Transcript Slide 1

9/5 Objectives (A day)
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Introduction of AP physics
Lab safety
Sign in lab safety attendance sheet
Classical Mechanics
Coordinate Systems
Units of Measurement
Changing Units
Dimensional Analysis
Lab safety General guidelines
1. Conduct yourself in a responsible
manner.
2. Perform only those experiments and
activities for which you have received
instruction and permission.
3. Be alert, notify the instructor immediately
of any unsafe conditions you observe.
4. Work area must be kept clean.
5. Dress properly during a laboratory activity.
Long hair should be tied back, jackets, ties,
and other loose garments and jewelry should
be removed.
6. When removing an electrical plug from its
socket, grasp the plug, not the electrical cord.
Hand must be completely dry before touching
an electrical switch, plug, or outlet.
7. Report damaged electrical equipment
immediately. Look for things such as frayed
cords, exposed wires, and loose connections.
Do not use damaged electrical equipment.
Please sign in lab safety attendance sheet
Classical Mechanics
• Mechanics is a study of motion and its causes.
• We shall concern ourselves with the motion of a
particle. This motion is described by giving its position
as a function of time.
Specific position & time → event
Position (time) → velocity (time) → acceleration
• Ideal particle
– Classical physics concept
– Point like object / no size
– Has mass
• Measurements of position, time and mass completely
describe this ideal classical particle.
• We can ignore the charge, spin of elementary particles.
Position
• If a particle moves along a
 straight line → 1-coordinate
 curve/surface → 2-coordinate
 Volume → 3-coordinate
• General description requires a coordinate system with an
origin.
– Fixed reference point, origin
– A set of axes or directions
– Instruction on labeling a point relative to origin, the
directions of axes and the unit of axes.
– The unit vector
Rectangular coordinates - Cartesian
Simplest system, easiest to visualize.
To describe point P, we use three coordinates: (x, y, z)
Spherical coordinate
• Nice system for motion
on a spherical surface
• need 3 numbers to
completely specify
location: (r, Φ, θ)
− r: distance between
point to origin
− Φ: angle between
line OP and z:
latitude = π/2 - Φ.
− θ: angle in xy plane
with x – longitude.
Time
• Time is absolute. The rate at which time
elapse is independent of position and
velocity.
Unit of measurement
Quantity/dimension
being measured
Length/[L]
SI unit (symbol)
meter (m)
British (derived)
unit
Foot (ft)
Time/[T]
second (s)
Second (s)
Mass/[M]
kilogram (kg)
Slug
Ele. Current/[I]
ampere (A)
Temperature[Θ]
Kelvin (K)
Amount of substance[N]
Mole (mol)
Luminous intensity[J]
Candela (cd)
International system of units (SI) consists of 7 base units. All
other units can be expressed by combinations of these base
units. The combined base units is called derived units
Physical Dimensions
• The dimension of a physical quantity specifies
what sort of quantity it is—space, time, energy,
etc.
• We find that the dimensions of all physical
quantities can be expressed as combinations of
a few fundamental dimensions: length [L],
mass [M], time [T].
• For example,
– Energy: E = ML2/T2
– Speed: V = L/T
Derived units
• Like derived dimensions, when we
combine basic unit to describe a quantity,
we call the combined unit a derived unit.
• Example:
– Volume = L3 (m3)
– Velocity = length / time = LT-1 (m/s)
– Density = mass / volume = ML3 (kg/m3)
SI prefixes
• SI prefixes are prefixes (such as k, m, c,
G) combined with SI base units to form
new units that are larger or smaller than the
base units by a multiple or sub-multiple of
10.
• Example: km – where k is prefix, m is
base unit for length.
• 1 km = 103 m = 1000 m, where 103 is in
scientific notation using powers of 10
SI uses prefixes for extremes
prefixes for power of ten
Prefix
Symbol
Notation
tera
giga
mega
kilo
T
G
M
k
1012
109
106
103
deci
centi
d
c
10-1
10-2
milli
micro
nano
pico
m
μ
n
p
10-3
10-6
10-9
10-12
Example: Convert the following
5 Tg
5 x 109
___________
kg
2 μm
2 x 10-6
___________
m
6 cg
___________
kg
6 x 10-5
7 nm
7 x 10-9
___________
m
4 Gg
___________
4 x 106
kg
Unit conversions
Note: the units are a part of the measurement as
important as the number. They must always be kept
together.
Suppose we wish to convert 2 miles into meters.
(1 miles = 1760 yards, 1 yd = 0.9144 m)
1760 yd
0.9144 m
2 miles x
1 mile x
1
yd
= 3218 m
example
• Convert 80 km/hr to m/s.
• Given: 1 km = 1000 m; 1 hr = 3600 s
km
80
x 1000 m
hr
1 km
x
1 hr
= 22
3600 s
m
s
Units obey same rules as algebraic variables and
numbers!!
Dimensional analysis
• We can check for error in an equation or expression by
checking the dimensions. Quantities on the opposite
sides of an equal sign must have the same dimensions.
Quantities of different dimensions can be multiplied but not
added together.
• For example, a proposed equation of motion, relating
distance traveled (x) to the acceleration (a) and elapsed
time (t).
1
x
2
at 2
Dimensionally, this looks like
L
L = 2 T2 = L
T
At least, the equation is dimensionally correct; it may still
be wrong on other grounds, of course.
Another example
use dimensional analysis to check if the equation is correct.
d=v/t
L = (L ∕ T ) ∕ T
[L] = L ∕ T2
Significant Figures (Digits)
• Instruments cannot perform measurements to arbitrary
precision. A meter stick commonly has markings 1 millimeter
(mm) apart, so distances shorter than that cannot be
measured accurately with a meter stick.
• We report only significant digits—those whose values we feel
sure are accurately measured. There are two basic rules:
– (i) the last significant digit is the first uncertain digit
– (ii) when multiply/divide numbers, the result has no more
significant digits than the least precise of the original
numbers.
The tests and exercises in the textbook assume there are 3
significant digits.
Scientific Notation and Significant Digits
• Scientific notation is simply a way of writing very large or
very small numbers in a compact way.
299792485 2.998 108
0.0000000010
878  1.088 109
• The uncertainty can be shown in scientific notation
simply by the number of digits displayed in the mantissa
1.5  10
2 digits, the 5 is uncertain.
1.50  103
3 digits, the 0 is uncertain.
3
Percent error
• Measurements made during laboratory work
yield an experimental value
• Accepted value is the measurements
determined by scientists and published in the
reference table.
• The difference between and experimental value
and the published accepted value is called the
absolute error.
• The percent error of a measurement can be
calculated by
(absolute error)
experimental value – accepted value
X 100%
Percent error =
accepted value
Lab period
• Lab report format
Class work
Homework
1. Read and sign the Lab Requirement Letter –
be sure to include both your signature and
your parent or guardian’s signature.
2. Read and sign the Student Safety Agreement
– both your signature and your guardian’s
signature.
3. Reading assignment: 1.1 – 1.6,
p. 29: #1.1, 1.3, 1.9
9/6 do now
•
The micrometer (1 μm) is often called the micron. How
many microns make up 1.0 km?
•
•
Homework questions?
Quiz tomorrow – on homework assignments
Objectives (B day)
• Sign up on mastering physics – do
assignments
• Math review – class work
Register Mastering Physics
(See instructions at mastering physics sign up info)
• Go to http://www.masteringphysics.com
• Register with the access code in the front of the
access kit in your new text, or pay with a credit
card if you bought a used book.
• WRITE DOWN YOUR NAME AND PASSWORD
• Log on to masteringphysics.com with your new
name and password.
• The VC zip code is 12549
• The Course ID: MPLABARBERA1010
Mastering physics due by 11:00 pm
tonight
9/7 do now (A day)
• quiz
9/7 Objectives (A day)
• Vector review
• Lab report requirement
There are two kinds of
quantities…
• Vectors have both magnitude
and direction
• displacement, velocity, acceleration
• Scalars have magnitude only
• distance, speed, time, mass
Two ways to represent vectors
Geometric approach
Vectors are symbolized graphically as arrows, in
text by bold-face type or with a line/arrow on top.
A
Magnitude: the size of the arrow
Direction: degree from East
θ
Algebraic approach
Vectors are represent in a coordinate system, e.g.
Cartesian x, y, z. The system must be an inertial
coordinate system, which means it is non-accelerated.
y
Magnitude: |R| = √x12 +y12
y1
o
p(x1, y1)
θ
x1
Direction: θ = tan-1(y1/x1)
x
Equal and Inverse Vectors
Equal vectors have the
same length and direction.
Inverse vectors have
the same length, but
opposite direction.
A
-A
Graphical Addition of Vectors:
“Head and tail ” & “parallelogram”
Head and tail method
E
Parallelogram method
E
C is called the resultant vector!
E is called the equilibrant vector!
Vector Addition Laws
 Commutative Law:
a+b=b+a
 Associative Law:
(a + b) + c = c + (b + a)
Subtract vectors: adding a
negative vector
Component Addition of Vectors
1) Resolve each vector into its x- and ycomponents.
Ax = Acos
Bx = Bcos
Ay = Asin
By = Bsin etc.
2) Add the x-components together to get Rx
and the y-components to get Ry.
Component Addition of Vectors
3) Calculate the magnitude of the
resultant with the Pythagorean
Theorem (|R| = Rx2 + Ry2).
4) Determine the angle with the
equation  = tan-1 Ry/Rx.
Algebraic Addition of Vectors
Ax = AcosA
Bx = BcosB
Ay = AsinA
By = BsinB
Rx = Ax + Bx
Ry = Ay + By
Bx
By
Ax
Ay
θB B
R
A
θA
θ
Rx
|R| = Rx2 + Ry2.
Ry
 = tan-1 Ry/Rx
• Homework
1. Reading assignment: 1.7 – 1.9
• p. 30
#31, 41, 43
Lab Period
• Lab 1: Vector Addition
• Objective: To compare the experimental
value of a resultant of several vectors to the
values obtained through graphical and
analytical methods.
• Equipment: A force table set
9/10 do now
vi
vf
The direction of the change in velocity is best shown by
A
B
C
B
E
Objectives (B day)
• Quiz corrections – count as a grade
• Homework questions?
• Unit vector
Unit vectors
• A unit vector is a vector that has a magnitude of 1, with
no units. Its only purpose is to point, or describe a
direction in space.
• Unit vector is denoted by “^” symbol.
• For example:
– ^
i represents a unit vector that points in the direction
of the + x-axis
j unit vector points in the + y-axis
– ^
– ^
k unit vector points in the + z-axis
y
^
j
^
k
z
^
i
x
• Any vector can be represented in terms of
unit vectors, i, j, k
Vector A has components:
Ax, Ay, Az
A = Axi + Ayj + Azk
In two dimensions:
A = Axi + Ayj
Magnitude and direction of the
vector
In two dimensions:
The magnitude of the vector is
|A| = √Ax2 + Ay2
The direction of the vector is
θ = tan-1(Ay/Ax)
In three dimensions:
The magnitude of the vector is
|A| = √Ax2 + Ay2 + Az2
Adding Vectors By Component
s=a+b
Where a = axi + ayj & b = bxi + byj
s = (ax + bx)i + (ay + by)j
sx = ax + bx; sy = ay + by
s = sxi + syj
s2 = sx2 + sy2
tanf  sy / sx
example
Given vector:
•
A =(3i + 4j ) m
Determine
1. The x, y, z component of A
2. the magnitude A
3. the direction of vector A with +x
example
a. Is the vector A = i + j + k a unit
vector?
b. Can a unit vector have any components
with magnitude greater than unity? Can it
have any negative components?
c. If A = a (3.0 i + 4.0 j ) where a is a
constant, determine the value of a that
makes A a unit vector.
Example – using unit vectors
Given the two displacement
A =(6 i + 3 j - k ) m
B =(4 i - 5 j + 8 k ) m
• Find the magnitude of the displacement
2A - B
2A - B =(8 i + 11 j - 10 k ) m
• Its magnitude = (√ 82 + 112 + 102 ) m = 17 m
• Class work
• Homework
– Read 1.9 #1.49; 1.69; 1.75
– Mastering physics – due Thu. 9/13, 11:00 pm
– Chapter 1 test is Friday
9/11 do now
Two forces F1 and F2 are acting at a point O, as
shown below.
Which of the fooling is very nearly the resultant
vector of F2 – F1?
a.
b.
F1
F2
c.
o
e.
d.
Objectives (A day)
• Homework questions?
• Multiplication of vectors
Multiplication of Vectors:
Product of a Scalar and a Vector
Vector
Product of Two Vectors
Scalar or Vector
Multiplication of Vector by
Scalar produce a vector
• Examples:
• momentum p = mv
• Net force F = ma
• Result
• A vector with the same direction,
a different magnitude and
perhaps different units.
Multiplication of vector by vector
produce a scalar
• Scalar product or dot product, yields a
result that is a scalar quantity
• Examples:
• work W = F  d
• Result
• A scalar with magnitude and no
direction.
Scalar product (Dot Product)
|C| = A  B
|C| = AB cos
|C| = AxBx + AyBy + AzBz
A

B
Commutative property
of scalar product
A∙B=B∙A
• Scalar product of parallel vectors:
A∙A = |A||A|cos0o = |A|2
A
A
• Scalar product of anti-parallel vectors:
A∙(-A) = |A||A|cos180o = -|A|2
A
-A
The sign of the product as a result
of the angle between two vectors
θ < 90o
A∙B > 0
θ = 90o
A∙B = 0
θ > 90o
A∙B < 0
Application of scalar product
• When a constant force F is applied to a body that
undergoes a displacement d, the work done by the force
is given by
W = F∙d
The work done by the force is
• positive if the angle between F and d is between 0 and
90o (example: lifting weight)
• Negative if the angle between F and d is between 90o
and 180o (example: stop a moving car)
• Zero and F and d are perpendicular to each other
(example: waiter holding a tray of food while walk
around)
Scalar product of unit vectors
Parallel unit vectors
Anti-Parallel unit vectors
i∙i=1
i∙j=j∙i=0
j∙j=1
j∙k=k∙j=0
k∙k=1
i∙k=k∙i=0
Mathematic meaning of scalar product
Projection of B on A
A∙B = A (Bcosθ)
A∙B = B (Acosθ)
Projection of B on A
B
θ
A
Bcosθ
Component of B along A
Vector multiplications obeys
distributive law
(A + B)∙C = A∙C + B∙C
example
A∙j = ?
A∙j = (Axi + Ayj + Azk)∙j = Ay
Component of A along y-Axis
Finding the angles with the scalar product
• Find the dot product and the angle between the
two vectors
A · B = |A||B|cosθ= AxBx + AyBy + AzBz
|A| = √Ax2 + Ay2 + Az2
|B| = √Bx2 + By2 + Bz2
cosθ =
A·B
|A||B|
=
AxBx + AyBy + AzBz
(√Ax2 + Ay2 + Az2 )(√Bx2 + By2 + Bz2 )
If cosθ is negative, θ is between 90o and 180o
example
A = 3i + 7k
B = -i + 2j + k
A∙B = ?
θ=?
example
• Find the scalar product A∙B of the two vectors in the
figure. The magnitudes of the vectors are A = 4.00 and B
= 5.00
θ=
130o
–
53.0o
=
77.0o
A∙B = (4.00)(5.00)cos 77.0o = 4.50
B
y
130o
A
θ 53.0o
x
Ax = (4.00)cos53.0o = 2.407; Ay = (4.00)sin53.0o = 3.195
Az = 0;
Bx = (5.00)cos130o = -3.214; By = (5.00)sin130o = 3.830
Bz = 0
A∙B = AxBx + AyBy + AzBz = 4.50
Finding the angles with the scalar product
• Find the dot product and the angle between the
two vectors
A = 2i + 3j + k
B = -4i +2j - k
cosθ = (A∙B) / (|A||B|)
A∙B = AxBx + AyBy + AzBz = -3
|A| = √Ax2 + Ay2 + Az2 = √14
|B| = √Bx2 + By2 + Bz2 = √21
cosθ = -0.175
θ = 100o
•Since cosθ is negative, θ is between
90o and 180o
homework
• Read 1.10; p. 31, #53, #55
Lab period
• Force table – unit vector, dot product,
9/12 do now
A particle, as shown below, moving from point to a point Q
on a curved path has respectively has v1 and v2
respectively. The direction of the average force on the
particle is best given by
v1
P
Q
v2
A
B
C
D
E
Objectives (B day)
• Vector product (Cross product)
Vector (Cross Product)
– Vector product or cross product,
yields another vector
•Application
•Work  = r  F
•Magnetic force
F = qv  B
•Result
•A vector with magnitude and a direction
perpendicular to the plane established
by the other two vectors
Vector product (cross product)
• The vector product of two
vectors A and B, also called the
cross product, is denoted by
C=AxB
• The vector product is a vector. It
has a magnitude and direction
Magnitude of C = A  B
C = AB sin  (magnitude)
A
Where θ is the angle from A toward B, and θ is
the smaller of the two possible angles.

B
Since 0 ≤ θ ≤ 180o, 0 ≤ sinθ ≤ 1, |A x B| is
never negative.
Note when A and B are in the same direction or in the
opposite direction, sinθ = 0;
The vector product of two parallel or anti-parallel
vectors is always zero.
Direction is determined by Right Hand Rule
Place the vector tail to tail, they
define the plane
AxB
B
θ
A
AxB
A x B is perpendicular to the plane
containing the vectors A and B.
Right-hand rule: we follow the
direction of the fingers to go from
the A to B, then the thumb points
in the direction of A x B
B
BxA=-AxB
A
θ
θ
A
B
BxA
Vector product vs. scalar product
• Vector product:
– A x B = ABsinθ (magnitude)
– Direction: right-hand rule-perpendicular to the A, B
plane
• Scalar product:
– A∙B = ABcosθ (magnitude)
– It has no direction.
• When A and B are parallel • When A and B are
perpendicular to each other
– AxB is zero
– A∙B is maximum
– AxB is maximum
– A∙B is zero
Calculating the vector product using
components
• If we know the components of A and B, we can
calculate the components of the vector product.
• The product of any vector with itself is zero
＊i x i = 0; j x j = 0; k x k = 0
• Using the right hand rule and A x B = ABsinθ
＊i x j = -j x i = k;
＊j x k = -k x j = i;
＊k x i = - i x k = j
A x B = (Axi + Ayj + Azk) x (Bxi + Byj + Bzk)
= AxByk - AxBzj
– AyBxk + AyBzi
+ AzBxj - AzByi
A x B = (AyBz – AzBy) i + (AzBx - AxBz) j + (AxBy – AyBx) k
If C = A x B then
Cx = AyBz – AzBy;
Cy = AzBx - AxBz;
Cz = AxBy – AyBx
The vector product can also be expressed in determinant
form as
AxB=
i
j
k
i
j
k
Ax
Ay
Az
Ax
Ay
Az
Bx
By
Bz
Bx
By
Bz
- direction
+ direction
A x B =(AyBz – AzBy) i + (AzBx - AxBz) j + (AxBy – AyBx) k
Multiplication of Vector by
Vector (Cross Product)
C=AB
A
C =

B
i j k
Ax Ay Az
Bx By Bz
Example 1.12
• Vector A has magnitude 6 units and is in the direction of
the + x-axis. Vector B has magnitude 4 units and lies in
the xy-plane, making an angle of 30o with the + x-axis
(fig. 1.32). Find the vector product C = A x B.
y
C = A x B = ABsinθ = (6)(4)sin30o = 12
From the right-hand rule, the direction
of C is along the z-axis, C = 12k
B
30o
A
C
z
We can also find C using
components of A and B
x
Example 1.12
• Vector A has magnitude 6 units and is in the direction of
the + x-axis. Vector B has magnitude 4 units and lies in
the xy-plane, making an angle of 30o with the + x-axis
(fig. 1.32). Find the vector product C = A x B.
y
We can also find C using
components of A and B
A = 6i
B
30o
B = 3.46i + 2j
A
C = (6i) x (3.46i + 2j) =12k
C
z
x
example
• Find the vector product A X B (expressed in unit vectors)
of the two vectors given in the figure.
A (3.60 m)
70o
30o
B (2.4 m)
C = A x B = ABsinθ = (3.60 m)(2.4 m)sin140o
(C = 5.6 k) m
From the right-hand rule, the direction of C is
along the z-axis,
homework
• Class work
• Homework
– Read: 1.10; p. 31 #1.59, 1.89
9/13 do now
• A particle of mass m at point P moves along the x-axis
and changes its velocity from vi at P to vf at Q, as shown
below. The direction of average acceleration is best given
by
d
meters
P
vi
vf
A.
B.
C.
D.
Q
t sec
E.
9/13 Objectives (B day)
•
•
•
•
•
AP Function Review
Chapter 1 review
Lab
Homework – function sheet
Chapter 1 test tomorrow
Lab period
• Lab 1 – part 4 – cross product
9/7 do now (A day)
• Antarctica is roughly semicircular, with a
radius of 2000 km. The average thickness
of its ice cover is 3000 m. How many cubic
centimeters of ice does Antarctica
contain? (ignore the curvature of Earth)
1.9 x 1022 cm3