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Math
For Reloaders
"Math
Definitions:"
Both Rational and Irrational numbers taken together are called
Real numbers:
Real
Numbers
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Irrational
Numbers Rrational
Numbers
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Non-integer Fractions
Terminating Decimals
Repeating Decimals Integers
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Negative
Integers Whole
Numbers
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Zero Positive Integers
(Natural Numbers)
Mathematical Symbols used:
Symbols Meaning example
= equals 576 = 576
+ Addition 526 + 224 = 750
- Subtraction 526 - 224 = 302
* Multiplication 526 * 224 = 117824
÷
or / Division 526 ÷ 224 = 2.3482
±
plus or minus 59º F ±5º F
=~ Approximately equal to =~ 3.14159 26536
>
Greater than 85 > 54
>
= Greater than or equal to 85 >= 54
»
Significantly Greater than 12385 » 5
<
Less than 54 < 85
<
= Less than or equal to 54 <= 85
«
Significantly Less than -85755754 « 858656
| | Absolute value of |-76| = 76
... Continuation 1 ÷ 3 = .333...
/ Per mph or m/h
i Imaginary number square root of -4 = 2i
Change in X
Sum of KE = (m * v² ÷ 2)
|~
or
|b~a Integral
between the limits
a and b 4 |~ sec3 Ø d Ø
or
|-1~1 dx ÷ dy
( ) Parentheses grouping
symbols (5 + 6) * 7 = 77
[ ] Bracket grouping symbols [(-10) ÷ 2]² = 25
PI =~ 3.14159 26536
º
Degree 59º F (Fahrenheit)
sin, cos, tan Trigonometric functions sin x, cos x, tan x
sin-1, cos-1, tan-1 Inverse Trigonometric Functions sin-1 x, cos-1 x, tan-1
x
Same as
csc x, sec x, cot x
Superscript Exponent 53 = 5 * 5 * 5 = 125
subscript Different type of the same variable Both are velocities but
Vo is different then V
Order of Operation:
Order Symbols Meaning example
Step One [ ], ( ) Perform operations on the inter
most grouping symbols first and
work your way to the outer most
grouping symbols, parentheses
( ), and brackets [ ]. x = [8 - ((12 - 2) ÷ (4 + 1))] *
4
x = [8 - (10 ÷ 5)] * 4
x = [8 - 2] * 4
x = 6 * 4
x = 24
Step Two xy, root of x, sin x Simplify all expressions like,
exponentials, logs, and function. x = [8 - ((24 - 6) ÷ Square
root of (42 + 9))] * tan 75.96375653
x = [8 - ((16 - 6) ÷ Square root of (16 + 9))] * tan 75.96375653
x = [8 - (10 ÷ Square root of 25)] * tan 75.96375653
x = [8 - (10 ÷ 5)] * tan 75.96375653
x = [8 - 2] * tan 75.96375653
x = 6 * tan 75.96375653
x = 6 * 4
x = 24
Step Three *, ÷ Do multiplication and division as
they occur from left to right. x = Square root of [((-8) + 192 ÷ 3)
* 4 - 4 + 160 ÷ 8 + 84 * 4]
x = Square root of [((-8) + 64) * 4 - 4 + 160 ÷ 8 + 84 *
4]
x = Square root of [56 * 4 - 4 + 160 ÷ 8 + 84 * 4]
x = Square root of [224 - 4 + 160 ÷ 8 + 84 * 4]
x = Square root of [224 - 4 + 20 + 84 * 4]
x = Square root of [220 + 20 + 336]
x = Square root of [240 + 336]
x = Square root of 576
x = 24
Step Four +, - Do addition and subtraction as
they occur from left to right. x = Square root of [((-8) + 192 ÷ 3)
* 4 - 4 + 160 ÷ 8 + 84 * 4]
x = Square root of [((-8) + 64) * 4 - 4 + 160 ÷ 8 + 84 *
4]
x = Square root of [56 * 4 - 4 + 160 ÷ 8 + 84 * 4]
x = Square root of [224 - 4 + 160 ÷ 8 + 84 * 4]
x = Square root of [224 - 4 + 20 + 84 * 4]
x = Square root of [220 + 20 + 336]
x = Square root of [240 + 336]
x = Square root of 576
x = 24
Special Note:
Exponential
Expressions The nth root of a number is the same as that number
raised to the reciprocal of the nth power. The nth root of x is
the same as x1/n also the same as x1/n where 1/n is expressed as
a decimal. For example: The eighth root of 6561 = 3, 65611/8 =
65611 ÷ 8 = 65610.125 = 3
Exponentials
Function sin2 x is different than sin x2; sin2 x is the same as
(sin x)2 and the sin x2 is the same as sin (x)2
Per or divided by
is given by the word per or the symbol / or the superscript or negitive subscript,
usually of 1, 2, or 3 as in
ft pre sec2 or mi pre hr2
ft/sec2 or mi/hr
ft*sec-2 or mi*hr-1
In this case there is not spaces around the multiplication symbol.
Work and Force is often given multiplied by
is given by the symbol * or -
ft*lb or ft-lb
N*m or N-m
In this case there is not spaces around the multiplication or division symbol.
Standards and Units:
There are two primary systems in the world today, the English and
the Metric system. The British or English system uses the standards
of the foot (ft.) for the length, the pound (lb.) for the force,
and the second (s) for the time. There are two metrics systems
in use today, The first and by far the most important is the
System International (French for International System), which
is abbreviated SI and uses the meter (m) for the length, the
kilogram (kg) for the mass, and the second (s) for the time.
This system used to be called the MKS (meter-kilogram-second)
system. The second is the cgs system and it uses the centimeter
(cm) for the length, the gram (g) for the mass, and the second
(s) for the time and is abbreviated just as cgs.
Second (s):
A second is defined, as the time required for a cesium-133 atom
to vibrate 9,192,631,770 times in a vacuum. There are 60 second
(sec) in a minute (min.), 60 min. in an hour (h.), 24 hours in
a day (one revolution around the Earth's axis), and 365 1/4 days
in a year (yr.: one full orbit around the Sun). This is why we
have an extra day every four years, called leap year, so we can
get caught up with time.
Light Speed:
Light speed is the speed of light traveling in a vacuum for one
second and that speed is 299,792,458 meters per second or 186,282.3971
miles per second and is indicated in physics and mathematical formulas
as a small c.
Light-year:
A light-year is the distance light travels in a vacuum in one year.
Light-year = 299,792,458 m/s * 60 * 60 * 24 * 365.25 = 17,987,547,480
m/min * 60 * 24 * 365.25 = 1,079,252,848,800 m/h * 24 * 365.25
= 25,902,068,371,200 m/day * 365.25 = 9,460,730,472,580,800 m/yr.
I'll let you figure out how many miles per year that is.
Parsec:
A parsec is equal to 3.26 light-years. Why such a strange number?
I'll leave that one up to Edward L. (Ned) Wright's Web Page on
distances.
Time Dilation:
Time Dilation simply stated is the faster an object is moving the
slower time will be measured. If you have a clock on earth it will
measure time by X amount while the same clock the faster it is
moving the slower the clock will run. An elementary particle which
have a very small mass (typically 10-30 to 10-27 kg.) like the
Muon, that requires little energy to be accelerated to speeds close
to c, has a lifetime of 2.2 of a microsecond at rest will have
a longer lifetime the closer to c it is traveling as predicted
by the time-dilation formula: t = to ÷ Square root of (1
- v² ÷ c²); where t is the change of time an object
has at velocity v and to is the change of time an object has at
rest. Since the Square root of (1 - v² ÷ c²) is
always less then 1, we see that t > to.
Meter:
The meter in 1983 was redefined in terms of the speed of light.
The meter is the length traveled by light in a vacuum during the
time interval of 1/299,792,458 of a second.
Inch (in):
An inch is defined as precisely 2.54 centimeters (cm.; there are
100 cm. in a meter, 1 cm. = .01 m.). 12 inches to a foot, 3 ft.
to a yard, 5280 ft. to a mile, and 6080 ft. to a nautical mile.
The length of a football field is 300 ft. or 100 yards.
Length Contraction:
Not only time intervals are different in different reference frames
but space intervals - lengths and distances - are different also,
according to the special theory of relativity. As an objects speed
increases its mass, gravity, and time intervals increases and its
length contracts or shortens only in the direction it is traveling
in according to the formula: L = Lo * Square root of (1 - v² ÷ c²);
where Lo is the length of the object at rest, and L will be the
objects length at speed v. As you can see from the formula, if
v were to equal c, the length of the object at rest would be multiplied
by zero and that would make the length of the object traveling
at the speed of light zero length, therefore the speed of light
is not possible.
Pi ():
Pi is defined as the ratio of the diameter of a circle divided
into its circumference. Today has been carried out to over 2.5
Billion places passed the decimal point.
PI out to 10,000 decimal places:
For our formulas we will only use the first 10 places passed the
decimal point, 3.14159 26536 (the last digit is a 6 because of
rounding off).
Degree (deg):
A degree is an angle of a circle that is divided into 360 arcs
of equal length. Each degree is broken down into 60 arcs of equal
length called minutes and each minute is further divided into 60
arcs of equal length called seconds. Each degree = 6.28318530717959 ÷ 360
= 0.017453292519943 rad. To convert from radian to degree is: Degree
(angle) = rad (angle) * 180 ÷ .
Radian (rad):
A radian is a unit arc, each having a length equal to the length
of the circle's radius. To find the angle Ø, in radians
of a circle is equal to the length of the circle circumference
divided by the radius of that circle; Ø = l ÷ r,
this is rotational motion. The circumference of a circle whose
radius has a length r is r * 2; thus Ø = l ÷ r =
2 * r ÷ r = 2 = approximately 6.2831853072 radians. 1 rad
= 360 ÷ 6.28318530717959 = 57.2957795130823º = 57º 17.7467707849372
min or 57º 17 min. and 44.8062470962331 sec. or approximately
57º 18 min. To convert from degrees to radians is: Rad (angle)
= degrees (angle) * ÷ 180.
Speed:
Speed is how far an object travels in a given time interval and
is defined as the Average speed = distance traveled ÷ time
elapsed. Velocity is used to signify both a magnitude (numerical
value) of how fast an object is moving and the direction in which
it is moving and is defined as the Average velocity = displacement ÷ time
elapsed. From this you can see that some times speed and velocity
are used interchangeably in ordinary language and I see no reason
to very from the ordinary language for our purposes.
Newton's First Law of Motion:
Every body continues in its state of rest or of uniform speed in
a straight line unless it is compelled to change that state by
forces acting on it. The tendency of a body to maintain its state
of rest or uniform motion in a straight line is called inertia.
As a result, Newton's first law is often called the law of inertia.
Mass:
Mass is a measure of the inertia of a body. The more inertia a
body has, the harder it is to change its state of motion. By dividing
its weight by the acceleration of gravity; M (mass) = F (weight
in pounds) ÷ a (acceleration of gravity on earth is 32.1734)
one can calculate an objects Mass. If the object weight is in grains
then 7000 has to be multiplied to the acceleration of gravity,
7000 grains per pound. That would make the formula, M = F ÷ (a
* 7000).
The
proper unit of mass in the SI (metric) system of unit is the
kilogram (kg). The actual standard is a particular
platinumiridium
cylinder whose mass is by definition one kilogram and is kepth
near Paris at the International Bureau of Weights and Measures,
1 kilogram = 0.0685 slug. In the cgs unit mass is the gram (g).
The English unit of mass is the slug and is defined as a mass that
will undergo an acceleration of 1 ft./sec.² (second squared)
when a force of 1 pound (lb.) is applied to it. Thus, 1 LB = 1
slug * ft./sec.², 1lb = 4.45 N, and 1 slug = 14.6 kilogram.
Einstein showed that the mass of an object increases as its speed
increases according to the formula: m = mo ÷ Square root
of (1 - v² ÷ c²). In this mass increase formula,
mo is the rest mass of the object, and m is the mass it will
be measured to have in which it moves at speed v. As an object
is accelerated to greater and greater speeds, its mass becomes
larger and larger. If v were to equal c, the denominator in this
equation would be zero and the mass m would become infinite.
To accelerate an object up to v = c would require an infinite
amount of energy, and so is not possible.
Force:
The Newton (N), in the SI unit, is the force required to accelerate
a mass of one kilogram one m/sec². Thus 1 N = 1 kg. * m./sec.².
Impulse is t (time) * F (force) and is measured in Newton's. 1
N = 0.225 lb. The dyne, in the cgs system of units, is defined
as the force required to accelerate a mass of one gram one cm./sec.².
Thus 1 dyne = 1 g. * cm./sec.². The pound, in the English
system of unit, is defined as the weight of a body with a mass
of 0.454 kg at a particular place on the earth where the acceleration
due to gravity is g = 32.1734 ft./sec.².
Density:
Density is mass per unit volume: Density (D) = Mass ÷ Volume.
Density is expressed in lb./cubic foot (lb./ft.³) or grams/cubic
centimeter (gm./cm.³).
Load Density:
Load density is the ratio of the weight of the powder charge to
the weight of fresh water that the cartridge case could hold up
to the base of the bullet.
Note: This is the same definition for the page on IMR Powders.
Use their definition when working out their problems not the following
definition.
This is my definition of Load density and I think it makes more
sense then the above definition of Load density. It is the ratio
of powder charge weight to the cartridge powder capacity weight,
of the same powder, to the base of the bullet when seated. Varying
either one, powder charge weight or bullet seating depth will change
the load density. A higher load density will produce higher chamber
pressure and vice versa. This change in chamber pressure is approximately
proportional to the square of the load density. This is the definition
that I will be using, unless otherwise stated.
Newton's Second Law of Motion:
The Acceleration of an object is directly proportional to the net
force acting on it and is inversely proportional to its mass. The
direction of the acceleration is in the direction of the applied
net force. A net force exerted on an object may make its speed
increase or if it is in a direction opposite to the motion, it
will reduce the speed. If the net force acts sideways on a moving
object, the direction as well as the magnitude of the velocity
changes. Thus, a net force gives rise to acceleration: a = F ÷ m
or F = m * a. Acceleration is the velocity that is changing and
is equal to velocity divided by time, a = V2 - V1 ÷ t2 -
t1. This same equation can give you time, t = V ÷ a and
velocity, V = a * t. Therefore: F = m * a = m * V ÷ t or
F * t = M * V.
Gravity:
Gravity is an acceleration and measured, by international agreement,
as 32.1734 ft./sec./sec. or 32.1734 ft./sec.². But the distance
traveled will be 16.0867 * t² on a free falling object. Gravitational
acceleration (g) decreases, with altitude, inversely with the square
of the distance to the earth's center, g = G ÷ re²,
where g is the gravitational acceleration, G is the gravitational
constant, re is the distance to the earth's center.
Gravitational constant:
The gravitational constant is G = 6.673290052 * 10-11 * N * m.²/kg.² in
the SI notation (or) 6.673290052 * 10-8 dyne * cm.²/g.² in
the cgs notation (or) 3.442654153 * 10-8 * lb. * ft.²/slug² in
the English notation.
Weight:
Weight is the amount of force exerted on an object do to the pull
of gravity. The English unit is the pound and is defined as the
weight or force of a body with a mass of .454 kg. at a particular
place on the earth where the acceleration due to gravity is g =
32.1734 ft./sec.². One pound is equal to 4.54 N. One gram
is equal to 15.4185 grains. One ounce is equal to 437.5 grains.
The weight of a rifle or handgun bullet is measured in grains (grs.).
It takes 7000 grs. to equal 1 pound. To convert from pound to grs.
you would divide 7000 into pounds and to convert grs. into pounds
you would multiply grs. by 7000. Grs. = pounds * 7000 and pounds
= grs. ÷ 7000.
Specific Gravity or Mass Density:
Specific gravity is the ratio of the density of a substance to
the density of fresh water at 39.2º Fahrenheit (4º Celsius).
Fresh water has a specific gravity of 1.0 and weighs 0.036091435
lb./in.³. By multiplying the specific gravity or mass density
of a substance by the weight of water, 0.036091435 lb./in.³,
you can find the weight of that substance in lb./in.³.
The
specific gravity of a bullet can be found by weighing a suspended
bullet by a thread from the pan of a balance
scale in air. Then
weigh the same bullet the same way but while the bullet is now
suspended under water. Now divide the dry weight of the bullet
by the difference of subtracting the second (wet) weight from the
first (dry) weight. The answer will be the specific gravity. Specific
Gravity = Dry weight ÷ (Dry weight - Water weight).
Sectional Density:
Sectional density is the ratio of a bullet's weight in pounds to
the square of its diameter in inches. SD (Sectional Density) =
W (weight of bullet in pounds) ÷ d² (bullet diameter
in inches squared).
Pressure:
Pressure is force acting on unit area: Pressure = Force ÷ Area
or P = F ÷ A. Pressure is expressed in lb./square inch (psi.,
pound/in.², lb./in.²) or kg./square centimeter (kg./cm.²).
Newton's Third Law of Motion:
Whenever one object exerts a force on a second object, the second
object exerts an equal and opposite force on the first object.
Or to put it another way. For every action there is an equal and
opposite reaction. This is what makes jets and rockets fly and
it is also what makes a gun recoil and you feel it as a kick.
Potential Energy (PE):
Potential energy is the energy of an object virtue of its position.
Gravitational PE = mg * y; where mg is weight and y is height or
PE2 - PE1 = mg * y2 - mg * y1 [gravitational PE]. Elastic potential
energy is proportional to the square of the amount of compression;
elastic PE = k * x² * ½.
Kinetic Energy (KE):
Kinetic energy is the energy of motion from the Greek word kinetikos,
meaning, "motion". An object in motion has the ability
to do work and thus can be said to have energy. KE = M * V² * ½ =
mg * y. For bullet energy (Bullets are in grains and there are
7000 grains to a pound. Therefore you must divide the bullet weight,
in grains, by 7000 to get pounds) the mass is M = F (in grains) ÷ a
* 7000. Substituting for the formula: mass "M" becomes
the weight of the bullet in grains "w" and we have KE
= w * V² ÷ (2 * 7000 * a (32.1734)) -- or -- w * V² ÷ 450427.6.
But, the scientific community uses 450400 for that value and so
will we.
Einstein showed that at high speeds the formula KE = M * V² * ½ is
not correct. You might think that using the above formula for the
increase in mass, m = mo ÷ Square root of (1 - v² ÷ c²)
would give KE = (1 * ½) * mo * v² <÷ Square
root of (1 - v² ÷ c²), but this formula, too,
is wrong. Einstein showed that the kinetic energy of a particle
is given by: KE = m * c² - mo * c²; where m is the mass
of the particle traveling at speed v, and mo is its rest mass.
Consistent with the idea that mass is a form of energy, Einstein
called mo * c² or moc² the rest energy of the object.
We can rearrange this to get, mc² = moc² + KE. We call
mc² the total energy E of the particle (assuming no potential
energy), E = mc², and we see that the total energy equals
the rest energy plus the kinetic energy: E = moc² + KE. For
a particle at rest in a given reference frame, its total energy
is Eo = moc² which is its rest energy. Here we have Einstein's
famous formula E = mc². For the kinetic energy can be written
in terms of the speed v of the object as: KE = mo * c² * ((1 ÷ Square
root of (1 - v² ÷ c²)) - 1). At very low speeds,
v << c, we see that it is approximately equal to the classical
equation of KE = mo * v² * ½. Indeed, the other equations
of the special relativity also reduce to their classical equivalents
at ordinary speeds: length contraction, time dilation, and mass
increase almost all disappears for v << c since the Square
root of (1 - v² ÷ c²) is approximately equal to
1.
The rotational kinetic energy is therefore I * ² * ½ where
I is the moment of inertia of the body and is its angular velocity.
Then for a bullet the formula would be I * ² / 450400. Then
the total kinetic energy is KE = M * V² * ½ + I * ² * ½ where
v is the linear velocity of the body, I is the moment of inertia
about its axis through the body, is the angular velocity about
its axis, and M is the total mass of the body. Then the total kinetic
energy for a bullet the formula would be KE = w * V² ÷ 450400
+ I * ² ÷ 450400
Work (W):
The word work has a variety of meanings in everyday language. But
in physics, work is given a very specific meaning to describe what
is accomplished by the action of a force when it makes an object
move through a distance. Also see my page entitled "Work,
Energy, and The State of Matter".
Specifically, the work done on a particle by a constant force (constant
in both magnitude and direction) is defined to be: The product
of the magnitude of the displacement times the component of the
force in the direction of the displacement. This can be written
as:
W = F * d * cos q
Where F is the constant force (both Kinetic Energy plus Potential
Energy), d is the net displacement of the particle, and q is
the angle between the directions of the force and the net displacement.
In the SI units work is measrured in newton-meters (N*m) and
a special name is given to this nuit, the joule (J): 1 J = 1
N*m. In the cgs system the unit of work is called the erg and
is defined as: 1 erg = 1 dyne*cm. And in the British units, work
is measured in foot-pounds. 1 J = 107 erg = 0.7376 ft*lb.
Acceleration:
Acceleration is a velocity that is changing during a change in
time. Acceleration = v ÷ t. By switching this equation around
you can fined velocity or time.
Momentum (p):
Momentum (p) is M (mass) * V (velocity). F is the net force applied
to an object and a change in p is the resulting momentum change
that occurs during the time interval change in t: F = change in
p ÷ change in t. Force acting on an object for a length
of time is called impulse, time * force. Therefore: Momentum (p)
= M * V = t * F = impulse.
A useful relation between the total energy E of a particle and
its momentum p can be derived. Since E = m * c² and p =
m * v, where m = mo ÷ Square root of (1 - v² ÷ c²),
we have:
E² = m² * c² = m² * c² * (c² + v² -
v²)
E² = m² * c² = m² * c² * v² + m² *
c² * (c² - v²)
E² = m² * c² = p² * c²+((mo² * c4
* (1 - v² ÷ c²)) ÷ (1 - v² ÷ c²))
or
E² = p² * c² + mo² * c4
E = p * c + mo * c²
For a body rotating about a fixed axis is called angular momentum
(L) and is defined as L = I * where I is the moment of inertia
and is the angular velocity. L = I * = (mr²) * (v ÷ r).
Rotational Motion:
Rotational motion is every point in a body rotating about a fixed
axis moves in a circle whose center is on the axis; the radius
of that circle is r, the perpendicular distance of that point from
the axis of rotation. A line drawn perpendicular from the axis
to any point sweeps out the same angle Ø in the same time.
Angular velocity () is defined as the rate of change of angular
position: = Ø ÷ t. All parts of a rigid body rotating
about a fixed axis has the same angular velocity at any instant.
Average angular velocity used to indicate how fast an object is
moving in a rotating direction and is defined as the Average angular
velocity = angular distance Ø ÷ time elapsed t, where Ø is
the angle through which the body has rotated in the time t. Instantaneous
angular velocity is defined as the small angle, Ø, through
which the body turns in a very short time interval t: = Ø ÷ t
[t very small, approaching zero]. Angular acceleration () is defined
as the rate of change of angular velocity: = ÷ t. Average
angular acceleration is defined as: = ( - o) ÷ t, where
o is the angular velocity initially and is the angular velocity
after the time t has passed. Instantaneous angular acceleration
is defined in the usual way: = ÷ t [t very small, approaching
zero]. Since is the same for all points of a rotating body, the
last two equations tells us that also will be the same for all
points. With measured in radians per second and t in seconds, will
be expressed as radians per second squared (rad ÷ s²).
If the body rotates with angular velocity , any particle will have
a linear velocity whose direction is tangent to its circular path;
the magnitude of its linear velocity, v, is v = l ÷ t. From
the equation Ø = l ÷ r, a change in rotation angle Ø is
related to the linear distance traveled by l = rØ. Hence
v = l ÷ t = r (Ø ÷ t), v = r. Although is
the same for every point in the rotating body at any instant, the
linear velocity v is greater for points farther from the axis.
The angular acceleration is related to the tangential linear acceleration
of a particle by = v ÷ t = r ( ÷ t), = r.
Angular
Linear
= o + t
v = vo + at
Ø = ot + .5t²
x = vo + .5at²
²
= o² + 2Ø
v² = vo² + 2ax
= ( + o) * ½
v = (v + vo) * ½
The angular acceleration will be proportional to the product of
the force times the lever arm. This product is called the moment
of the force, or Torque (). The angular acceleration of an object
is found to be directly proportional to the net applied torque,
: . The lever arm is defined as the perpendicular distance of
the axis of rotation from the line of action of the force, =
r * F sin Ø. The quantity m * r² represents the rotational
inertia of a particle and is called its moment of inertia. The
sum of the various torque's is just the total torque, so we have
= ( mr²) . Example, if we have 3 particles we use = mr² (for
the first particle) + mr² (for the second particle) + mr² (for
the third particle), m is the mass and r is the radians the mass
is from the rotational axis. is in radian/seconds² and the
moment of inertia in kg * m² in the SI units.
Radius of Gyration (k):
The radius of gyration of an object is defined as if all the mass
of the object were concentrated at a specific distance from the
axis; it would have the same moment of inertia as the original
object. The moment of inertia of any object can be written in terms
of its radius of gyration as, I = Mk²
Atmospheric Pressure:
Atmospheric pressure acts on all bodies and objects with a force
of about 14.7 psi or 1.03 kg./cm.² at sea level which is equal
to 1 atmosphere (ATM). Pressure is often expressed in ATM.
Partial Pressure:
The partial pressure means that in a mixture of gases, each gas
exerts part of the total pressure. The partial pressure contributed
by a single gas is in direct proportion to its percentage of the
total volume of the mixture.
Seawater:
Seawater contains about 75 elements that occur in nature, of these,
the four most abundant elements are oxygen, hydrogen, chlorine,
and sodium. Seawater is always slightly alkaline because it contains
several alkaline earth minerals, sodium, calcium, magnesium, and
potassium. The temperature of seawater varies from about 30º F
to 86º F. Seawater exerts a force of 0.445 psi. per foot (1
kg./cm.² per 9.75 meters) of depth and 0.432 psi. per foot
(1 kg./cm.² per 10 meters) of depth in fresh water. Seawater
weighs more then fresh water because of all the added elements.
Earth:
The earth's mass is 5.98 * 1024 kg. and the radius is about 6,380,000
meters or 20,926,400 ft., the earth's diameters then is 12,760,000
m., and the earth's circumference is 40,086,722.26 m. By taking
40086722.26 ÷ 24 = we get the earth's rotation, 1670280.09416667
meters, per hour. Then taking the earth's rotation per hr ÷ 60
= the earth's rotation, 27838.0015694444 meters, per minute. Taking
the earth's rotation per minute ÷ 60 = the earth's rotates,
463.966692824074 meters or 1522.20043577452 feet, per second at
the equator. If we take the rotation of the earth in radians, 6.28318530717959
per day, and divide by 24 we get the earth's rotational rate in
rad per hour and by dividing that by 60 we get the earth's rotational
rate in rad per minute and by dividing that by 60 we get the earth's
rotational rate in rad per second. We should have a rotational
rate of 0.000072722052166 rad per second. You can do the same for
degrees.
Temperature:>
Temperature refers to how hot or cold an object is. Molten steel
is said to have a high temperature whereas a cold tray of ice is
said to have a low temperature. Heat is energy in transit; it always
flows from a substance at a higher temperature to the substance
at a lower temperature. Many properties of matter change with temperature.
Most materials expand when heated and contract when cooled. Water
expands when changing from a liquid to ice and from a liquid to
a gas (water vapor). The electrical resistance of matter also changes
with temperature. There are three numerical scale commonly used
to day. The most common scale is the Celsius scale (C), named in
honor of the Swedish astronomer Anders Celsius the inventor of
this scale. The most common scale in the United States is the Fahrenheit
scale (F), named for the German scientist G. D. Fahrenheit the
inventor of this scale. The most important scale in scientific
work is the absolute, or Kelvin scale (K). The Celsius scale sets
the freezing point of water at 0º and boiling point of water
at 100º. The Fahrenheit scale is based on mercury and sets
the freezing point of water at 32º and boiling point of water
at 212º. But, the Kelvin scale system is based on the theoretical
temperature, that the volume of a gas decreases as the temperature
decreases, at which a gas volume reaches zero. To convert from
Celsius to Fahrenheit is F = C * 9 ÷ 5 + 32. But there are
two ways to convert from Fahrenheit to Celsius. The first way is
C = (F - 32) * 5 ÷ 9. The second way also shows the linear
relationship between Fahrenheit and Celsius temperatures and is
C = F * 5 ÷ 9 - (160 ÷ 9). Note: [(160 ÷ 9)
comes from taking the equivalent of absolute Zero in Fahrenheit,
-459.67, multiplying it by 5 than dividing it by 9 than adding
the freezing point of water in Kelvin (K), 273.15, which is also
zero degrees Celsius. (160 ÷ 9) = (-459.67 * 5 ÷ 9)
+ 273.15 = 17.777777777…. On a calculator the last digit
would be an 8 because of rounding up.] Any temperature on the Celsius
scale can be changed to Kelvin by adding 273.15 to it: K = C +
273.15 or Kelvin to Celsius is C = K - 273.15. Lets tie in Fahrenheit
and Kelvin to see what type of conversion formula we can come up
with. We will need to convert Kelvin to Celsius then to Fahrenheit:
F = (K - 273.15) * 9 ÷ 5 + 32. Now, lets look at converting
Fahrenheit to Kelvin. The first way is to use C = (F - 32) * 5 ÷ 9
and now we have K = (F - 32) * 5 ÷ 9 + 273.15 and the second
way is to use C = F * 5 ÷ 9 - (160 ÷ 9) and we get
K = F * 5 ÷ 9 - (160 ÷ 9) + 273.15. Nothing to it!!!
Absolute Zero:
Absolute Zero is a theoretical temperature. All gases shrink in
volume proportional to the temperature of that gas. Now, all gases
will liquefy at a specific temperature for that gas. But, by extending
the straight line graph to a temperature at which the volume of
any gas theoretically shrinks to zero is this Absolute Zero temperature.
Absolute zero has been determined, at present, to be measured at
-273.15° C which correlates to -459.67° F. The absolute
scale is the Kelvin and absolute zero is 0° Kelvin (K).
Trigonometric functions:
Sin (angle) = length of side opposite (angle) ÷ length of
hypotenuse.
Cos (angle) = length of side adjacent to (angle) ÷ length
of hypotenuse.
Tan (angle) = length of side opposite (angle) ÷ length of
side adjacent to (angle).
Csc (angle) = length of hypotenuse ÷ length of side opposite
(angle).
Sec (angle) = length of hypotenuse ÷ length of side adjacent
to (angle).
Cot (angle) = length of side adjacent to (angle) ÷ length
of side opposite (angle).
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