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Physics for Rolfers

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Translator
Pages: 2-14
Year: 1989
Notes on S.I

Notes on Structural Integration – November 1989 – 89/1

Volume: 89/1

The fundamental new concept which Ida Rolf has introduced in connection with the well-being of human beings is to look at the influence of gravity, the physical weight of the body and its parts. From here she went on to interrelate physical to other factors, from anatomy to physiology to neurology and so on, all the way to metaphysics.

But the new angle remains the aspect of gravity which probably nobody before her has followed up with such a consequence in this context. It seems obvious that with her background as a scientist she understood gravity in the strict scientific context.

Gravity is a basic and central issue of Rolfing. Structural Integration can only make sense in the context of the reality of physical weight, how the body reacts to it and by what means it is transmitted. The basic premise supposes the existence of mechanical forces acting throughout the body-structure and that they are subject to the applicable laws of physics. Sometimes I have a slight suspicion that some people reject this premise. I suggest a self-test in two parts. From the premise above I contend that an arm has a physical weight, induced by gravity. First hold out one arm stretched horizontally in the frontal plane and let it there for five minutes. Then hold out the other arm the same way but bend the lower arm about the elbow-joint towards yourself, the hand above the articulatio humeri. Both times the supported weight at the joint is the same but the outstretched arm takes the center of gravity further out, the increased bending-torque leads to a faster fatigue. The differing sensation in the shoulder-girdle after the test can be quantatively explained by the basic premise. The test is scientific because it can be falsified.

This article tries to lay out some basic principles of elementary physics. One, perhaps a small one, of the many prerequisites upon which Rolfing has to build. First the more elementary concepts of the mathematical description of motion shall be summarized as tools for the understanding of Newton’s definition of gravity and some other selected topics.

Displacement, Velocity, Acceleration

In order to understand the concept of gravity and weight it is necessary to develop the preeconditional concept of acceleration. Acceleration is used to describe movement which in turn is basically defined by the notions time, displacement and velocity. This sequence is in line with the historical evolution of mechanics. It might have to do with the hierarchy of human perception, the sensual experience of motion seems to be primary to the want of knowing where it stems from.

The units required and used to describe displacement, velocity, acceleration and later on force and energy are:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-1.jpg’>

(*) the abbrevations for the units are always set in angular brackets.

For colloquial considerations the mathematical descriptions used are not always as stringent as they should be but always sufficient to follow the argument.

Displacement

Quantitative descriptions of motion are based on measurements and calculations of positions, displacements, velocities and accelerations.

It must be distinguished between translational motion and rotational motion (Fig.1). In translational motion every part of an object moves in the same direction, though the direction can be a curve. In rotational motion an object turns about a fixed axis which can either lay within or without the object itself. The general case will be a combined motion. Then the mathematical descriptions always deal with the two types separately.

The following arguments about velocity and acceleration are confined to translational motions along a straight line. This is sufficient to explore the principles.

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-2.jpg’>

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-3.jpg’>

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-4.jpg’>

Velocity

The terms velocity and speed are used synonymously. Velocity is defined in terms of the displacement, or the change in the position of, or the distance covered, by an object in a specified interval of time, or the elapsed time.

The graph (Fig.2) shows the relation between displacement and elapsed time along a straight line. It is possible to figure out that after 5 seconds the object is 50 meters away from the origin 0. This is an interpolation. Or we can assume that at 100 meters from the origin 0 the time elapsed would be 10 seconds. This is an extrapolation.

For a constant velocity the plot shows a straight line, the motion is uniform and ‘s’ is proportional to ‘t’. Velocity is therefore defined as the ratio of displacement to elapsed time.

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-5.jpg’>

Acceleration

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-6.jpg’>

Again, the aim is to find a relationship between displacement, time and speed for this more complex motion, to define it at any chosen point. What we can do now is to calculate the average velocity during time intervals. Taking the velocity between t(0)= 0 and t(1)= 2 and t(2)=4 as v(2) and so on, we get:

and so on:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-7.jpg’>

Taking the increments ∆s and ∆t small enough, e.g. microscopically small, gives no longer an average velocity but an exact value at a discrete instant along the curve. This is then called the instantaneous velocity, represented by the tangent to the curve at the chosen point. The slope, or the steepness, of the tangent to the curve corresponds to the magnitude of the velocity. Fig.3 shows the tangent at t=5 seconds. The steepness, or slope, of a curve changes continuously along its course. A tangent is defined as the straight line touching the curve at a specified point, its slope coinciding with the slope of the curve at this discrete location.

Plotting the velocity of the same motion against the elapsed time shows a uniform change, represented by a straight line (Fig.4). It is a motion with constant acceleration, the characteristic behavior of objects experiencing a constant force. The speed increases uniformly.

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-8.jpg’>

In general terms, the relation between change of speed to change of time ∆v to ∆t is linear, the ratio of change of velocity to elapsed time is called acceleration, denoted by ‘a’. Acceleration is a measure for the change, increase or decrease, of speed in a motion.

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-9.jpg’>

This equation represents the general form, valid for uniform as well as non-uniform acceleration. A uniform, or constant, acceleration results in a linearly changing velocity as in Fig.4. For this more handy case, which will suffice to describe the phenomena of gravitation, the equation for the acceleration can be simplified.

With a = ∆V / ∆t constant along a motion it doesn’t matter how large the increments for ∆V and ∆t are. We can take the whole quantity of v and t to write:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-10.jpg’>

Thus the graph describes a motion of constant acceleration. The unit sec(2) for square-seconds looks a bit strange. The result of a=10[m/sec(2)] may be read as 10 meters per second, per second. It expresses that the speed changes every second by 10 meters per second. In this description of the uniformly accelerated motion the displacement, or the distance covered, has disappeared. It would now be interesting to see this lost quantity retrieved.

In uniform, unaccelerated motion the relation between displacement ‘s’, velocity ‘v’ and time ‘t’ is:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-11.jpg’>

It is possible to mathematically prove that this applies to all types of motions, uniform and nonuniform. In any velocity/time graph the area under the curve is always equivalent to the distance covered.

This means for the motion with constant acceleration, using the general formula to calculate areas of triangles (Fig.5b):

s = [v x t]/2

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-12.jpg’>

With the help of this last equation the variables( s, v, t, a) of a uniformly accelerated motion can be interrelated:

we have a = v/t → v = a x t → t = v/a

and s = 1/2vt

substituting v by a x t in s = ½ vt → s = ½ at x t = ½ at(2)

turning it around and solving it for t in terms of ‘s’ and ‘a’:

t(2) = 2 s/a → t = √(2s/a)

furthermore, with t = v/a substituted in s = ½ vt:

s = ½ (v/a) x v = l/2 v(2)/a → v(2) = 2as → v = √(2s/a)

The following table shows a recapitulation of the relationships between ‘s’ , ‘t’ ,’v’ and ‘a’ for uniform and accelerated motions. The multiplication-sign ‘x’ is omittet.

uniform motion s = vt → accelerated motion s = ½ vt = ½ at(2)

uniform motion v = s/t → accelerated motion v = at = √(2as)

uniform motion t = s/v → accelerated motion t = v/a = √(2s/a)

Incidentally, all this has been compiled by the Bishop of Lisieux, France, around 1360 A.D.

The next paragraph will deal with the reciprocal relationship between acceleration and force. The above equations will then allow to connect and describe the interaction. Neither speed, acceleration, force nor gravity can be measured directly. Time and displacements in space are the only variables accessible to direct quantification. This means that all other concepts have to be derived from and tied to the experience able and measurable reality.

With all that knowledge it is now easy to make a small detour to the mathematical discipline called ‘calculus’. Assuming the motion of an object subjected to the uniform acceleration of the magnitude ‘g’, we can depict the following sequence of curves with ‘a’, ‘v’ and ‘s’ each as a function of the elapsed time:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-13.jpg’>

The mathematical operation to get from top to bottom of this sequence is called ‘integration’, from bottom to top ‘differentiation’. A further interesting detail: the shaded area in the a/t-graph is equivalent to the instantaneous velocity at the time ‘t’ (v=gt), the shaded area in the v/t-graph, as seen before, to the distance covered up to the time ‘t’ (s= ½ vt).

A numerical example, involving s,t,v and a: on the leaning tower of Pisa stands Galileo Galilei, eighty meters above the ground, experimenting with dropping stones and other objects to the ground. On hitting the ground a stone has reached a velocity of 40[m/s]. What was the acceleration and the time elapsed between dropping and impact?

from v(2) = 2as→ a= v(2)/2s

a = (40×40)/160 = 1600/160 = 10 [m/sec(2)] and t = v/a = 40/10 = 4[sec]

It’s by no means obvious that the acceleration in free fall is constant, it can’t be deduced through hard thinking. Galileo repeated this experiment innumerable times from different heights with differently sized objects of different materials. He found that indeed the acceleration is constant, thereby laying the foundation for Newton’s laws about motion, gravity and inertia.

Newton’s Laws of Motion

The above prelude served to define and hopefully clarify the concept of acceleration, the base for the concept of gravity.

The expression ‘Law’ is perhaps misleading. Its not Fig.7: that Newton got access to principles laid down in some blue-print for the universe and therefore not comparable to the incident when God revealed the ten commandements to Moses. Their background is the curiosity why material things behave as they do, their formulation the result of hypotheses deduced from general observations secured by practical experiments. A real breakthrough of Newton’s laws was the introduction and deployment of the concept ‘force’ in a systematic and consistent manner.

In a preface to his works, Newton set up, as he called them, ‘rules of reasoning in philosophy’. Just to Cite two of them:

– We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. (Rule of parsimony)

– Therefore to the same natural effects we must, as far as possible, assign the same causes.

The First Law of Motion

This law is also called ‘The Law of Inertia’.

– Every material body persists in its state of rest or uniform, unaccelerated motion in a straight line, if and only if it is not acted upon by a net force.

In other words, if there are no forces on an object or if the net force is zero (when several forces cancel themselves out), then:

– an object at rest remains at rest.

– an object in motion continues to move with constant speed and unchanged direction.

This is not in accordance with the subjective human everyday experience. Anybody who has ever been involved in taking a piano up the staircase to the fourth floor will rather concur with the Aristotelian view that all objects move only if some force causes the movement to occur. Thereby a cart detached from the pulling horse would quickly come to rest. It was argued that this was because the natural state of all material objects is being at rest, consistent with the postulated universal law that all things have an inherent bias for their natural state. Air blown into water will quickly ascend to the surface, back to its natural and obvious place above the water (Anybody can verify the hypothesis experimentally). For the relentless movement of the celestial bodies the existence of a divine ‘prime mover’ was assumed. And so on.

An example is a car moving at constant velocity (Fig.7). The acting forces are:

The forward force FE exerted by the engine

The backward forces:

– Air resistance FA

– Friction from power-transmission FT

– Friction between tires and road FR

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-14.jpg’>

The Second Law of Motion

This is about the rate of change of velocity, the acceleration.

– The net force acting on a given material body is directly proportional to, and in the same direction as, its acceleration.

Proportionality between force and acceleration means that they are directly related, modified by some proportionality constant. Thus with force as ‘F’, acceleration as ‘a’ and the proportionality constant as ‘k’:

F= k x a

Experiments showed the constant k depending solely on the ‘bulk’ of the material bodies. With the same applied force, a brick of double the size would experience only half the acceleration. The name mass (m) was given to this quality of bulk, with the synonyms inertia and inertial mass, i.e. ‘m’ was assumed to be the proportionality constant.

The relevant equation reads then:

F= m x a

For practical applications it was necessary to calibrate one of them, ‘F’ or ‘m’. A standard mass-quantity was created out of platinum and attributed with the mass of 1 kilogram [kg]. Again, the unit kilogram is not a divine concept, it is kept in a museum in Paris. The calibration is arbitrary, any other mass would have served the same purpose because mass is a linear concept.

The unit for force follows from the equation F=ma. The multiplication of [m/sec(2)] for ‘a’ with [kg] for ‘m’ results in [mkg/sec(2)] for ‘F’. A unit which is called ‘Newton’ [N].

Gravity

The most remarkable of all forces is the ubiquitous gravitational force. A colloquial synonym could perhaps be the word ‘heavyness’. Its existence can’t be explained, only the phenomena of its sources and effects can be conceptualized and measured. All material bodies carry a static field of gravitational force (not energy!). It is an inherent property of matter and decreases with the distance from the surface. Forces only arise when two or more material objects come within range of their mutual fields of gravity. The accruing gravitational force depends on the mass of the objects and on their distance.

The gravitational force between two objects of the respective mass m(1), and m(2), their centers separated by a distance ‘r’is calculated as:

F = [G x m(1) x m(2)]/r(2)

with G = 6.67 x 10(-11)[Nm(2)/kg(2)] as the universal gravitational constant, an experimentally found proportionality factor, valid for all materials.

An example: Find the gravitational force acting between a woman with a mass of 60 kg and a man with a mass of 80 kg, the distance between their centers being 0.2 meters (this is about as close as they can get).

F = [6.67 x 60 x 80]/[(10(11) x 0.2(2)] = 8 x 10(-16)[N]

Expressed in colloquial units the attractional force amounts to 0.0008 grams or a bit less than the thousandth part of a gram. It can’t be gravity which holds them together.

The same calculation for the earth and the woman would come up with a force of about 600 [N] acting between the two.

It’s surprising to note that gravitational forces are always mutual. Not only the earth exerts a force on the woman but also the woman on the earth.

With the equation F = m x a the gravitational force can be found e.g. by the experiment of Galileo.

As postulated by the second law of motion the force of gravity is only dependent on the mass, the acceleration ‘a’ being the constant factor. On the surface of the earth ‘a’ is around 9.8 [m/sec(2)], always directed to its center. Since it is such a special acceleration it was given the letter ‘g’.

Ordinary material objects experience a force exerted by the earth of:

F(grav) = mg = m x 9.8[m/sec(2)]

An object with a mass of 50[kg]:

F(grav)= 50 x 9.8 = 490[N]

Again, F(grav) being such a special force was given the name Weight (W):

W=mg

In everyday use the gravitational constant ‘g’ is omitted and the mass in [kg] is taken to indicate the weight. Mass in the gravitational context is called gravitational mass.

Inertia

Besides gravitational mass material objects possess the quality of inertia. The inertia can be shown in an experiment, taking an object with the measured gravitational mass of 1 kg and pushing it with a constant force F= 3[N] (A 0.3 kg) over a frictionless, horizontal plane. No gravitational forces influence this arrangement (Fig.8).

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-15.jpg’>

The inertial mass is equivalent to the gravitational mass. It is not obvious that the two should be identical. For a long time this was thought to be accidental until Einstein could clear up the mystery.

Note: The weight of an object is a gravitational force with the direction towards the center of the earth, a ‘vector quantity’.

Mass is an inherent property with no direction, a ‘scalar quantity’.

The Third Law of Motion

This law is about action and reaction

– Whenever two bodies A and B interact so that body A experiences a force (contact, gravity, magnetic etc.), then body B experiences simultaneously an equally large and oppositely directed force.

A single object can neither exert nor experience any force at all. Forces spring up only in interaction and then immediately cancel themselves. Whereby it’s always arbitrary to call one the action and the other the reaction, there is no cause and effect, the two forces cause each Fig.10 other simultaneously.

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-16.jpg’>

F(1) represents the weight of the person, or the force exerted by the earth, fixed at the center of gravity and pointing downward. The instant reaction is F(2), the pull of the person on the earth, equal in magnitude. This is in accordance with the third law of motion. In accordance with the second law of motion the two objects should now show an accelerated movement towards each other. Since this does not usually happen there must be more forces present to prevent this. A second set of forces acts where the two bodies contact. F(3) as the upward pressure by the surface of the earth against the feet of the standing person and the equal pressure F(4) exerted by the feet downwards on the surface of the earth. These are now elastic forces. It would be a systematic mistake to hold F(3) as a reaction to F(l), the weight of the person to the pressure experienced on the soles of the feet. But it is of course the subjective sensual experience. The ground prevents us from accelerating towards the center of the earth, thereby providing the vectoral lift which enables us to stand upright. Imagine somebody falling down from the leaning tower of Pisa. It’s certainly gravity that is responsible for his fall but it doesn’t do anything for his stability.

Related Concepts

Equilibrium (I)

According to the first law of motion an object is said to be in equilibrium when no net force acts upon it. This is the case when no forces act or, more often, it occurs because two or more forces add up to zero, or balance.

The type of equilibrium – stable, unstable, indifferent- is determined by noting how the object reacts to a slight disturbance (Fig.10):

at A:

the ball will roll down the hill → unstable

at B: the ball will roll back to its original place → stable

at C: the ball will rest at any place near C → indifferent

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-17.jpg’>

Vectors

Many quantities do not only have a magnitude but also a direction. They are vector-quantities. Examples are velocity, force, acceleration, in contrast to scalar-quantities like time, mass, heat etc. which have no direction. Magnitude expresses a quantity, direction a quality.

A set of rules has been established for their mathematical treatment.

Addition and Subtraction:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-18.jpg’>

For the purpose of addition and subtraction vectors can be moved as long as direction and magnitude are conserved.

They further must have identical lines of action or intersect in the same plane.

Parallel vectors can’t be added, they result in a torque about P:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-19.jpg’>

Vector Components:

Vectors can be broken up in components along convenient directions:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-20.jpg’>

In the case of an object sliding or rolling down an inclined plane of an angle Ө against the horizontal:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-21.jpg’>

F(grav): the weight-vector pointing vertically down

F(N): the force between the surfaces of plane and object, acting perpendicular to the slope

F(S): the force accelerating the object down the plane.

mathematically: FN= F(grav) x cos Ө, F(s)= F(grav) sin Ө

Torque

The situation of needing to unscrew a rusted nut with a wrench will illustrate the concept:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-22.jpg’>

Per definition, the torque τ is the product of F times the perpendicular distance r (lever arm) to the point P of action:τ = F x r

The forces in the system are:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-23.jpg’>

F and F’ are equal but in opposite directions, the system is therefore in translational equilibrum. The friction τ’ of the rust on the nut sustains the rotational equilibrum.

The importance of the condition ‘perpendicular’ is shown in Fig. 17. If the angleis not a right angle, only the component F(y) will produce a torque, the component (F)x merely pushes against the nut. We have then:

τ = (F x sin Ө) x r

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-24.jpg’>

Fig. 18 shows an alternate way to find the torque.

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-25.jpg’>

r’ = r x sin Ө and thus τ = F x (r x sin Ө)

which is the same as above.

Couples

A pair of forces with equal magnitude but opposite directions and different parallel lines of action, separated by a distance ‘d’, are called a couple (as in real life!). Torque is usually related to a point of action. The individual forces can be slid back and forth along their lines of action without changing the magnitude of the torque.

Fig.19 illustrates the concept. The point of action P can be situated either between the forces or anywhere outside.

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-26.jpg’>

τ = τ(1) + τ(2), where τ(1) = – F x a and τ(2) = F x b (clockwise movements are defined as negative, counterclockwise movements as positive)

thus τ(1) = F x b ? F x a = F x (b – a)

and as can be seen from Fig.19: b – a = d

therefore = F x d

The torque of a couple is independent from the location of the point of action, it is solely defined by F and d.

Equilibrium (II)

The definition of equilibrium in accordance with the second law of motion can now be extended to include rotational motions. The two conditions are then:

– The net force on the object must be zero:
F(net) = 0 (Translational condition)

– The net torque on the object computed about any convenient point must be zero: τ(net) = 0 (Rotational condition).

Fig.20 shows a balance with unequal arms. With m(1) = 3 kg, what is the value for ma holding the balance in equilibrium and for F(3) supporting the arms?

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-27.jpg’>

The condition for translational equilibrium:

F(1) + F(2) + F(3) = 0 → F(3) = – F(1) ? F(2)

from F(1) = m(1) x g = 3 x g and F(2) = m(2) x g follows:

F3 = – 3 x g ? m(2) x g

The condition for rotational equilibrium:

τ(1) + τ(2) + τ(3) = 0

choosing ‘O’ as the convenient action point to calculate the torques:

τ(1) = F(1) x r(1) = 3g x 4 = 12g

τ(2) = -2 x r(2) = – m(2)g x 6 – 6m(2)g

τ(3) = F(3) x 0 = 0

adding (2),(3) and (4):

12g – 6m(2)g + 0 = 0 → m(2) = 2

substituting m(2) by 2 in (1):

F3= -3g – 2g = -5g

What anybody could have guessed by just looking at it.

The Center of Gravity

Forces are frequently of gravitational origin. For the computation of effects of larger bodies it is often necessary to localize its point of action. That’s where the concept of ‘center of gravity’ comes in: The effect of the weight of an object is equal to that due to a concentrated infinitely small object of the same weight placed at a point called ‘Center of Gravity’ (CG). The CGs of uniformly dense symmetrical objects is at their geometrical centers. For other objects the CG can be calculated mathematically or located experimentally.

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-28.jpg’>

To find the torque τ(B) at the cross-section a-a (Fig. 21a) induced by the weight ‘W’ requires the identification of the relevant lever-arm ‘r’. Taking the above definition, the CG representing the weight of the object will have to be in its middle. The lever-arm ‘r’ is then equal to the distance between the CG and cross-section a-a. This could be verified by an experiment (Fig. 21b). Letting the object hang from a string fixed to the cantilever at the assumed point will result in the same magnitude of τ(B).

The weight will bend the cantilever slightly downwards, the effect of ‘W’ at a-a is therefore called ‘bending-torque’:

τ(B) = W x r

For a doughnut-shaped object the GG is not even in the object itself, by bending over a man can move it outside his body (Fig. 22)

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-29.jpg’>

For more complicated two and three-dimensional objects the CG can be found experimentally by suspending it at two different points. A suspended object at rest always hangs so that its CG is directly and vertically below the point of suspension O (Fig 23a). In any other configuration a rotational torque would appear, induced by the displacement between the weight ‘W’ and the suspending force F(s). In Fig. 24 the object would rotate counterclockwise until CG falls beneath O and the torque disappears.

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-30.jpg’>

Suspending the object from a second point O’ again brings the CG to a vertical line through O’. This then defines the CG spatially at the intersection of the two lines (Fig 23b).

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-31.jpg’>

Fig. 25 illustrates the mathematical approach. A line a-a is drawn through the CG of a random object. Each side is thought to be broken down into minute particles ‘dm’. The addition of all individual products ‘dm’ times its distance ‘r’ from the line a-a, done separately for both right and left side, will show identical sums:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-32.jpg’>

(L)Σ r x dm = (R)Σ r x dm

The CG of a mobile element is a simple example. In order to hang balanced CG and point of suspension must exactly coincide. For Fig.26 this requires:

W(L) x r(L) = W(R) x r(R) → l x 6 = 3 x 2

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-33.jpg’>

Thus the CG is foremost a concept of balance and therefore of a higher order than symmetry. Symmetry is a geometrical issue about shapes, indicating that two halves can be mirrored, whereas balance extends the viewpoint by the qualitative step of introducing forces. In connection with human structures it is possible to speak of left/right symmetry and balance but only of front/back balance.

Stability

The condition for a supported object to be in balance requires its CG to lie above the base area defined by its supports:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-34.jpg’>

W ? N(A) ? N(B) = 0 / meets the translational condition

W x d ? N(A) x l = 0 / meets the rotational condition computed about point B

If the condition for stability is not fulfilled:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-35.jpg’>

W ? N(A) ? N(B) = 0 / meets the translational condition

W x d + N(B) x l = 0 / this condition can’t be met.

W and N(B) by turning counterclockwise about A generate both a positive torque which will always be larger than zero. The object starts turning.

Levers

The lever is probably one of the most frequent applications of torque. A lever in its simplest form consists of a rigid bar used in conjunction with a fulcrum at O. Three types of levers can be defined according to the relative positions of the load force F(L) the applied force F(A) and the fulcrum (Fig.29).

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-36.jpg’>

Type 1:

As in a scale the necessary force F(A) to counteract the force F(L) can be calculated by using the condition for rotational balance:

F(A) x d(A) ? F(L) x d(L) = 0

→ F(A) x d(A) = F(L) x d(L)

→ F(A) = F(L) x d(L)/d(A)

with F(L) = 4000 N (≈ 400 kg), d(L) = 0.2 m and d(A) = 2 m:

F(A) = 4000 x 0.2 = 400 N (≈ 40 kg)

the relation d(A)/d(L) = 2/0,2 = 10 is called the ‘mechanical advantage’ of the lever.

Type 2:

The condition for rotational balance:

F(L) x d(L) – F(A) x d(A) = 0 → F(A) = F(L) x d(L)/ d(A)

Type 3:

The condition for rotational balance:

F(L) x d(L) – F(A) x d(A) = 0 → F(A) = F(L) x d(L)/ d(A)

with the same numerical assumptions as for type 1:

F(A) = 4000 x 2/0,2 = 40000 N (≈ 4000 kg)

the mechanical advantage: d(A)/ d(L)= 0.2/2 = 0.1

The forearm is an example of the lever-type 3:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-37.jpg’>

The elbow-joint represents the fulcrum, the pull of the biceps the applied force counteracting the downward pull of the forearm’s gravity-induced weight W=mg. The schematic drawing (Fig.31) shows the appearing forces:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-38.jpg’>

F(B): force exerted by the biceps
F(H): compressional force on upper arm
W: weight of forearm, assumed as W = 30 N

Distances are given in centimeters, assumed for an average arm.

The conditions for translational and rotational equilibrum will yield the magnitude of F(B) and F(H)

translational condition:

F(H) + W ? F(B) = 0 → F(H) = F(B) – 30

rotational condition (about the fulcrum):

τ(B) ? τ(W) = 0; τ(H) = 0 (for obvious reasons)

→ F(B) x 5 ? W x 15 = 0

F(=) [30 x 15]/5 = 90N(≈ 9kg)

substituting F(B) in equation (1):

F(H) = 90 ? 30 = 60N(≈ 6kg)

Adding a load F(L) = 100 N (≈ 10 kg) on the hand increases the necessary pull of the biceps:

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-39.jpg’>

translational condition:

F(H) + W + F(L) ? F(B) = 0

→ F(H) = F(B) – W ? F(L) = F(B) – 30 – 100 = F(B) ? 130

rotational condition:

F(B) x 5 – W x 15 – F(L) x 35 = 0

→ F(B) = [(W x 15) + (F(L) x 35)]/5 = [(30 x15) + (100 x 35)]/5 = 790N (≈ 79kg)

substituting (F)B in equation (1):

F(H) = 790 – 130 = 660 (≈ 66 kg)

Or in English, the biceps brachialis has to exert a pull of 79 kg to hold against the weight of the forearm plus the applied load of 10 kg. The pressure in the joint against the humerus amounts to 66 kg.

Elasticity and Plasticity

Any real material, from rubber band to concrete, will always be deformed at least slightly when forces are applied. Internal electrical and magnetic molecular forces hold materials together and allow the gradual deformation. The effects of the applied forces can be measured and categorized without knowing what exactly happens within and among the molecules.

Physical science distinguishes between stress and strain. ‘Stress’ is a measure for the force applied to produce deformations in a material, whereas ‘strain’ is used to describe the deformation itself.

Stress is defined as the force per unit area like m(2), cm(2) or mm(2). It is usually denoted by the greek letter ‘σ’. A force F = 720 N applied to a femur with a cross-sectional area of A = 6 cm(2) results in a stress or σ of:

σ = F/A = 720/6 = 120 [N/cm(2)] (≈ 12 kg/cm(2)

Three kinds of stresses are usually defined. ‘Tension stress’ as the force producing lengthening of an object, ‘compression stress’ for shortening forces and ‘shear stress’ corresponding to the application of forces acting in a scissor like manner.

Strain describes the deformation. A rubber band of the length ‘l’ subjected to the force F will stretch a distance ∆ l. A band of the double length 2 x l, under the same force, will stretch by 2∆1. This fractional change in length is then denoted by the greek letter ϵ:

ϵ = ∆dl/1

The relation between stress and strain for a material under tension can only be deduced experimentally. A typical example for the behavior of a so called ductile material under tension stress is shown in Fig.33.

<img src=’https://novo.pedroprado.com.br/imgs/1989/1031-40.jpg’>

O to A: the region of elastic behavior, σ and ϵare proportional. A relaxation of the stress to zero will let the material regain its original shape. The slope of the line from O to A expresses an inherent property of the tested material, the elastic modulus ‘E’. E = σ/ϵ

A: linear limit

B: elastic limit

B to C: the region of plastic behaviour. A material stretched beyond B will not regain its original shape. At B’ a relaxation back to σ = 0 will leave an irreversible residual deformation ∆ ϵ, a subsequent application of stress to a behavior along the dashed line

C: ultimate tension strength

D: fracture point

Note that the region of plasticity shows a very flat incline, a small increment of stress leads to a large deformation. The magnitude of this region is a measure of the ductility or the plastic deformability of a material. Assuming soft tissue to be a material of a similar characteristic this would then mean that the applied force for its lengthening has to be just slightly over the elastic limit to effect a progressive deformation.

Within the elastic region stress and strain are proportional, hence the applied force is linearly related to the elongation. This can be derived from the equation E = σ/ϵ, or reformulated σ = E x ϵ. With σ = F/A and ϵ = ∆1/1 this becomes:

F/A = E x ∆/1 → F= (A x E)/1 x ∆1 I

For a given structure ‘A’, ‘E’ and ‘1’ are definable physical properties and can therefore be condensed to:

k = (A x E)/1, and thus: F = k x ∆l

where ‘k’ is called the ‘spring constant’. For every material showing an elastic behavior the specific value for ‘k’ can be determined experimentally.

Physics for Rolfers

As everybody will notice, this is neither all the physics there is nor all that a Rolfer should know. It is a start and a base for further inquiries into concepts linked to moving bodies: momentum and energy, the principles of their conservation, the storage of energy as potential, kinetic or spring energy, rotational movements and so on.

Gravity is a constant companion of human beings, inducing, by virtue of its physical existence, relentlessly changing forces the moving body has to cope with. Besides that the process of Rolfing involves at times the application of force. And here we should always be very much aware and clear about by what they are counteracted in the complicated system of movable, turnable, elastic, plastic, brittle, stiff multifunctional parts which constitute the skeletal and myofascial system of the body. Or else we are, as once a Rolfer put it, just pushing around the flesh.

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