paddle, a
function of force. And unless you increase the acceleration of the paddle, then
simply, mathematics would control a resulting smaller force. But here is the
beauty of it all. By reducing the surface area you made it possible to achieve
a greater force because you will be capable of establishing a substantially
greater acceleration as the board approaches and WHACKS into the submissive's bottom.
So really the lesson to be learned here is, "Can I increase F in greater magnitudes through both
decreasing mass m and increasing acceleration a?" The answer is just about
always YES. However, too many holes can in fact cause the implement to fail, as
I will explain in a moment.
It's also important to understand the striking differences
that will result in a paddle with holes than one without holes. To understand
this, let's look at the relationship, Pressure P equals Force F
divided by the cross sectional area A
P = F/A
A dimensional analysis justifies this relationship
through common references. For example, the pressure in your automobile tires
in measured in force per unit area, pounds per square inch, psi. In the case of
the paddle, the painfully fixed and controlling variable F will be distributed over the
resulting reduced area remaining from the removal of the holes. Thus, we need
to discard the gross cross sectional area of the paddle's blade and rather,
examine the net effective area. The net effective area will be smaller and
therefore, the punishing uniform force will have to result in increased
pressure points (more concentrated forces in a smaller area).
Finally, using Newton's Third Law: The mutual forces
of action and reaction between two particles are equal, opposite, and collinear
- it can be justified then that both the ppaddle and the butt are being equally
spanked.
Flexibility can add an extra wicked stinging
sensation. Depending on the type of material being used, elastically yielding
flexibility limits will be dictated. A one-fourth inch thick plywood paddle can
demonstrate massive authority by allowing it to flex. It just so happens that
when you decrease the gross sectional cross area by drilling even the smallest
of holes through a wooden surface, you ultimately end up dramatically reducing
the internal stresses of the paddle itself. This is because you have broken the
continuous bonds holding the paddle together. Thus, an absence of resistance
will permit the paddle to flex until enough internal fibers are tensioned
enough to stop further flexing. But remember, too much flexing can result in an
over extension of the stress-strain curve past the inelastic upper limit and
into the inelastic range, thus irreversible damage could be done to the
utensil. However, Safety for the buttocks must always take superior
significance over a material article.
For those of you that wish to ascertain a more in depth investigation
into practices and principles that are applicable to paddle design,
construction, and usage, please reference the below on-line sources:
The Isaac Newton Institute for
Mathematical Sciences
Engineering
Mechanics: Statics; by William F. Riley
& Leroy D. Sturges
Engineering
Mechanics: Dynamics; by William F. Riley & Leroy D. Sturges
Harvard University /
Department of Physics
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