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The Physics of the Paddle Swat


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#1 Catharsis By Her Hand

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Posted 26 May 2011 - 08:43 PM

The Physics of the Paddle Swat

ABSTRACT

This research note from iSwat (Institute Spanking Wisdom And Technology) uses classical mechanics to analyze the paddle swat. It reaches several conclusions:
  • Pain is proportional to the speed of the paddle upon impact of the butt;
  • Pain is proportional to the "stiffness" of the butt (i.e., its hardness) - a butt that is tight from bending over will experience more pain than if it is softer, as from standing erect (and... hard asses hurt more than fat asses!);
  • Pain is inversely proportional to paddle weight - light paddles hurt more;
  • Pain is the same for a paddle hitting a larger area (e.g., both cheeks) as for that paddle hitting a smaller area (i.e., one cheek), as long as impact speed and butt tightness are the same.
Experimental research from community practitioners, with reporting of results, is encouraged.

INTRODUCTION

A spanking is replete with physics, as well as physio-neurology. Physics governs:
  • speed of the paddle that the arm imparts for a swat;
  • amount of impact;
  • distribution of the impact energy across the buttocks;

while neurology governs:
  • pain that a particular swat inflicts;
  • the buildup of pain from multiple blows during a spanking;
  • the long-term accommodation of higher levels of spanking pain.

Over the longer term, cognitive psychology governs such effects as:
  • effect of severe disciplinary spanking in deterring behavior;
  • effect of stress-relief spanking in releasing stress.

This treatise attempts to address the first four bullets above. The analysis is oversimplified, but I will try to point out factors that I know I am ignoring.

ASSUMING PAIN IS PROPORTIONAL TO IMPACT PER AREA

The pain of a swat (we will abbreviate pain as P in our upcoming equations) is postulated to be proportional to the impact, I, per unit area, A.

P = I/A

Area applies to the size of the patch of buttocks that the paddle swat hits. The same power of swat over a large area will hurt less than that same swat over a smaller area. (Exceptions, qualifications, and admissions of oversimplification will be noted in parenthetical expressions. One oversimplification is in stating that the swat covers a particular area. A planar surface, e.g., paddle, touches a sphere, or butt globe, in exactly one point. As its stroke continues, the area that it touches becomes larger as the flesh is compressed, so a single 'area of impact' is an oversimplification. Another oversimplification lurks here... as the area becomes smaller, it vanishes to nothing and becomes a point. Pressure on small areas can occur between nerves and not be felt as severely, and a small point will penetrate skin rather than strike it.)

Impact is defined as the change in force per unit time. Force (F), as we all know, equals mass (m) times acceleration (a, not the same as area A), where acceleration equals change in velocity, and velocity corresponds to change in position per unit time (e.g., miles per hour). In imitation of real physicists, we will indicate the change in a quantity per unit time by placing an apostrophe (') after that quantity. Hence, velocity = change in position (x) per unit time, or v = x'. Likewise, a = v' and F = ma = mv' = mx''. So:

I = F' = (ma)' = ma'

To maximize impact, and therefore pain, one seeks to maximize the change in acceleration. While being swung, the paddle moves with increasing speed until it strikes the buttocks. Its speed upon striking the buttocks is called s (pronounced, "ass.") When it strikes, it immediately slows down, and then it rebounds by moving in the opposite direction. The distance over which it slows to a stop and reverses direction, d, is the distance between its position of initial impact and its position when the resistive force of the buttocks, as compressed by the paddle, becomes strong enough to reverse the paddle direction.

BUTT COMPRESSION AND HOOKE'S LAW

To model the compressive resistance of the butt, we apply Hooke's Law, which states for a spring, its force is proportional to its displacement... at rest, it has zero force, but the more you pull (or for compressive force here, push) it, the harder you have to pull, because the more it pulls back. Let us apply that model to buttocks flesh:

The drop in paddle velocity from its speed upon first touching the buttocks, s, to zero occurs over the ass depression distance d, as described above. The time over which this change in v occurs is the time that it takes the paddle to travel that depression distance, d. However, when the paddle first touches the skin, there is no opposing force, so its deceleration is zero. The further it proceeds, the more its progress is resisted. The degree of resistance is proportional to the distance of depression at any one instant. Such linear restoring force results in "simple harmonic motion" of the paddle, i.e., it slows down according to the sine function of the proper frequency, which begins at its impact velocity s but then in curved manner governed by the sine function, slows down increasingly quickly (yes, really!) to zero. It then bounces back in the reverse direction, continuing the change in acceleration (and hence the pain-producing impact).

To continue, we have to characterize the spring constant of the butt. If I press a one-inch square piece of wood against the butt, how hard do I have to push it to depress the butt one inch? That number, in pounds per inch of depression, is the spring constant for a patch of with area equal to 1. If I double the area, I will have to push twice as hard to get the same depression. Hence, our effective spring constant is kA.

Let us call that spring constant k (at the end, I will provide a highly entertaining method of measuring your own butt's k!).

For a spring of constant k, hit with a mass of m moving at a velocity of s, it turns out that

d = [square root (m/k)] * s.

The maximum depression distance increases as the square root of the paddle mass; it decreases as the square root of the spring constant, and it increases in direct proportion to the impact velocity.

EQUATION OF MOTION OF PADDLE DURING IMPACT

Also, the position as a function of time of the paddle upon impact is:

x(t) = d { sin [((k/m)**(1/2) )*t] },

where d is as expressed just above. The quantity (k/m)**(1/2) gives the angular frequency of the sinusoidal oscillation -- for high "stiffness" k, the frequency (bounce back speed) increases; for higher mass m, the paddle wants to keep plowing ahead and bounce back is slower.

Hence, substituting for d and using our effective spring constant that includes impact area, kA,

x(t) = {[square root (m/kA)] * s } * {sin [((kA/m)**(1/2))* t]}

To determine the change in force, or impact, we simply take the third derivative of x(t) with respect to t (or t''', where first derivative gives velocity; second derivative gives acceleration; third derivative gives change in acceleration which, when multiplied by m, gives change in force).

I = {m*(kA/m)**(3/2)]{cos [((kA/m)**(1/2))* t(final, initial)]}},

where we use the notation t(final, initial) to indicate that we evaluate the expression for the final time, i.e., when the paddle bounces back from the skin it just hit, and subtract the value of the expression at the initial time, when the moving paddle first touches the skin.

So how long does the paddle contact the skin (that is, what is the value of t(final) - t(initial) )? It is simply the time that it takes the argument of the cos to progress through the value pi ("argument" is the quantity within the parentheses, [((kA/m)**(1/2))*t]). So if the argument is pi, then t = pi * (m / kA)**(1/2). For higher spring constant or greater area pushing the paddle back, the time is shorter (as indicated by kA in the denominator); for a heavier paddle, it takes more time for the restoring force of the ass to push the paddle back (hence m is in the numerator).

In fact, since cos (0) = 1 and cos (pi) = -1, the contact time is given by the constant times the difference between cos (pi) and cos (0), which is -1 -1 = -2, so the impact becomes:

l = - [(kA/m)**(3/2)] * {[square root (m/kA)] * s } * (-2), or, simplifying,

l = 2s[(kA/m)].

CONCLUSIONS

We recall our presumption that pain P is proportional to impact per unit area, so finally we find:

P = 2sk/m.

This is the major result of this work. It states that the higher the speed of impact, s, the greater the pain (makes sense!). It says that the higher the spring constant, the greater the pain. Recall that the spring constant is high if the butt is firm. If you are standing straight, your butt is softer than if you are holding your ankles with your knees bent, which is the more painful position for a paddling. Fat people have softer butts (and less pain) than thin people.

But wait... m (mass of the paddle) is in the denominator! That means that heavy paddles hurt less than light ones! Why?? It harks back to our postulate that pain is proportional to impact, which in turn is the change in force. A heavy paddle will want to keep going further and will take more time to reverse its direction than a light one, hence the third derivative of its position (or equivalently, its change in acceleration) is less than for a light paddle!

Another surprise is that the area, A, of the impact portion of the paddle does not appear. A paddle striking both cheeks apparently hurts the same as if it strikes one cheek, just as long as its impact velocity is the same!

EXERCISE FOR THE STUDENT

A break is hereby inserted to allow the student to go to the bathroom with a ruler and a one-inch-cross-section stick, a postal scale, and to figure out with the scale how hard he has to push the stick to make it push butt flesh in one inch. If the student does not have a stick, he may use another method - wet the butt (or better yet, paint it with red lipstick). Sit upon a white surface with full weight (pick up your feet), then get up and measure the surface area of where your butt has touched. Now, you have to calculate the distance of depression needed to splay your butt cheeks out to the size measured. ("Yikes.. oh, hi, honey! What am I doing with your lipstick? er... umm.. I am doing Science! Yes, I will buy you a new one.") For a waist size of 36", your waist diameter is about 10.5". Let us equate that to your hip diameter as well, then realize that the diameter of each butt cheek might be half that, or 5 inches. For example, if you have two red five-inch-diameter circle marks and you weigh 200 lbs, your butt was depressed 2-1/2" by 100 lbs (each cheek bears 1/2 your weight) and so you butt has an average spring constant of 0.002 feet / lb. Using the depression method, I measured an 1/8" depression for a 1 lb force over 2 sq in., which translates to 0.01 feet / lb.

ANOTHER EXERCISE FOR THE STUDENT

Film a spanking by a pro with a high-frame-rate camera, to determine what velocity the paddle reaches before hitting the butt. This gives the parameter s in the analysis above. Be sure and measure the velocity of the portion of the paddle that hits the butt, not just the velocity of the hand. For a circular path, the strike area on the paddle will travel farther and at greater velocity. For extra credit - recognize that the classic Cracker Barrel paddle weighs 5.25 oz, almost exactly the same as a baseball. Both are propelled by an arm. An excellent fast-ball pitcher will throw a ball at 100 mph. A pro spanker hand may not go that fast, but due to the lever arm leading to the strike area of the paddle, that strike area may go that fast. For extra credit, receive swats with the Cracker Barrel paddle, then stand at home plate and moon the pitcher at a pro game (Justin Verlander of the Tigers is one who can pitch at 100 mph.) Have the pitcher hit your butt with a few fast ball pitches. Report on which hurt more.

COMING SOON

Enjoyed this treatise? Visit iSwat's sister organization, Institute Sexual Cognitive Research Expertise and Wisdom (iScrew) for the appearance of The Physics of Making Love. Includes many cheerful facts about the coefficient of friction and fluid viscosity.
"Most men fantasize about receiving a hard disciplinary punishment but aren't quite prepared for what it involves... It is a tremendous amount of pain applied in a manner that is beyond their limits in order to teach them a proper lesson." Miss J, lead of Women-Spanking-Men.com

#2 PET5

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Posted 26 May 2011 - 09:21 PM

I shouldn't be surprised that someone put so much time into trying to figure this out, but I am.

#3 sillyme

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Posted 26 May 2011 - 09:23 PM

I shouldn't be surprised that someone put so much time into trying to figure this out, but I am.


this is great, thx for sharing

#4 Guest_kiwi_*

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Posted 26 May 2011 - 11:18 PM

I wish I had as much free time as you do, Catharsis! This is great :lol:
I wonder what would happen if a student were to submit something like this as a physics assignment...

#5 OwwItHz

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Posted 27 May 2011 - 04:14 AM

Some people don't have enough to do. ;)

#6 Catharsis By Her Hand

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Posted 27 May 2011 - 04:34 AM

Some people don't have enough to do. ;)

Remember, OwwItHz and others, I had a 90-minute, 1,000 swat spanking a week ago... I had lots of time during the spanking, and she let me use paper and pencil during the spanking to work it out.

Folks, if you are inspired by this post, perform and post your own research. What is k? What is s? See the Exercises for the Student!
"Most men fantasize about receiving a hard disciplinary punishment but aren't quite prepared for what it involves... It is a tremendous amount of pain applied in a manner that is beyond their limits in order to teach them a proper lesson." Miss J, lead of Women-Spanking-Men.com

#7 Guest_2bspanked_*

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Posted 27 May 2011 - 06:04 AM

My research:

I bent over

She said "I'm gonna spank your "S"

I said "K"

It hurt

The End
:lol:
2b

#8 Guest_SinglegalAZ_*

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Posted 27 May 2011 - 06:52 AM

My research:

I bent over

She said "I'm gonna spank your "S"

I said "K"

It hurt

The End
:lol:
2b

As Tal would say.... LMDAO!!! :lol:

#9 Guest_NeedsOTK214_*

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Posted 27 May 2011 - 06:56 AM

That about sums it up, 2B. Very well put.

#10 Catharsis By Her Hand

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Posted 27 May 2011 - 08:59 AM

"O, spankos, spankos," said Cathy, graciously waving his tail. "'The unexamined life is not worth living' saith Socrates. Likewise, 'The unexamined spanking is not worth giving,' saith Catharsis."

"You boldly give, and you meekly accept, spankings without ever, ever asking, 'Why do spankings hurt?'. Never in the history of Spankingdom has such comprehensive research been placed before you, yet yea it is like unto placing pearls before swine.

"You and your ilk content yourselves with seeing what is and doing today what you did yesterday. I, on the other hand, see what is not and ask, 'Why not?' Thanks to the lack of curiosity that curses ye Acceptors, spankings now are the same as spankings given in caveman days. They have not yet benefitted from such innovations as Digital Electronics, Wireless Technology, or Lady Gaga.

"Why, it is just like the urban-dwelling child who, when asked where milk comes from, answers, 'The Supermarket!' She has no knowledge of the dairy or the cow. Yea, ye remain content in your cluelessness, innocent in your ignorance, delighted in your dooflessness! You think that Spankings Grow On Trees.

"Yet I and my Institute, iSwat, labour despite your ridicule to improve the spankings of our future generations, no longer confining them to spankings of Caveman Days. Yes, when we we are done, no more will you doubters be able to smugly proclaim, 'Spanking... So simple a Caveman can do it.' When iSwat has completed its mission, our nation's Universities will offer graduate programs that specialize in spanking."
"Most men fantasize about receiving a hard disciplinary punishment but aren't quite prepared for what it involves... It is a tremendous amount of pain applied in a manner that is beyond their limits in order to teach them a proper lesson." Miss J, lead of Women-Spanking-Men.com

#11 Catharsis By Her Hand

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Posted 27 May 2011 - 09:48 AM

(sigh)...

I'll bet these naysayers don't even know what the spring constant of their own butt is! (hint: it is measured in pounds per foot).
"Most men fantasize about receiving a hard disciplinary punishment but aren't quite prepared for what it involves... It is a tremendous amount of pain applied in a manner that is beyond their limits in order to teach them a proper lesson." Miss J, lead of Women-Spanking-Men.com

#12 MissBam

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Posted 27 May 2011 - 10:24 AM

Cathy.

Step away from the keyboard and drop your trousers.
Never let your morals get in your way of doing what's right.

#13 Loving Witness

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Posted 27 May 2011 - 10:29 AM

Catharsis..... I knew I shouldn't have devoted so much time to spanking your a** with those 1000 swats. I think I may have impacted your brain as well as your butt. LOL

And for that.......... I think you've recovered enough for another session in AssBusting101.

This time you shall be bent over with butt cheeks firm, so that your pain is drastically increased and your ability to write has been stripped away.

Anxiously awaitng,
Loving Witness

By allowing our bodies to be stripped and beaten, our wills to be humbled and our tears to flow in the presence of a Loving Witness, we are literally re-creating the Descent of the Goddess with each punishment.


#14 Catharsis By Her Hand

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Posted 27 May 2011 - 12:07 PM

Catharsis..... I knew I shouldn't have devoted so much time to spanking your a** with those 1000 swats. I think I may have impacted your brain as well as your butt. LOL

And for that.......... I think you've recovered enough for another session in AssBusting101.

This time you shall be bent over with butt cheeks firm, so that your pain is drastically increased and your ability to write has been stripped away.

Anxiously awaitng,
Loving Witness

(in meek, small, penitent voice):
Yes, ma'am, Loving Witness

I am sorry, Loving Witness.

Standing bare in the corner, bent over, butt cheeks firm, awaiting your chastisement, Loving Witness.

Thank you, ma'am... may I please have another?

"Most men fantasize about receiving a hard disciplinary punishment but aren't quite prepared for what it involves... It is a tremendous amount of pain applied in a manner that is beyond their limits in order to teach them a proper lesson." Miss J, lead of Women-Spanking-Men.com

#15 Guest_2bspanked_*

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Posted 27 May 2011 - 03:58 PM

(sigh)...

I'll bet these naysayers don't even know what the spring constant of their own butt is! (hint: it is measured in pounds per foot).



Okay, I'll explain things a bit slower, because I know this stuff can difficult to grasp:

Regarding Spring Constant:

My research indicates that her desire and ability to spank is pretty much the same regardless of the time of year, although I have noticed a certain enthusiasm exudes as winter comes to a close.

Regarding "pounds per foot":

My research and clinical trials show that there is a direct correlation between the number of pounds per foot, and the intensity and duration of the spanking - therefore I no longer pound my feet when told to bend over.


:lol:
2b

"The overly examined life can be exhausting" - Mrs. Socrates

#16 Catharsis By Her Hand

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Posted 27 May 2011 - 05:12 PM

Okay, I'll explain things a bit slower, because I know this stuff can difficult to grasp:

Regarding Spring Constant:

My research indicates that her desire and ability to spank is pretty much the same regardless of the time of year, although I have noticed a certain enthusiasm exudes as winter comes to a close.

Regarding "pounds per foot":

My research and clinical trials show that there is a direct correlation between the number of pounds per foot, and the intensity and duration of the spanking - therefore I no longer pound my feet when told to bend over.


:lol:
2b

"The overly examined life can be exhausting" - Mrs. Socrates

:lol: :lol: :lol:

I LOVE it!

C
"Most men fantasize about receiving a hard disciplinary punishment but aren't quite prepared for what it involves... It is a tremendous amount of pain applied in a manner that is beyond their limits in order to teach them a proper lesson." Miss J, lead of Women-Spanking-Men.com

#17 Guest_candygirl04_*

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Posted 27 May 2011 - 06:04 PM

Forget my bottom....now my head hurts.

#18 Guest_SoftNGentleSpirit_*

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Posted 27 May 2011 - 08:43 PM

:huh: This newbie is pondering "Oh my...Do I have to know THIS!? And all this time I thought it was an applied art."

#19 OwwItHz

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Posted 28 May 2011 - 04:25 AM

I think you need a grant to work on the "hole theory of paddle design" to settle, once and for all, whether there's a benefit to making holes in the paddle blade and if so, what's the benefit?

Some claim it's to reduce air resistance, resulting in a higher velocity at the point of contact. Others say it has to do with how the skin stretches to fill each hole rather than just around the edge of the paddle blade. Then there's the issue of the air cushion between the paddle and the surface being struck and how the holes provide a release of the build-up of air pressure.

Surely your research has uncovered some facts about this.

#20 Catharsis By Her Hand

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Posted 28 May 2011 - 06:13 AM

I think you need a grant to work on the "hole theory of paddle design" to settle, once and for all, whether there's a benefit to making holes in the paddle blade and if so, what's the benefit?

Some claim it's to reduce air resistance, resulting in a higher velocity at the point of contact. Others say it has to do with how the skin stretches to fill each hole rather than just around the edge of the paddle blade.

OwwItHz, you have just hit upon The Burning Question of Spankingdom. I shall deploy the vast resources of iSwat upon it forthwith!

As a stopgap to hold you over while years of research are conducted, I offer these observations:

  • Holes in a paddle allow release of the bow wave of compressed air that forms a cushion just as the paddle is about to strike the butt, hence preventing a last minute drop in impact speed, s;
  • Holes with vertical walls indeed provide a cutting surface into which butt flesh can exude during a blow, cutting itself on the sharp edge -- chamfered holes, drilled to diagonal orientation with a countersink bit, avoid this cutting edge, and some paddles, such as those from Walt's Paddles, provide the spanker with the choice of vertical hole walls on one side of the paddle and diagonal hole walls on the other;
  • Holes shorten the life of the paddle. My able spanker and research associate, Loving Witness, has broken three paddles on me (two within the last six weeks). Of those, two had holes in them.

    As promised above, I will immediately write a grant request to the government for funds to complete research upon your excellent topic. Thank you!

"Most men fantasize about receiving a hard disciplinary punishment but aren't quite prepared for what it involves... It is a tremendous amount of pain applied in a manner that is beyond their limits in order to teach them a proper lesson." Miss J, lead of Women-Spanking-Men.com




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