Chapter 1: Static Forces

“Mechanics” is a branch of physics which concerns with the effect of forces.

Force amF

onaccelerati :

mass :

a

m

2sec

m ][ ;kg ][ am

Dimensions:

Fundamental quantities in physics:

1. length 2. mass 3. time

Aristotle (384 ~ 322 BC)

“The animal that moves makes its change of position by pressing against that which is beneath it…Runners run faster if they swing their arms for in

extension of the arms there is a kind leaning upon the hands and the wrist.”

“The animal that moves makes its change of position by pressing against that which is beneath it…Runners run faster if they swing their arms for in

extension of the arms there is a kind leaning upon the hands and the wrist.”

Length

Length

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.

4. It is called the SI system (Système International) of units.

Length: meter (m)

Mass: kilogram (kg)

Time: second (s)

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

1. 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.

2. The cylinder is 39 cm in height and in diameter.

3. 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

Table 1.2, p. 5

1. Before 1960, the standard of time was defined in terms of the average length of a solar day in the year 1900.

2. The basic unit of time, the second, was defined to be 1/86 400 of the average solar day.

3. 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.

4. The second is now defined as 9 192 631 770 times the period of oscillation of radiation from the cesium atom.

Cesium fountain atomic clock

Table 1.3, p. 6

Torque

A body is in equilibrium if the vector sum of the forces and the torques acting on the body is zero.

0i

iF

0i

i

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.

Equilibrium and Stability

Equilibrium Considerations for the Human Body

56%

The center of gravity of an erect person with arms at the side is at approximately 56% of the person’s 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.

Stability of the Human Body under the Action of an External Force

m 5.1 aa FT

The counterclockwise torque Ta about the point A produced by the applied force Fa is

m 1.0WTw

The opposite restoring torque Tw due to the person’s weight is

Assuming that the mass of the person is 70 kg, his weight W is

(N) newton 6868.970 mgW

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

m-N 6.68m 5.1 aF

Therefore, the torque required to topple an erect person is

N 7.455.1

6.68aF

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.

One can also increase the stability against a topple force by spreading the legs.

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.

Skeletal Muscles

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.

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

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.

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

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.

For all three types of levers, the force F required to balance a load of weight W is given by

2

1

d

WdF

The mechanical advantage M of the lever is defined as

1

2

d

d

F

WM

M (Class 1): > or < 1

M (Class 2): > 1

M (Class 3): < 1

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.

2

1

2

1

d

d

L

L

2

1

2

1

d

d

v

v

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.

M

1

This is a Class 3 lever.

The Elbow

The direction of the reaction force Fr shown is a guess. The exact answer will be provided by the calculations.

coscos rm FF

sinsin rm FWF

x component of the forces

y component of the forces

The torque about the fulcrum must be zero

WFm cm 40sin cm 4 WFm 10sin

With θ = 72.6◦, the muscle force Fm is WWW

Fm 5.10954.0

10

6.72sin

10

With a 14 kg weight in hand, N 14408.9145.10 mF

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 ϕ.

cos6.72cos1440 rF

sin8.9146.72sin1440 rF

N 430cos rF

N 1240sin rF

orN 1320rF

9.70

Model of the Elbow

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).

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.

0cos71cos Rm FF

0sin71sin RLm FWWF

(x components 0f the force = 0)

(y components 0f the force = 0)

0cm 18

cm 10cm 7)sin(

W

WF LR (torque about point A = 0)

The Hip

θ

Since WL = 0.185 W, WFR 31.2sin

WFm 59.1 3.77 WFR 37.2

For a person with mass 70 kg the weight is 686 N and the force on the hip join is 1625 N.

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)

Normal Case

Injured Case

Limping

“24And Jacob was left alone, and a man wrestled with him until the break of the dawn. 25And when the man saw that He did not prevail against him, He touched the socket of his hip; and the socket of Jacob’s hip was dislocated as he wrestled with Him. 26And the man said, Let Me go, for the dawn is breaking. But Jacob said, I will not let You go unless You bless me.” (Genesis 32: 24 ~ 26)

Vertebral column

Cervical vertebrae

Thoracic vertebrae

Lumbar vertebrae

Sacrum

Pivot

The Back

the fifth lumbar vertebra.

A

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.

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.

the fifth lumbar vertebra.

A

N 3201 W

N 160 2 W

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.

Ballet Dancing