fundamental tools chapter 1. fundamental tools expectations after this chapter, students will: ...
TRANSCRIPT
Fundamental Tools
Chapter 1
Fundamental Tools
Expectations
After this chapter, students will: understand the basis of the SI system of units distinguish between units and dimensions be able to perform dimensional analyses distinguish between fundamental and derived units be able to convert a quantity to different units know standard powers-of-ten prefixes be able to solve right triangles
Fundamental Tools
Expectations (continued)
After this chapter, students will: distinguish between vector and scalar quantities be able to resolve vectors into orthogonal components be able to add and subtract vectors know how vectors can be multiplied by scalars know how vectors can be multiplied by vectors know how many significant figures are in a given
number
What is Physics?
The mapping of mathematics onto the material world.
A mathematical description of the interactions of space, time, matter, and energy.
An experimental science: theory is judged by how well it predicts the results of experiments.
Numbers and Units
“I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of Science, whatever the matter may be.”
--- William Thomson (Lord Kelvin) 1824 - 1907
Numbers and Units
Quantity: a characteristic of an object or material that can be expressed quantitatively (in numbers)
examples: height, weight, volume, density
Dimension: the name of the class or category of units that express a physical quantity
examples: length, mass, time, velocity
Numbers and Units
Unit: a reference standard with an agreed-upon definition that allows quantities to be specified by comparison to it.
examples: meter, second, pound
Types of Units
Base unit: the units of the quantities length, mass, and time.
examples: meter, kilogram, second
Derived units: units made by combining other units
examples: meters / second, kilogram·meters / (second)2
Systems of Units
SI (Système International d’Unités)
Base units: length: meter (m)
mass: kilogram (kg)
time: second (s)
CGS (“small metric”)
Base units: length: centimeter (cm)
mass: gram (g)
time: second (s)
Systems of Units
BE (“British engineering”)
Base units: length: foot (ft)
mass: slug (sl)
time: second (s)
In this class, we’ll use SI units almost exclusively.
Significant Figures
The uncertainty in our knowledge of the numerical value of a physical quantity is indicated by the number of significant figures we use to express that value.
To determine how many significant figures are in a number (example: 0.0000149 m)
Write the number in proper scientific notation: 1.49×10-5 m.
Count the digits in this part of the number: 3 digits
Note: for “proper” scientific notation: a × 10n,
1 ≤ a < 10
Unit Conversions
Basic principles: 1 is the multiplicative identity. Multiplying any quantity
by one does not change the value of that quantity. If a fraction’s numerator and denominator are equal, the
fraction is equal to one.
Example: convert 3.45 years to seconds
s 101.09 min 1
s 60
hr 1
min 60
day 1
hr 24
year 1
days 365.24years 45.3 8
fractions equal to one
Dimensional Analysis
A consistency check for mathematical relationships in physics.
A formula or equation that “passes” a dimensional analysis may or may not be correct.
However …
A formula or equation that “fails” a dimensional analysis cannot be correct.
Dimensional Analysis
What is a dimensional analysis? How can I do one?
Substitute the dimensions represented by each variable for that variable in the equation to be analyzed.
Algebraically simplify the equation: exponentiate, multiply, divide, add, subtract, cancel.
In simplest terms, both sides of the equation should have the same dimensions.
Dimensional Analysis
(Simple) example: your geometry is a little fuzzy. But you’re pretty sure that the surface area of a sphere is given by:
Check it:
… and the analysis fails. That formula can’t be correct.
3
3
4rA
... No ??? lengthlength
length3
4length
32
32
Common Powers-of-Ten
power of ten
prefix symbol example
109 giga - G GHz
106 mega - M MW
103 kilo - k km
10-2 centi - c cm
10-3 milli - m mm
10-6 micro - m
10-9 nano - n nm
Right-Triangle Trigonometry
Basic relationships
a
b
c
a
b
222
tan
cos sin
bac
b
a
c
b
c
a
a
aa
Right-Triangle Trigonometry
Basic relationships
a
b
c
a
b
a
b
c
a
c
b
b
bb
tan
cos sin
Vector and Scalar Quantities
Scalar: completely specified by a magnitude (size)
Vector: completely specified by both a magnitude and a direction
Examples:
Distance (scalar): the airport is 15 km away from here.
Displacement (vector): the airport is 15 km southwest from here.
Vector and Scalar Quantities
Scalar: speed, temperature, time, mass, energy, volume, area, length
Vector: velocity, acceleration, momentum, force
Note that the ability to take on + or – values does not make a quantity a vector. Example: Celsius or Fahrenheit temperature.
Vector Properties
Symbol: an arrow (line segment with a point) arrow length shows vector magnitude arrow points in vector direction
Mathematical notation: bold-font letter A arrow on top of letter “hat” on top of letter (usually a unit vector)
A
i
A
Vector Mathematics
Vectors can be: Added Multiplied
by a scalar by another vector (in two different ways)
Subtracted Divided
Vector Addition
Here’s a graphical look at vector addition: we want to add A and B.
+x
+y
A
B
Vector Addition
First, we note that we can translate a vector to any other location without changing it (either magnitude or direction).
+x
+y
A
A
A
A
Vector Addition
So, we translate B so that its “tail” coincides with A’s “point.”
+x
+y
A
B
Vector Addition
Now, we draw a third vector from the beginning point of A (its “tail”) to the ending point of B (its “point”).
That third vector is the sum: A + B.
+x
+y
A
B
A+B
Vector Multiplication
There are three kinds of multiplication that can be done with vectors.
First: multiplication by a scalar.
Magnitude of the product vector: magnitude of the factor vector times the scalar.
Product vector direction: same or opposite the factor vector direction
VbVb
Vector Multiplication
There are three kinds of multiplication that can be done with vectors.
Second: scalar product of two vectors (“dot product”).
The scalar product is zero
if the vectors are perpendicular;
a maximum value when they
are parallel +x
+y
A
B
cosABBA This kind of vector
multiplication is commutative
Vector Multiplication
There are three kinds of multiplication that can be done with vectors.
Third: vector product of two vectors (“cross product”).
Direction: perpendicular to bothA and B, and in accordancewith the right-hand rule
The vector product is zeroif the vectors are parallel;a maximum value when theyare perpendicular
)(magnitude sinABBAC
+x
+y
A
B
C
This kind of vector multiplication is NOT commutative
Vector Subtraction
This time, we wantA – B.
Graphically:
+x
+y
A
B
Vector Subtraction
Our first step is to muliplyB by the scalar -1,producing – B:
+x
+y
A
-B
Vector Subtraction
And now we move – B to the point of A, just aswe did before:
+x
+y
A
-B
Vector Subtraction
And we draw in thesum: A + (-B)=A – B.
+x
+y
A
-BA-B
Vector Addition by Components
Any vector can be expressed as the sum of two vectors, both orthogonal to the coordinate axes.
One is the X component, and one is the Y component.
+x
+y
A
AX
AY
YX AAA
Vector Addition by Components
Simple right-triangle trigonometry allows us to calculate the magnitudes of these components:
sin
cos
AA
AA
Y
X
+x
+y
A
AX
AY
Vector Addition by Components
Example: we want to add vectors A and B.
BAC
+x
+y
A
B
A
B
Vector Addition by Components
First: resolve A and B into components.
(Replace A and B with component vectors AX, AY, BX, and BY, all orthogonal to the coordinate system.)
+x
+y
AB
BY
BX
AY
AX
A
B
Vector Addition by Components
The components of the sum, C, are the sums of the components of A and B.
Since the X components are either parallel or antiparallel, their magnitudes add algebraically.
The same is true of the Y components.
+x
+y
AB
BY
BX
AY
AX
A
B
YYY
XXX
BAC
BAC
Vector Addition by Components
BAY
YYY
BAX
XXX
BAC
BAC
BAC
BAC
sinsin
coscos
+x
+y
BY
BX
AY
AX
(magnitude)
(magnitude)
Vector Addition by Components
BAY
YYY
BAX
XXX
BAC
BAC
BAC
BAC
sinsin
coscos
(magnitude)
(magnitude)
+x
+y
CY = AY + BY
CX = AX + BX
Vector Addition by Components
Pythagoras’ theorem yields the magnitude of C:
The direction of C:
22YX CCC
X
YC C
Carctan
+x
+y
CY =
AY +
BY
CX = AX + BX
C
C
Vector Addition by Components
A couple of things to remember: You are free to define your coordinate system so
that it makes your life easier. These are always correct:
as long as you measure counterclockwise from the +X direction.
sin
cos
AA
AA
Y
X