Download PH 201-4A spring 2007 PH 201 4A spring 2007

PH 201-4A
201 4A spring 2007
Introduction and Mathematical
Concepts
Lectures 1,2
Chapter 1 and Appendix A&B
(Cutnell & Johnon, Physics 7th edition)
1
The Nature of Physics
¾ The science of physics has developed out of the efforts of men and
women to explain our physical environment. These efforts have
been so successful that the laws of physics now encompass a
remarkable variety of phenomena, including planetary orbits, radio
and TV waves, magnetism and lasers, to name just a few.
¾ The exciting feature of physics is its capacity for predicting how
nature will behave in one situation on the basis of experimental data
obtained in another situation. Such predictions place physics at the
heart of modern technology and, therefore, can have a tremendous
impact on our lives (e.g. the medical profession uses X-rays,
ultrasonic, and magnetic resonance methods for the obtaining
images of the interior of the human body, and physics lies at the
core of all these).
¾ Because physics is so fundamental, it is a required course for
students in a wide range of major areas. During studies of physics
you will learn how to see the world through the “eyes” of physics,
reason as a physicist
h i i d
does, and
d will
ill llearn h
how to apply
l physics
h i
principles to a wide range of problems.
2
PHYSICS
Physics is the study of the physical universe.
universe It is founded on experimentation: we
ground the science directly to nature through observations that entail the measurement
of physical quantities. Physics focuses on issues that are truly essential to the way
nature works.
Data is the objective record of our observations.
A law is a statement of a recurring pattern of events.
Theory is the explanation of phenomena.
An agreed-upon measure of physical quantity is a unit.
An unchanging embodiment of a unit that serves as a primary reference is a standard.
Today, the scientific community follows the Systeme International (SI), a program in
which the majority of measurements are based on the following units:
•Length = meter (m)
•Mass = kilogram (kg)
•Time = second (s)
•Electric current = ampere (A)
•Temperature = kelvin (K)
3
Powers of Ten and Scientific Notation
In science, very large and very small decimal numbers are conveniently expressed in
terms of powers of ten.
100 = 1
101 = 10
10-1 = 1/10 = 0.1
102 = 100
10-2 = 1/102 = 0.01
103 = 1000
10-3 = 1/103 = 0.001
Example:
Earth’s
Earth
s radius = 6,380,000 m = 6.38 x 106 m
The factor of ten raised to the six power is ten multiplied by itself six times. The
decimal place should be moved six places to the right to obtain the number without
the power of ten.
Numbers expressed with the aid of powers of ten are said to be in scientific notation.
Calculations that involve the multiplication and division of powers of ten are carried
out acco
according
d g to the
t e following
o o
g general
ge e a rules:
u es
1/10n = 10-n
10n x 10m = 10n+m
10n/10m = 10n-m
Where n and m are any positive or negative numbers.
4
Length
Metric System (decimal approach)
All quantities are divided into 10, 100, 1000…
Multiples and submultiples of the meter (m)
Kilometer
1 km = 103 m
Meter
1m
Centimeter
1 cm = 10-2 m
Millimeter
1 mm = 10-3 m
Micron
1 µm = 10-66 m
Nanometer
1 nm = 10-9 m
Angstrom
1 Å = 10-10 m
Fermi
1 fm = 10-15 m
Multiples and submultiples of the foot
Mile
1 mile = 5280 feet = 1609.38 m
Yard
1 yard = 3 ft = 0.9144 m
Foot
1 ft
f = 0.3048 m
Inch
1 in. = 2.540 cm
The meter is the length traveled by a light wave in a vacuum in a time interval of
1/299 92 4 8 s
1/299,792,458
5
Some distances and sizes
6
Time
Conceptually,
p
y, time is the measure of the rate at which change
g occurs.
In 1967, the SI unit of time, the second (s), was defined as the interval required for
9,192,631,770 vibrations of the cesium-133 atom measured via an atomic beam clock.
3000 years ago the Egyptians divided day and night into 12 equal hours.
Babylonian arithmetic used 60 at its number base – began tradition of subdividing things
into 60 equal parts.
14th century mechanical clock – hour was divided into 60 minutes, 60 sec. = 1 minute
7
Multiples and Submultiples of the Second
8
The meter, the second, and the kilogram are the fundamental units or base units for
length, time and mass.
Any other physical quantity can be measured by introducing a derived unit
constructed from some combination of the base units.
9
Significant Figures
Measurement is very different from counting, although both associate numbers with
notions. We can count the number of beans in a jar and know it exactly. But we
cannot measure the height of a jar exactly.
There is no such thing as an exact measurement.
Practically, measurements are made to some desired precision that suits the
experimenter’s
i
t ’ purposes ((often
ft determined
d t
i d by
b the
th limitations
li it ti
off the
th available
il bl
instruments).
Example:
p
A rod whose length is being measured with a ruler.
The object is between 4.1 and 4.2 cm long. Because there are no finer divisions than 1 mm on
the scale, fractions of a millimeter will have to be estimated. We conclude that the rod is 4.15 cm
g The 4 is certain,, as is the 1,, but the 5 might
g be in error as much as ±1 and trying
y g to arrive
long.
at any more figures would be meaningless.
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11
12
13
14
15
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The role of units in problem solving
Since any quantity, such as length, can be measured in several different units, it is
important to know how to convert from one unit to another.
Example: Someone is 2.00 yard tall. Using the fact that 1 inch is exactly 2.54
cm, how tall is the person in centimeters?
Reasoning: When converting between units we write down the units explicitly in the
calculations and treat them like any algebraic quantity. In particular we will take
advantage of the following algebraic fact: Multiplying or dividing an equation by a
factor of 1 does not alter the equation.
Solution:
-Since 3.00 feet = 1.00 yard, it follows that (3.00 feet)/(1.00 yard) = 1
-Since 12.0 inches = 1.00 foot, it follows that (12.0 inches)/(1.00 foot) = 1
-Since 2.54 cm = 1.00 inch, it follows that (2.54 cm)/(1.00 inch) = 1
-Using these factor to multiply the equation:
“Height
Height = h = 2.00
2 00 yards
yards” we find that
h = (2.00 yd)(1)(1)(1) = (2.00 yd) * (3.00 ft)/(1.00 yd) * (12.0 in)/(1.00 ft) * (2.54 cm)(1 inch) = 182.88 cm
After rounding to three significant figures: h = 183 cm
17
Units as a problem solving aid
• Guides the use of conversion factors
•Provides an internal check to eliminate certain kinds of errors if the units are carried
along during each step of the calculation and treated like any algebraic factor.
Example:
p
The tank of a car contains 2.0 gallons of gas to start with and that gas is added at a rate of 7.0
gallons/minute. The total amount of gas in the tank 96 seconds later can be obtained by adding the
amount put into the tank to the amount present initially.
Total amount =
Initial amount of gas + Gas added =
2.0 gallons + (7.0 gallons/min)(96 sec) =
2.0 gallons + 672 gallons * sec/min ≠ 2.0 + 672 = 674
because the units for the two added terms are not the same
Solution:
Total amount =
2 0 gallons + 7.0
2.0
7 0 gallons/min * 1 min/60 sec * 96 sec =
2.0 gallons + 11 gallons = 13 gallons
•Only quantities that have exactly the same units can be added or subtracted.
The procedure of carrying along the units serves as automatic reminder to convert all data in
•The
calculations into a consistent set of units.
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Dimensional Analysis
•In physics, the term dimensional analysis is used to refer to the physical nature of a
quantity and the type off unit used to specify
f that quantity.
•Dimensional analysis is used to check mathematical relations for the consistency of
their dimensions.
Example:
E
l A car starts ffrom rest and
d accelerates
l
to a speed
d “v”
“ ” in
i a time
i
“t”.
“ ” We
W wish
i h to
calculate distance ‘x’ traveled by the car, but we are not sure whether the correct relation
is x = ½vt2 or x = ½vt.
We can determine the correct relation by checking the quantities on both sides of the
•We
equals sign to see if they have the same dimensions. If the dimensions are not the same,
the relation is incorrect.
x = ½vt2 cannot be correct
x = ½vt
[L] = [L/T] * [T]2 = [L][T]
[L] = [L/T] * [T] = [L]
19
Trigonometry
The sine,, cosine,, and tangent
g
of an angle are numbers
without units.
The Pythagorean Theorem states that the square of the hypotenuse of a right
triangle is equal to the sum of the squares of the other two sides: h2 = ho2 + ha2
Inverse trigonometric functions:
θ = sin-1(ho/h)
θ = cos-1(ha/h)
θ = tan
tan-1(ho/ha)
1(ho/ha)
- θ equals the angle whose sine is ho/h
- the use of “-1” does not mean take the reciprocal
- another way to express the inverse trig. functions is to use arc sin,
arc cos, and arc tan
20
Problem:
The drawing shows sodium and chloric ions positioned at the corners of a cube that is
part of the crystal structure of sodium chloride (common table salt). The edge of the
cube is 0.281 nm (1 nm = 10-9 m) in length.
(a) Find the distance in nm between the sodium ion located on one corner of the cube
and the chlorine ion located on the diagonal at the opposite corner.
(b) What is the value of the angle θ in the drawing?
21
The nature of physical quantities: Scalars and Vectors
Scalars
A scalar quantity is one that can be described by a single number (including any
units) giving its size or magnitude.
T
Temperature
t
20°C
Mass
84 kg
Volume
50 m3
Time
11 seconds
Vectors
A quantity that deals inherently with both magnitude and direction is called a vector
quantity.
22
Displacement vector: a change of the position. The vector is graphically
represented by an arrow or directed line segment.
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The displacement vector has a length and a direction.
Any two line segments of identical length and direction represent equal vectors
regardless of whether the endpoints of the line segments are the same.
Vector A – quantity that has a magnitude and direction
Scalar – any quantity that has magnitude but no direction
24
Vector addition and subtraction
Two displacements carried out in succession result in a net displacement which can be
regarded as the sum of the individual displacements.
25
Vector Addition
Tail-to-Head Method
The directed line segment
connecting the tail of A to the
h d off B is
head
i the
h resultant
l
Parallelogram Method
We place the tail of B on the tail of A and
draw a parallelogram with A and B as two
sides. The diagonal is then the resultant.
1. Order of addition makes no difference
2. The magnitude of the resultant is usually less
than the sum of the magnitudes.
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3. The negative of a given vector A is a vector of the same magnitude, but opposite
direction.
Th subtraction
The
bt ti
off two
t
vectors
t
A and
d B is
i defined
d fi d as the
th sum off A and
d -B
B
4. A vector can be multiplied by any positive or negative number
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Vector and Scalar Components of a Vector
We describe the
position by means of
the displacement
vector from the origin
to the point.
X and Y components
of the position vector
For an arbitrary vector A the definition of the
components is analogous to the definition of the
components of the position vector
The addition or subtraction of two vectors can be performed by adding or subtracting
their components.
components
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The Component Method of Vector Addition
1. For each vector
determine the X
and Y components
2. Find algebraic
sum of the X
components
Find algebraic
sum of Y
components
3. Use the X and Y components of vector C and the Pythagorean theorem to determine
the magnitude of vector C
4. Use either the inverse (sine, cosine or tangent) function to find the angle θ that
specifies the direction θ of the vector C
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Ch. 1
Problem 28
A basketball player runs a pattern consisting of three segments. The corresponding
three displacement vectors A, B and C have equal magnitudes of 7.0 m. Displacement
vector A is directed forward and parallel to one side of the court, vector B is directed
forward at a 45 degree angle with respect to the side of the court, and vector C is
directed forward and parallel to the side of the court. With a scale drawing, use the
graphical technique to find the magnitude and direction of the displacement vector for a
straight line dash between the starting and finishing points.
Reasoning:
The displacement vector R for a straight
straight-line
line dash between
the starting and finishing points can be found by drawing the
vectors A, B and C to scale in both magnitude and orientation
in a tail-to-head fashion and connecting the tail of A to the
head of C. The magnitude of the resultant can be found by
measuring its length and making use of the scale factor.
Similarly, the direction of the resultant is found by measuring
the angle θ it makes with the side of the court.
Solution:
By construction and measurement the magnitude of the
resultant vector R is 20 m. The angle θ made with side of the
court is 15 degrees.
30
Example: Addition of vectors by means of components
A pilot flies her route in two straight line segments. The displacement vector A for the
first segment
g
has a magnitude
g
of 243 km and direction of 50 degrees
g
north of east. The
displacement vector B for the second segment has a magnitude of 57.0 km and direction
of 20 degrees south of east. The resultant displacement vector is R = A + B. What is the
magnitude and direction of vector R? Use the component method and specify the
direction relative to due east.
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Components of a Vector
The magnitude of the force vector F is 280 N. The X component of this vector is
directed along the +X axis and has a magnitude of 150 N.
N The Y component points
along the +Y axis.
a) Find the direction of vector F relative to the +X axis
b) Find the component of vector F along the +Y axis
32
The Components of a Vector:
Ch. 1 Problem 37
Resolving a vector into its components
The vector
Th
t A iin th
the d
drawing
i
has
h a magnitude
it d off 750 units.
it Determine
D t
i the
th magnitude
it d and
d
direction of the X and Y components of the vector A relative to
a) The black axes, and
b) The colored axes
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