Chapter 1: Static Forces

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Chapter 1 Static Forces
Mechanics is a branch of physics which concerns
with the effect of forces.
Force
Dimensions
Fundamental quantities in physics 1. length
2. mass 3. time
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Aristotle (384 322 BC)
The animal that moves makes its change of
position by pressing against that which is
beneath itRunners run faster if they swing their
arms for in extension of the arms there is a kind
leaning upon the hands and the wrist.
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The animal that moves makes its change of
position by pressing against that which is
beneath itRunners run faster if they swing their
arms for in extension of the arms there is a kind
leaning upon the hands and the wrist.
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• 1. The law of physics are expressed in terms
  basic quantities.
• 2. In mechanics, the three basic quantities are
  length (L), mass (M), and
• time (T). All other quantities in mechanics
  can be expressed in terms of these
• three.
• 3. An international committee has agreed on a
  system of definitions and
• standards to describe fundamental physical
  quantities.
• It is called the SI system (Système
  International) of units.

Length meter (m) Mass kilogram (kg) Time
second (s)
Length
Length
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• In A.D. 1120. King Henry I of England decreed
  that the standard of length in his country would
  be the yard and that the yard would be precisely
  equal to the distance from the tip of his nose to
  the end of his outstretched arm.
• Similarly, the original standard for the foot
  adopted by the French was the length of the royal
  foot of King Louis XIV. This standard prevailed
  until 1799.
• In 1799 the legal standard of length in France
  became the meter, defined as one ten-millionth of
  the distance from the equator to the North Pole.
• In 1960, the length of the meter was defined as
  the distance between two lines on a specific bar
  of platinum-iridium alloy stored under controlled
  condition.
• With the strong demand of scientific accuracy,
  the definition of the meter was modified to be
  equal to 1 650 763.73 wavelengths of orange-red
  light emitted from a kryton-86 lamp.
• In October 1983, the meter was redefined to be
  the distance traveled by light in a vacuum during
  a time interval of 1/299 792 458 second.

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• The SI unit of mass, the kilogram, is defined as
  the mass of a specific platinum-iridium alloy
  cylinder kept at the International Bureau of
  Weights and Measures at Sèvres, France.
• The cylinder is 39 cm in height and in diameter.
• The mass of a carbon-12 atom is taken to be 12
  atomic mass units (12 u).

1u 1.660 540 2 ( 0.000 001 0) 10-27 kg
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Table 1.2, p. 5
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• Before 1960, the standard of time was defined in
  terms of the average length of a solar day in the
  year 1900.
• The basic unit of time, the second, was defined
  to be 1/86 400 of the average solar day.
• In 1967, the second was redefined to take
  advantage of the great precision with an atomic
  clock. The clock will neither gain nor lose a
  second in 20 million years.
• The second is now defined as 9 192 631 770 times
  the period of oscillation of radiation from the
  cesium atom.

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Cesium fountain atomic clock
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Table 1.3, p. 6
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Torque
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Equilibrium and Stability
A body is in equilibrium if the vector sum of the
forces and the torques acting on the body is zero.
The position of the center of mass with respect
to the base of support determines whether the
body is stable or not.
The wider the base on which the body rests, the
more stable it is that is the more difficult it
is to topple it.
In Fig. 1.2a the torque produced by its weight
tends to restore it to its original position.
In Fig. 1.2b the same amount of angular
displacement of a narrow-based body results in a
torque that will topple it.
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Equilibrium Considerations for the Human Body
The center of gravity of an erect person with
arms at the side is at approximately 56 of the
persons height measured from the soles of the
feet. The center of gravity shifts as the person
moves and bends. The act of balance requires
maintenance of the center of gravity above the
feet.
When carrying an uneven load, the body tends to
compensate by bending and extending the limbs so
as to shift the center of gravity back over the
feet.
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Stability of the Human Body under the Action of
an External Force
The counterclockwise torque Ta about the point A
produced by the applied force Fa is
The opposite restoring torque Tw due to the
persons weight is
Assuming that the mass of the person is 70 kg,
his weight W is
The restoring torque produced by the weight is
therefore 68.6 newton-meter (N-m). The person is
on the verge of toppling when the magnitudes of
these two torques are just equal that is Ta Tw
or
Therefore, the torque required to topple an erect
person is
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By bending the torso the center of gravity will
be shifted away from the point A and as the
result will the restoring torque be increased.
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One can also increase the stability against a
topple force by spreading the legs.
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Skeletal Muscles
The skeletal muscles producing skeletal movements
consist of many thousands of parallel fibers
wrapped in a flexible sheath that narrows at both
ends into tendons. The tendons, which are made of
strong tissue, grow into the bone and attach the
muscle to the bone.
1. Most muscles taper to a single tendon. But
some muscles end in two or three tendons these
muscles are called, respectively, biceps and
triceps. 2. Each end of the muscle is attached to
a different bone. 3. In general, the two bones
attached by muscles are free to move with respect
to each other at the joints where they contact
each other.
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There is a great variability in the pulling force
that a given muscle can apply. The force of
contraction at any time is determined by the
number of individual fibers that are contracting
within the muscle. When an individual fiber
receives an electric stimulus, it tends to
contract to its full ability. If a stronger
pulling force is required, a larger number of
fibers are stimulated to contract.
Experiments have shown that the maximum force a
muscle is capable of exerting is proportional to
its cross section. From measurements, it has
been estimated that a muscle can exert a force of
about 7 106 dyn/cm2 of its area.
7 106 dyn/cm2 7 105 Pa
1 atm 1.013 105 Pa
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Levers
To compute the forces exerted by muscles, the
various joints in the body can be conveniently
analyzed in terms of levers. Such a
representation implies some simplified
assumptions. We will assume that the tendons are
connected to the bones at well-defined points and
that the joints are frictionless.
A lever is a rigid bar free to rotate about a
fixed point called the fulcrum.
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GIVE ME A PLACE TO STAND AND I WILL MOVE THE EARTH
A remark of Archimedes quoted by Pappus of
Alexandria Collection or Synagoge, Book VIII, c.
AD 340
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There are three classes of levers. In a Class 1
lever, the fulcrum is located between the applied
force and the load. In a Class 2 lever, the
fulcrum is at one end of the bar the force is
applied to the other end and the load is
situated in between. A Class 3 lever has the
fulcrum at one end and the load at the other. The
force is applied between the two ends. Many of
the limb movements of animals are performed by
Class 3 levers.
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For all three types of levers, the force F
required to balance a load of weight W is given by
M (Class 1) gt or lt 1 M (Class 2) gt 1 M
(Class 3) lt 1
The mechanical advantage M of the lever is
defined as
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A force slightly greater than what is required to
balance the load will lift it. As the point at
which the force is applied moves through a
distance L2, the load moves a distance L1.
These relationships apply to all three classes of
levers. It is evident that the excursion and
velocity of the load are inversely proportional
to the mechanical advantage.
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The Elbow
This is a Class 3 lever.
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Model of the Elbow
The direction of the reaction force Fr shown is a
guess. The exact answer will be provided by the
calculations.
x component of the forces
y component of the forces
The torque about the fulcrum must be zero
With a 14 kg weight in hand,
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Model of the Elbow
The direction of the reaction force Fr shown is a
guess. The exact answer will be provided by the
calculations.
Now we are in a position to evaluate Fr and ?.
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The Hip
This figure shows the hip joint and its
simplified lever representation, giving
dimensions that are typical for a male body. The
hip is stabilized in its socket by a group of
muscles, which is represented in figure b as a
single resultant force Fm. When a person stands
erect, the angle of this force is about 71?with
respect to the horizon. WL represents the
combined weight of the leg, foot, and thigh.
Typically, this weight is a fraction (0.185) of
the total body weight W (WL 0.185W).
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The Hip
We will now calculate the magnitude of the muscle
force Fm and the force FR at the hip joint when
the person is standing erect on one foot as in a
slow walk. The force W acting on the bottom of
the lever is the reaction force of the ground on
the foot of the person. This is the force that
supports the weight of the body.
?
(x components 0f the force 0)
(y components 0f the force 0)
(torque about point A 0)
Since WL 0.185 W,
For a person with mass 70 kg the weight is 686 N
and the force on the hip join is 1625 N.
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Limping
Persons who have an injured hip limp by leaning
toward the injured side as they step on that
foot. As a result, the center of gravity of the
body shifts into a position more directly above
the hip joint, decreasing the force on the
injured area. Calculations for the case show
that the muscle force Fm 0.47W ( as opposed to
the normal case of Fm 1.59W) and that the force
on the hip joint is FR 1.28W (as opposed to the
normal case of FR 2.37W)
Injured Case
Normal Case
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Pivot
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The Back
When the trunk is bent forward, the spine pivots
mainly on the fifth lumbar vertebra. We will
analyze the forces involved when the trunk is
bent at 60 from the vertical with the arms
hanging freely. The pivot point A is the fifth
lumbar vertebra. The lever arm AB represents the
back. The weight of the trunk W1 is uniformly
distributed along the back its effect can be
represented by a weight suspended in the
middle. The weight of the head and arms is
represented by W2 suspended at the end of the
lever arm. The rector spinalis muscle, shown as
the connection D-C attached at a point two-third
up the spine, maintains the position of the back.
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The Back
For a 70-kg man, W1 and W2 are typically 320 N
and 160 N, respectively. To hold up the body
weight, the muscle must exert a force of 2000 N
and the compressional force of the fifth lumbar
vertebra is 2230 N. This example indicates that
large forces are exerted on the fifth lumbar
vertebra.
It is not surprising that backaches originate
most frequently at this point, the fifth lumbar
vertebra.
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When standing on tiptoe, the total weight of the
body is supported by the reaction force at point
A. This is a Class 1 lever with the fulcrum at
the contact of the tibia. The balancing force
is provided by the muscle connected to the heel
by the Achilles tendon. Calculations show that
while standing tiptoe on one foot the
compressional force on the tibia is 3.5 W and the
tension force on the Achilles tendon is 2.5 W.
Standing on tiptoe is a fairly strenuous position.
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Ballet Dancing