Lecture1 Vectors
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Transcript of Lecture1 Vectors
WELCOME TO PHYSICS 101Introduction to Physics 1 !
• Learning to think and reason about the physical world
- Conceptual Understanding- Make connections with real-world applications
• Build a “physical intuition”
• Problem-Solving skills
• This course is not about memorization or mindless use of mathematicswe will use math to think about relationships between quantities.
• This is a course that is very different from high school physics.
Class Themes
What is Physics all about ?• Provides a quantitative understanding of physical
phenomena in our universe
• Based on experimental observations and mathematical analysis
• Used to develop theories that explain the phenomena being studied and relate to other established theories
• Experiments refine our theories about our universe
To be quantitative….• Need standards of measurement for different physical
quantities
• Use SI System for measuringa. length
1 meter = distance travelled by light in vacuum in a certain fraction of second.
b. mass1 kilogram = mass of a cylinder kept in
Int’l Bureau of Weights/Standardsc. time
1 second = a certain # times the period of radiation oscillation of a cesium atom
• Scalar - a physical quantity that is specified by a positive or negative number with a unit.Ex. Temperature, Volume, Mass, Time, Energy
Rules of ordinary arithmetic are used to manipulate scalar quantities
• Vector - is a physical quantity that must be described by a magnitude (number) and unit, plus a direction.
Scalars vs. Vectors
Example of a Vector• A particle travels from A
to B along the path shown by the dotted red line– This is the distance
traveled and is a scalar
• The displacement is the solid line from A to B– The displacement is
independent of the path taken between the two points
– Displacement is a vector
Other vectors: velocity, acceleration, force, momentum
Vectors: Notation and Properties• When handwritten, use an arrow: • When printed • Magnitude | |
Ar
Ar
Ar
Equality of Two Vectors
• Two vectors are equal if they have the same magnitude and the same direction
• All of the vectors shown are equal
Adding Vectors• When adding vectors, their directions must be taken into
account• Units must be the same
(can’t add velocity and acceleration vectors)
Two Methods: 1. “Graphical” Method
- use scale drawings2. “Component or Analytical” Method
- more convenient
Adding Vectors GraphicallyEstablish a scale and coordinate systemDraw vectors to be added “tip to tail”.Measure the resultant R and itsdirection, relative to A or B
θ
Or Use Trigonometry (Sine and Cosine Laws)
Adding Multiple Vectors
“Tip-to-tail” Method is Repeated
Note: Vector Addition is Commutative
A + B = B + A
(Ordering doesn’t matter)
Subtracting Vectors• Negative of a vector
– reverses the vector’s direction
• Continue with standard vector addition procedure
Multiplying/Dividing a Vector by a Scalar
• The result is a vector, “scaled” by the scalar factor• Negative scalar multiplication reverses direction.
Dot/Scalar and Vector Cross Products – very important vector multiplcation will be studied in future chapters
A
2A
- A
Components of a Vector
Rectangular components– projections of the vector
along the x- and y-axes
A vector A can be represented as the sum of its components Ax and Ay .
A = Ax + Ay
θ= cosAxA θ= sinAyA
x
y12y
2x A
AtanandAAA −=θ+=
One can further simplify this with the use of “unit vectors”.
Unit Vectors• A unit vector is a dimensionless
vector with a magnitude of exactly 1.• Unit vectors are used to specify
a direction
kand,j,i
• provide the basis for representing any vectorin the space.
Unit Vectors – why use them ?• The symbols
represent unit vectors in the x, y and z directions
• They form a set of mutually perpendicular vectors
kand,j,i
Unit Vectors in Vector Notation
• is the same as Ax
is the same as Ay etc.
• The complete vector can be expressed as
i
j
kjiA ˆAˆAˆA zyx ++=r
xAr
yAr
Adding Vectors Using Unit Vectors
• Since• Then
• Then Rx = Ax + Bx and Ry = Ay + By
( ) ( )( ) ( )jiR
jijiRˆBAˆBA
ˆBˆBˆAˆA
yyxx
yxyx
+++=
+++=r
r
x
y12y
2x R
RtanRRR −=θ+=
BARrrr
+=
Adding Vectors with Unit Vectors
Adding Vectors Using Unit Vectors – Three Directions
• Using
• Rx = Ax + Bx , Ry = Ay + By and Rz = Az + Bz
etc.RRtanRRRR x1
x2z
2y
2x
−=θ++=
BARrrr
+=
Lecture Problem 1-8
Suppose your hair grows at the rate 1/32 inch per day. Find the rate at whichit grows in nanometers (nm) per second. Because the distance between atoms in a molecule is on the order of 0.1nm, your answer suggests how rapidly layers of atoms are assembled in this protein synthesis.
50x400xHair
400x
Lecture Problem 1-8
Suppose your hair grows at the rate 1/32 inch per day. Find the rate at whichit grows in nanometers (nm) per second. Because the distance between atoms in a molecule is on the order of 0.1nm, your answer suggests how rapidly layers of atoms are assembled in this protein synthesis.
(1/32) inch/day x .0254 m/inch x 109 nm/m x 1 day/24 hr x 1 hr/60 min x 1 min/60 sec
9.2 nm/s ~ 90 layers of atoms/second
50x400xHair
400x
Lecture Problem 1-45Consider the two vectors: A = 3i – 2j and B = -i – 4j. Calculate
a. A + B c. |A + B| e. the directions of A + B, A - Bb. A – B d. |A – B|
Lecture Problem 1-45Consider the two vectors: A = 3i – 2j and B = -i – 4j. Calculate
a. A + B c. |A + B| e. the directions of A + B, A - Bb. A – B d. |A – B|
Solution:a. A + B = 3i – 2j + -i – 4j = 2i – 6j
b A – B = 3i – 2j – ( -i – 4j) = 4i + 2j
c. |A + B| = √(22 + 62) = 6.32
d. |A – B| = √(42 + 22) = 4.47
e. direction of A + B: tan θ = -6/2 θ = -71.6° = 288°
direction of A – B: tan θ = 2/4 θ = 26.6°
Question: What to do if there are 3 vectors to add or subtract ?