Download Chapter 2 Basic physical concepts

Chapter 2 Basic physical concepts
Science is concerned with measurement and prediction of the behavior of natural systems.
This chapter introduces a number of basic physical quantities: a set of consistent units (the
MKS (Meter-Kilogram Second, also called SI for Système International d'units) for
measurement of the fundamental quantities length, time and mass; and the notions of
speed, velocity, force, pressure, work and energy, quantities that can be derived from
the fundamental quantities. Another measurable quantity of great importance in the
atmosphere is temperature, which is a measure of the kinetic energy of molecules in a
gas. We introduce the relation between pressure, temperature and density called the
Perfect Gas Law, a simple equation of central importance in atmospheric studies. The
objective is to develop a level of familiarity with the physical concepts underlying these
ideas and to become proficient at carrying out simple calculations.
2.1 Fundamental quantities: length, time and mass
Length and time are familiar physical quantities, and methods for measuring them are
taken for granted. Each is measured by comparison with a standard, which defines, by
international agreement, the units of measurement. Once consensus is reached on the
standard units for length, time and mass, the units for other physical quantities can be
determined in terms of these units, hence the moniker "fundamental quantity".
The length of an object is measured using a meter stick, which may be regarded as the
standard. This was originally an actual physical piece of metal kept in a safe place in Paris.
The standard meter could then be divided into useful segments for measuring small
objects, or replicated and placed end-to-end to measure large distances, for example,
1/100th being a centimeter (cm), or ×1000 a kilometer (km). Today the standard meter is
defined using a laser, but the basic concept hasn't changed.
Measurements of time are analogous: originally the basic time interval was an average
earth day, divided by 24 for hours or by 86400 to get a second. Today the unit of time is
defined much more accurately using a laser, but again the concept hasn't changed. Using
laser light has the enormous advantage that anyone, anywhere, can buy the equipment
needed to define standard lengths or time.
The mass of an object is a measure of the amount of matter in it, i.e. the number of
molecules, atoms, etc. An object has the same mass whether it is on the surface of the
earth, in space, or on the moon. Therefore we say that mass is an intrinsic property. Mass
is easily measured using a balance as illustrated in Figure 2.1 (left panel). If there are
equal masses on both sides of the balance, the beam will be level, whether we make the
measurement on the surface of the earth, on an airplane, or on the moon. The standard of
mass, 1 kg, is defined as the mass of 1 liter (10-3 m3) of water at a temperature of 0° C.
An alternate definition may be used, that 6.02×1023 atoms of carbon have a mass of
exactly 12 gm (0.012 kg). The number 6.02×1023 is called Avogadro's number; it defines
a mole of atoms or molecules. A mole of molecules has mass equal to the molecular
weight in grams. These apparently obscure relationships give a consistent standard scale
for measuring mass on macroscopic and molecular scales.
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1 kg
2 kg
1 kg
MASS is a
PROPERTY OF THE OBJECT
(#molecules in it)
ass is measured using a balance
to compare masses of two objects
Weight = mg
Mass x force of gravity (mg)
Depends on gravity
Not a property of the object
Measured (force) by deflection of a spring
Figure 2.1 Measuring mass using a balance, and weight using a spring
2.2 Derived quantities: velocity, acceleration, force, pressure, energy
Important quantities for studies of the atmosphere, that can be expressed in terms of the
fundamental quantities length, time, and mass, include speed, velocity, acceleration, force,
pressure, and energy (work). Table 2.1 summarizes the definitions of several important
derived quantities, most of which are familiar. Speed is the distance traveled by an object
per unit time: speed = distance ÷ time. Speed can be measured for example by a radar gun
that determines the distance to the car when the trooper presses the button, then repeats
the measurement at a fixed later time. A computer in the unit computes how far the car
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has moved and divides by the time interval to compute speed, which the trooper compares
to a standard (e.g. 55 miles hr-1 = 24.7 m s-1 in MKS units; see the Appendix for a
summary of units of measure).
Velocity is the speed of an object in a specified direction. Like many quantities of physics,
velocity is a vector, meaning that it has a direction and a magnitude, both of which must
be known to determine velocity. In Science A-30 we will not deal explicitly with vectors,
to keep things simple. Usually this will be satisfactory, but sometimes our discussions
require thinking about direction as well as magnitude. For example, suppose we toss a
basketball at a brick wall. Assume it bounces off the wall without losing any energy (see
below), returning at the same speed as when we threw it. The direction has changed,
however, and the ball exerted a force on the wall when this happened. This is how
molecules exert force on a surface [the golf ball atmosphere demonstration in class shows
this phenomenon].
Acceleration is the rate of change of velocity, i.e. how much an object increases or
decreases its speed, or changes direction, in a unit time. If we drop a ball it accelerates
towards the center of the earth due to the intrinsic attraction of masses for each other.
The acceleration of an object due to gravity has special importance for
Table 2.1 Physical properties defined in terms of the fundamental properties.
Quantity, definition
distance
velocity=
time
change of velocity
acceleration=
time
force=mass×acceleration
weight= mass×acceleration of gravity
= force exerted by gravity on an
object
work = force×distance
kinetic energy = energy of motion
Temperature (related via Boltzmann's constant, k,
to the mean kinetic energy
of a gas molecule)
pressure = force per unit area
Formula
Dimensions (units)
L/t
m s-1
v/t
m s-2
F=m a
kg ms-2 ≡ Newtons (N)
F=mg
(same as force; g=9.8 ms-2)
E=F L
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E= mv2
2
kg m2 s-2 ≡ Joules (J)
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mv2 = kT
2
P = F/A
(same as work)
[non-dimensional, units=degrees
Kelvin; k=1.38 10-23 J/K]
kg m-1 s-2
atmospheric processes, and we reserve a special symbol, g, for this acceleration. The
value of g is 9.8 m s-2, i.e. if there is no friction to slow it down, an object that we drop
will be traveling 9.8 m s-1 after 1 second, 19.6 m s -1 after 2 seconds, etc. (Our ball could
get a speeding ticket in only 3 seconds.)
Force was first defined by Sir Isaac Newton (1642-1727) as the product of
mass×acceleration,
F = m×a
(Eq. 2.1)
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The basketball in the example above experiences an acceleration when its direction of
motion changes. Likewise, a ball on a string that is whirled in a circle is constantly
changing its direction and therefore being accelerated, which can be experienced directly
as a force on the string. The unit of force is kg m s-2 : 1 kg m s-2≡ 1 Newton (N). Thus an
object with mass 1 kg weighs 9.8 N.
What is the difference between the mass and the weight of an object?
The weight of an object is the force on it due to gravitational acceleration (g, 9.8 m s-2),
weight = mg, a result of the intrinsic attractive force of two masses (the earth and the
object) for one another. In everyday life we more often measure weight than mass; usually
using a spring or similar device (see Figure 2.1). How far a spring extends when an object
is weighed depends on the acceleration of gravity acting on the object. If we take the
spring to the moon, where the gravitational acceleration is much smaller than on the earth,
the object will weigh less (the spring will not extend as far), even though its mass has not
changed. If we use a balance to compare the mass of an object to standard masses, the
measurement does not depend on the value of g, although the measurement might be hard
to make if g were very small, as in outer space. This example illustrates why weight is not
an intrinsic property of the object, but mass is intrinsic.
2.3 Pressure
Pressure and the overlying weight of fluid
When we speak of pressure (P) we mean the force (F), or weight, of an object distributed
over a surface area (A) (P=F/A); pressure has units N/m2. The mass of air molecules
above each square meter is about 10,000 kg m-2, or 10 metric tons (1 metric ton =1000
kg). The weight of the weight of these molecules exerts a pressure on the surface of the
earth. This pressure is about 100,000 N m-2, i.e. g×104 kg m-2. Hence the weight of
atmosphere on the surface of your desk, with an area of about 1 m2, is about the same as
that of a city bus.
The pressure of the atmosphere is often measured using a column of mercury (a
barometer, demonstrated in class), and atmospheric pressure is often reported using the
unit atmosphere, defined as the weight (per square meter) of a column of mercury exactly
76 cm deep (1 atmosphere (atm) ≡ 1.013⋅105 Newton m-2).
A container of water 10 m high×1 m wide×1 m deep holds 10 cubic meters of water, or
10,000 kg of mass. Thus the weight of water on the bottom of the container is equivalent
to that of the whole column of the atmosphere, and the water adds about 1 atm of
pressure to that of the overlying atmosphere. Thus an individual swimming 10 m
underwater feels a pressure of 2 atm (1 atm from the weight of the atmosphere and 1 atm
from the weight of the water) on his/her body. Scuba divers breathe compressed air with a
pressure matched (by the regulator) to the total pressure of overlying water + atmosphere,
to avoid having the pressure of the water collapse their lungs.
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The crucial difference between pressure and force is demonstrated by considering a skier
on deep powder. The skier's mass (m), and therefore weight (m×g) changes very slightly
with or without skis, so the force that the skier exerts on the snow is nearly unchanged by
putting on skis. Nevertheless, without skis, the skier sinks into the snow, but with skis the
skier rides on top of the snow. The reason is that the area on which the weight is exerted,
and thus the pressure, does change. For example, take a skier with mass 60 kg (weight
588 N, or 132 lbs) and size 8 boots, which have a surface area of about 0.03 m2, which
gives a pressure of 196,000 N/m2 (pressure = force/area = 5880/0.03). If the skier puts on
skis with area 0.3 m2, then the pressure exerted on the snow is 10 times less (the same
force spread over 10 times the area). Anyone who has lost a ski in deep powder knows
that this makes a big difference in terms of the ability of the snow to support the skier.
Pressure and molecular motions
Why doesn't a desk collapse under the weight of 10 tonnes of atmosphere? To answer this
we have to understand how a gas (the atmosphere) exerts pressure on a surface. Suppose
we have air molecules in a container. The molecules in the gas are moving all the time. We
already pointed out that when the molecules hit a solid surface, they bounce off. This
change in direction of motion is a change in the velocity, an acceleration, and hence a
force is exerted on the surface. The force depends on the mass and velocity of each
molecule, and on the number of molecules.
Experiments carried out in the 18th and 19th centuries produced two important empirical
laws called Boyle's Law and Charles' Law. They showed that there is a very simple
formula relating the temperature, pressure, and number of molecules in a volume. These
laws are combined into one very important relationship, the Perfect Gas Law,
PV = NkT
(Eq. 2.2a)
where P is pressure in volume V, N the number of molecules in the volume, and T the
absolute temperature (Kelvin). k is Boltzmann's constant (1.38×10-23 J K-1).
Since we don't have confined volumes in the atmosphere, we usually use the Perfect Gas
Law in a different form,
P = nkT
(Eq. 2.2b)
where n (= N/V, the number density) is the number of molecules per unit volume.
Meteorologists prefer to use a macroscopic quantity, the mass density ρ (ρ ≡ mn, m is the
mass per molecule) for which the Perfect Gas Law is given as P=ρR'T, where R'=k/m.
The mass density ρ of air is the number of kilograms contained in a volume of 1 cubic
meter.
The pressure forces described by the Perfect Gas Law are exerted in all directions in a
fluid such as the atmosphere, since the molecules are moving randomly in all directions.
Your desk mentioned is immersed in the fluid (air), so the pressure of the atmosphere
below and inside the desk balances the pressure of the overlying atmosphere, preventing
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the desk from collapsing. (The collapsing-drum demo illustrates this point. Atmospheric
pressure easily crushes a strong steel container when the air is removed from the inside.)
2.4 Energy and Temperature
Newton also defined the work done on an object, or energy given to an object when a
force is applied to it. Work is the product of the applied force times the distance over
which the object moves with the force applied. A very important example is the energy
needed to lift an object against gravity. To raise an object of mass m to height h, we must
supply energy mgh. An object falling distance h acquires kinetic energy (the energy of it's
motion) equal to mgh. The object is raised, or falls, under the force of gravity.
Ludwig Boltzmann (1844-1906) derived the relationship between the kinetic energy of
molecules and the temperature (absolute) that appears in the Perfect Gas Law: a molecule
in a gas has on average kinetic energy
E = 3/2 k T
(Eq. 2.3)
This critical discovery makes the connection between the empirical laws of Boyle and
Charles, the concept of pressure in terms of the weight of fluid, and the notion that
pressure is exerted on an object by the force of molecules of a gas (or liquid) bouncing off
the solid body. The constant k in this equation is called Boltzmann's constant and T is the
absolute temperature. Note that, in this equation, when the absolute temperature T is 0,
molecules have zero kinetic energy, i.e. they stop moving.
Temperature is measured using a confusing variety of scales. The only scale for the
Perfect Gas Law is the Kelvin scale, where 0 is absolute zero, corresponding to no
molecular motion. For everyday life the world (except the US) uses the Celcius scale
defined by the ice point (0 C) and the boiling point of water under 1 atm pressure (100°
C). The value of Boltzmann's constant, k = 1.38×10-23 Joules/Kelvin, is chosen so that the
temperature on the absolute Kelvin scale is measured in steps (degrees) the same size as
degrees in the traditional Celcius (or centigrade) scale, which make 0 Kelvin ≡ -273.15°
Celcius (0°C
273.15 K; T(K)=T(C)+273.15). The Fahrenheit scale, used only in the
US, is related to the Celcius scale by T(F)=1.8 T(C)+32. (Students must be careful not to
confuse these temperature scales, and be able to convert one to the other as needed).
Ù
2.5 Pressure differences and fluid motions
Let us consider a thought experiment that illustrates how differences in pressure in a fluid
causes fluid to move, in a direction that tends to eliminate the pressure difference. Figure
2.2 depicts two examples of pairs of tanks filled with water, and connected by a pipe. The
atmosphere exerts a pressure of 1 atm on the surface of the water in each tank of each
pair, and the weight of the water provides additional pressure. The pressure on the bottom
of each tank is the sum of atmospheric and water pressure. The picture on the left shows
a case where the water depth is the same in both tanks. We know from everyday
experience that there will not be any flow of water between the tanks. This experiment
shows why we defined pressure as force per unit area: it doesn't matter how wide the
reservoir of the tank is, but only how much fluid lies above a unit area of the bottom of the
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tank. Since the height of the fluid in the two tanks is the same, pressures at the bottom of
the tanks are equal, and the pressure forces are balanced and cancel each other. The fact
that there is zero pressure difference between the tanks implies no net force in the fluid to
make it move, and no tendency for fluid to flow from one tank to the other.
Figure 2.2. Tanks with connected reservoirs filled with water and air (upper right portion of right
panel). (left) Water levels are equal, and therefore the pressures are equal, in both sides. The fact that the
left reservoir has larger area and volume is irrelevant. (right) Pressure is higher in the left reservoir than
in the right, and water will flow from left to right until the levels of water are equalized.
In the experiment on the right, the pressure is higher at the bottom of the tank with the
higher column of water; the pressure difference exerts a force making water flow through
the connecting pipe, and flow will continue until water levels are the same on both sides.
Thus from everyday we experience, we know that the flow of fluid is caused by pressure
gradients, i.e. differences in pressure, not on the total pressure. This principle holds in the
simple tanks considered in Figure 2.2, and in the atmosphere and the oceans.
2.6 Summary of Main Points of the Chapter
• Atmospheric pressure is the weight of the atmosphere above a given altitude.
• At the surface atmospheric pressure is equal to 100 000 N m-2, equivalent to the
pressure of a column of 10 m of water or 0.76 m of mercury, 1 bar or 1000 mbar
(mb).
• Fluid pressure (in the atmosphere) acts in all directions
• Fluids move from high pressure to low pressure. Winds and ocean currents result from
pressure differences in the atmosphere or ocean.
• The Perfect Gas Law gives the relationship between the pressure, P,, temperature, T,
number of molecules, N, and volume, V, of a gas: PV = NkT (k=1.38 10-23
J/(molecule⋅K); it is also expressed as or P=nkT, where n is number density (N/V), or
P = ρR'T, where ρ is mass density (kg m-3) and R'=287.5 J/(kg ⋅K) for dry air.
Table 2.2 Summary of demonstrations used in lectures on physical principles
‰ One kilogram mass on a balance. We discussed how a balance works and why equal
masses remain equal, independent of the force of gravity.
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‰ One kilogram mass on a spring. We discussed how the spring works as a device to
measure weight or force, and why this measurement is not independent of gravity. The
1kg mass weighed 9.8 Newtons.
‰ Effect of circular motion (acceleration due to changing direction of the velocity)on the
force exerted by a ball on a string. When the ball is swung around in a circular path, it
can ligt a weight that hs greater mass than the ball. The mass of the ball has not
changed, but its body force has.
‰ Pressure on a plate exerted by bounding molecules (the "ping pong ball atmosphere"):
a net force (pressure) is exerted by ping pong balls as they bounce off a suspended
plate, even though each makes only momentary contact.
‰ Collapsing 55-gallon drum: The enormous force of atmospheric pressure on the can
collapses it even though it is very strong--the surface area is several square meters and
it cannot withstand the force of several 10's of tons when the air is removed from the
inside using a vacuum pump.
Table 2.3: Appendix: units, conversion factors, physical constants
BASIC UNITS
Length
Mass
Time
Temperature
MKS system
meter (m)
kilogram (kg)
second (sec)
degree Kelvin ( K)
cgs system
centimeter (cm)
gram (g)
second (sec)
degree Kelvin ( K)
meter sec−1
meter sec−2
Newton (1 N= 1 kg m sec−2 )
Newton meter−2
Joule (= 1 kg m2 sec−2 )
centimeter sec −1
centimeter sec−2
dyne (= 1 g cm sec−2 )
dyne centimeter−2
erg (= 1 g cm2 sec−2 )
kg m−3
m−3
gm cm−3
cm−3
°K =.°C + 273.15
1 bar = 1000 millibars (mb) =
105 N m−2
1 mb = 1000 dyne cm−2
1 Angstrom (Å)=10 −8 cm
1 erg =10-7 Joules
°C =(°F - 32)(5/9)
1 atm = 1.013 bars
Acceleration of gravity g=
9.8 m s-2
Boltzmann’s const k= 1.38 10-23
Joule/K
Atmospheric gas const R’= 287.5
Joule/(kg K)
Earth’s angular velocity
(Ω)=7.27 10-5 s-1
(K =degrees Kelvin)
FORMULAS
volume of a sphere = 4/3π r3
area of a sphere = 4π r2
Composite Units
Velocity
Acceleration
Force (F=ma)
Pressure
Energy (e.g., ½ mv2) and Work
(Force × distance)
Mass density (ρ)
Number density (n)
CONVERSIONS
1 m = 100 cm
1 kg = 1000 g
1 mb = 100 N m −2
1 joule = 1 N m
1 atm = 76 cm Hg
CONSTANTS
Avogadro’s number = 6.02 1023
1 erg = 1 dyne cm
1 Å=10-4 micron (µ)
1m3 =103 liters = 106 cm3
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CHAPTER 2 BASIC PHYSICAL CONCEPTS
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2.1 Fundamental quantities: length, time and mass
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2.2 Derived quantities: velocity, acceleration, force, pressure, energy
Table 2.1 Physical properties defined in terms of the fundamental properties.
What is the difference between the mass and the weight of an object?
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2.3 Pressure
Pressure and the overlying weight of fluid
Pressure and molecular motions
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2.4 Energy and Temperature
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2.5 Pressure differences and fluid motions
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2.6 Summary of Main Points of the Chapter
Table 2.2 Summary of demonstrations used in lectures on physical principles
Table 2.3: Appendix: units, conversion factors, physical constants
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