Welcome
Course:
AP Physics
Room:
207
Teacher:
Mrs. LaBarbera
Email: diana.labarbera@valleycentralschools.org
Post session: Tue. – Fri.
Objectives
•
•
•
•
Introduction of AP physics curriculum
Lab safety
Sign in lab safety attendance sheet
Chapter 1 - units, physical quantities, and
vectors
Chapter 1- units, physical
quantities, and vectors
1. Know the fundamental quantities and units of
mechanics.
2. Be able to determine the number of significant figures in
calculations.
3. Differentiate between vectors and scalars
4. Be able to add and subtract vectors graphically.
5. Be able to determine the components of vectors and to
use them in calculations
6. Know the unit vector and be able to use them with
components to describe vectors
7. Know the two ways of multiplying vectors.
1.1 The Nature of Physics
• Physics is an experimental science.
• Theories are formed through observation
and experiments. However, no theory is
ever regarded as the final and ultimate
truth.
• All theories can be revised by new
observations.
• All theories have a range of validity.
Percent error
• Measurements made during laboratory work
yield an experimental value
• Accepted value are 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
1.2 Solving Physics Problems
I SEE
•
•
•
•
Identify the relevant concepts – determine
target variable and the given quantities.
Set up the problem – choose equations based
on the known and unknown from Identify step.
Execute the solution – “do the math”
Evaluate your answer – “Does the answer
make sense?”
1.3 Standards and Units
• SI Fundamental Quantities And Units Of Mechanics
Quantity Standards
SI unit (symbol)
Time
Time required for 9,192,631,770 cycles
of cesium microwave radiation
second (s)
Length
Distance light travels in vacuum in
1/299,792,458 seconds
Meter (m)
Mass
The mass of a particular cylinder of
platinum-iridium alloy kept at
International Bureau of Weights and
Measures a Servres, France
kilogram (kg)
All other units can be expressed by combinations of these
fundamental (base) units. The combined base units is called
derived units.
Derived units
• Like derived dimensions, when we
combine base 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
T
1012
giga
G
109
mega
M
106
kilo
k
103
deci
d
10-1
centi
c
10-2
milli
m
10-3
micro
μ
10-6
nano
n
10-9
pico
p
10-12
• The British System
– Length:
– Force:
1 inch = 2.54 cm
1 pound = 4.448221615260 N
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,
– The dimension for Energy: E = ML2/T2
– The dimension for Impulse: J = ML/T
1.4 Unit consistency and
conversions
• 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
T
=L
At least, the equation is dimensionally correct; it may still
be wrong on other grounds, of course.
Example
use dimensional analysis to check if the equation is correct.
d=v/t
L = (L ∕ T ) ∕ T
[L] ≠ L ∕ T2
Conversion Strategies: I SEE
• Identify the target units and the known conversion
factors
• Setup the problem using the given units and conversion
factors to determine the unknown. Note units can be
multiplied or divided like numbers.
• Execute: do the math
• Evaluate: “Does the answer make sense?”
Example: we wish to convert 2 miles into meters. (given
conversion factors:1 miles = 1760 yards, 1 yd = 0.9144 m)
1760 yard 0.9144m
2mile 

 3218m
1mile
1yard
Note: the units are a part of the measurement as important as the
number. They must always be kept together.
Example 1.1
• The official world land speed record is 1228.0
km/h, set on 10/15/1997, by Andy Green in the
jet engine car Thrust SSC. Express this speed in
m/s.
Example 1.2
• The world’s largest cut diamond is the First Star
of Africa. Its volume is 1.84 cubic inches. What
is tis volume in cubic centimeers? In cubic
meters?
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!!
Example
Dimensional Analysis is simply a technique you can use to
convert from one unit to another. The main thing you have to
remember is that the GIVEN UNIT MUST CANCEL OUT.
Suppose we want to convert 65 mph to ft/s
or m/s.
miles 1hour 1 min 5280 ft  65 11 5280 

65
1 60  60 1
hour 60 min 60 sec 1mile
ft
95
s
1meter  95 1 
1 3.281
3.281 ft
29 m / s
ft
95
s
1.5 Uncertainty and Significant
Figures
• 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  10 8
0.0000000010878  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
3
1.50  10 3
2 digits, the 5 is uncertain.
3 digits, the 0 is uncertain.
Example 1.3
The rest energy E of an object with rest mass m is given by
Einstein’s equation
E = mc2
Where c is the speed of the light in vacuum (c = 2.99792458 x 108
m/s). Find E for an object with m = 9.11 x 10-31 kg.
Test Your Understanding 1.5
• The density of a material is equal to its mass divided by
its volume. What is the density (in kg/m3) of a rock of
mass 1.80 kg and volume 6.0 x 10-4 m3
1. 3 x 103 kg/m3
2. 3.0 x 103 kg/m3
3. 3.00x 103 kg/m3
4. 3.000x 103 kg/m3
5. Any of these
1.6 Estimates and orders of
magnitude
Estimation of an answer is often done by rounding
any data used in a calculation.
Comparison of an estimate to an actual calculation
can “head off” errors in final results.
Example 1.4
• You are writing an adventure novel in which the hero
escapes across the border with a billion dollars’ worth of
gold in his suitcase. Is this possible? Would that amount
of gold fit in a suitcase? Would it be too heavy to carry?
(given 1 g of gold ≈ $10.00 and density of gold ≈ 1 g/cm3)
Test Your Understanding 1.6
• What is approximate number of teeth in all
the mouths of everyone at VC?
1.7 vectors and vector additions
• There are two kinds of quantities…
• Vectors have both magnitude and direction
• displacement, velocity, acceleration
• Scalars have magnitude only
• distance, speed, time, mass
Vectors
• Vectors show magnitude and direction, drawn as a ray.
Equal and Inverse Vectors
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
y1
o
Magnitude: R = √x12 +y12
p(x1, y1)
θ
x1
x
Direction: θ = tan-1(y1/x1)
Vector addition
• Vectors may be added graphically, “head to tail.” or
“parallegram
Commutative properties of
vector addition
R  A  B  C  ( A  B)  C  A  ( B  C )
Resultant and equilibrant
B
A+B=R
A
R
E
R is called the resultant vector!
E is called the equilibrant vector!
Subtract vectors: adding a
negative vector
example
• At time t = t1, and object’s velocity is given by the vector
v1 a short time later, at t = t2, the object’s velocity is the
vector v2. If the magnitude of v1 = the magnitude of v2,
which one of the following vectors best illustrates the
object’s average acceleration between t = t1 and t = t2
v2
v1
v2
v2 –v1
v1
-v1
v2 -v1
A
B
C
D
E
v2
Example 1.5
• A cross-country skier
skies 1.00 km north and
then 2.00 km east on a
horizontal snow field.
How far and in what
direction is she from the
starting point?
Test Your Understanding 1.7
• Two displacement vectors, S and T, have magnitudes S =
3 m and T = 4 m. Which of the following could be the
magnitude of the difference vector S -T? (there may be
more than one correct answer)
1. 9 m
2. 7 m
3. 5 m
4. 1 m
5. 0 m
6. -1 m
1.8 Components of vectors
• Manipulating vectors graphically is insightful but difficult when
striving for numeric accuracy. Vector components provide a numeric
method of representation.
• Any vector is built from an x component and a y component.
• Any vector may be “decomposed” into its x component using A*cos θ
and its y component using A*sin θ (where θ is the angle the vector A
sweeps out from 0°).
  
A  Ax  A y
sin  
Ay
A
cos  
Ay
A
The sign of the component depends on the
angle from 0o
Y is positive
X is negative
Y is negative
X is negative
Example 1.6.
• a) what are the x and y components of vector D? the magnitude
of the vector is D = 3.00 m and the angle α = 45o.
• b) what are the x and y components of vector E? the magnitude of
the vector is E = 4.50 m and the angle β = 37.0o.
Doing vector calculations using components
• Vector addition strategies
1) Resolve each vector into its x- and ycomponents.
Ax = Acos
Ay = Asin
Bx = Bcos
By = Bsin
etc.
2) Add the x-components together to get
Rx and the y-components to get Ry.
Rx = Ax + Bx
Ry = Ay + By
3) Calculate the magnitude of the
resultant with the Pythagorean
Theorem R  R 2  R 2
x
y
4) Determine the angle with the
equation  = tan-1 Ry/Rx.
Finding the direction of a vector sum by looking at the individual components
•
Multiplying a vector by a scalar
• Multiplying a vector by a positive scalar changes the
magnitude (length) of the vector, but not its direction.
A
D =2A
2A is twice as long as A
Dx = 2Ax, Dy = 2Ay
• Multiplying a vector by a negative scalar changes the
magnitude (length) of the vector and reverse its
direction.
A
D = -3A
-3A is three times as long as A
and points in the opposite
direction.
Dx = -3Ax, Dy = -3Ay
Example 1.7
• Three players are brought to the center of a large, flat field, each is
given a meter stick, a compass, a calculator, a shovel, and the
following three displacements:
– 72. 4 m 32.0o east of north
– 57.3 m 36.0o south of west
– 17.8 m straight south
• The three displacements lead to the point where the keys to a new
Porsche are buried. Two players start measuring immediately, but the
winner first calculates where to go. What does she calculate?
Example 1.8
• After an airplane takes off, it travels 10.4 km west, 8.7
km north, and 2.1 km up. How far us it from the takeoff
point?
Test Your Understanding 1.8
• Two vectors A and B both lie in the xy-plane.
a. Is it possible for A to have the same magnitude
as B but different components?
b. Is it possible for A to have the same components
as B but a different magnitude?
1.9 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
using unit vector representation
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
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?
^
^
A
=
a
(3.0
i
+
4.0
j ), where a is a
c. If
constant, determine the value of a that
makes A a unit vector.
Example 1.9
Given the two displacement
D =(6 ^
i + 3^
j -^
k) m
^
^
E =(4i - 5 j + 8 ^
k) m
• Find the magnitude of the displacement
2D - E
^
^
^
=(8
i
+
11
j
10
k) m
2D - E
• Its magnitude = (√ 82 + 112 + 102 ) m = 17 m
Test Your Understanding 1.9
• Arrange the following vectors in order of
their magnitude, with the vector of largest
magnitude first.
a.A = (3i + 5j – 2k) m
b.B = (-3i + 5j – 2k) m
c.C = (3i – 5j – 2k) m
d.D = (3i + 5j + 2k) m
1.10 Products of Vectors
1. A scalar Product
 
C  A  B  A B cos f
C = AxBx + AyBy + AzBz
• Scalar product or dot product, yields a result that is
a scalar quantity.
• Example: work W = F  d the Result is a scalar with
magnitude and no direction.
• Scalar product is commutative:
   
A B  B  A
  
   
C  ( A  B)  C  A  C  B
1.25
 
C  A  B  A  B cos f
C  A  B//
C  A//  B
The sign of the
scalar product
• Scalar product of same vectors:
A∙A = |A||A|cos0o = |A|2
A
A
• Scalar product of opposite vectors:
A∙(-A) = |A||A|cos180o = -|A|2
A
-A
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)
Calculating the scalar product
using components
Parallel unit vectors
perpendicular 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
C  A  B  Ax B x  Ay B y  Az B z
example
A = Axi + Ayj + Azk
A∙j = ?
A∙j = (Axi + Ayj + Azk)∙j = Ay
Component of A along y-Axis
Example 1.10 Calculating a scalar
product
• Find the vector product A∙B of
the two vectors in the diagram.
The magnitudes of the vectors
are A = 4.00 and B = 5.00
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
 
Ax Bx  Ay B y  Az Bz
A B
cos  

A B ( A 2  A 2  A 2 )( B 2  B 2  B 2 )
x
y
z
x
y
z
If cosθ is negative, θ is between 90o and 180o
example
A = 3i + 7k
B = -i + 2j + k
A∙B = ?
θ=?
Example 1.11
• Find the angel between the two vectors:
A = 2i + 3j + k
and
B = -4i + 2j - k
The vector product
Termed the “cross product.” Result is a vector
with magnitude and a direction perpendicular to
the plane established by the other two vectors.
Direction is determined by Right Hand Rule
Place the vector tail to tail, they
define the plane
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
BxA=-AxB
θ
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.
 
C  A  B  A  B
 
C  A  B  B  A
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
– AxB is zero
– A∙B is maximum
• When A and B are
perpendicular to each other
– 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
Example 1.12
• Vector A has a magnitude of 6 units and is in the
direction of +x axis. Vector B has a magnitude of 4
o
units and lies in the xy-plane, making an angle
of
30

with the +x-axis. Find vector product A  B
Check Your Understanding 1.10
• Vector A has magnitude 2 and vector B has magnitude
3. the angle φ between A and B is known to be either 0o,
90o, or 180o. For each of the following situations, state
what the value of φ must be. (in each situation there may
be more than one correct answer.)
1. A∙B = 0
2. A x B = 0
3. A∙B = 6
4. A∙B = - 6
5. │A x B│= 6