Lecture1 Vectors
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Transcript of Lecture1 Vectors
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WELCOME TO PHYSICS 101Introduction to Physics 1 !
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• 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
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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
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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
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• 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
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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
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Vectors: Notation and Properties• When handwritten, use an arrow: • When printed • Magnitude | |
Ar
Ar
Ar
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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
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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
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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)
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Adding Multiple Vectors
“Tip-to-tail” Method is Repeated
Note: Vector Addition is Commutative
A + B = B + A
(Ordering doesn’t matter)
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Subtracting Vectors• Negative of a vector
– reverses the vector’s direction
• Continue with standard vector addition procedure
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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
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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
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θ= cosAxA θ= sinAyA
x
y12y
2x A
AtanandAAA −=θ+=
One can further simplify this with the use of “unit vectors”.
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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.
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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
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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
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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
+=
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Adding Vectors with Unit Vectors
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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
+=
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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
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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
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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|
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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 ?