chapter 3 vectors in physicsnsmn1.uh.edu/rbellwied/classes/spring2013/ch3_notes.pdf · summary of...
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Chapter 3
Vectors in Physics
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Units of Chapter 3• Scalars Versus Vectors
• The Components of a Vector
• Adding and Subtracting Vectors
• Unit Vectors
• Position, Displacement, Velocity, andAcceleration Vectors
• Relative Motion
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3-1 Scalars Versus VectorsScalar: number with units
Vector: quantity with magnitude and direction
How to get to the library: need to know how farand which way
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3-2 The Components of a VectorEven though you know how far and in whichdirection the library is, you may not be ableto walk there in a straight line:
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3-2 The Components of a VectorCan resolve vector into perpendicularcomponents using a two-dimensionalcoordinate system:
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3-2 The Components of a VectorLength, angle, and components can becalculated from each other usingtrigonometry:
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3-2 The Components of a Vector
Signs of vector components:
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3-3 Adding and Subtracting VectorsAdding vectors graphically: Place the tail of thesecond at the head of the first. The sum pointsfrom the tail of the first to the head of the last.
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3-3 Adding and Subtracting Vectors
Adding Vectors Using Components:
1. Find the components of each vector to beadded.
2. Add the x- and y-components separately.
3. Find the resultant vector.
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3-3 Adding and Subtracting Vectors
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3-3 Adding and Subtracting Vectors
Subtracting Vectors: The negative of a vector isa vector of the same magnitude pointing in theopposite direction. Here, .D= A B!
r r r
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Example: Vector A has a length of 5.00 meters and pointsalong the x-axis. Vector B has a length of 3.00 meters andpoints 120° from the +x-axis. Compute A+B (=C).
A x
y
B
120°
C
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adj
opp
cos
sintan
hyp
adjcos
hyp
oppsin
==
=
=
!
!!
!
!
( )
( ) m 50.160cosm00.360cos 60cos
m 60.260sinm00.360sin60sin
!=°!=°!="!
=°
=°=°="=°
BBB
B
BBB
B
xx
y
y
and Ax = 5.00 m and Ay = 0.00 m
A x
y
B
120°60°
By
Bx
Example continued:
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The components of C:( )
m 2.60m 2.60 m 00.0
m 3.50m 1.50 m 00.5
=+=+=
=!+=+=
yyy
xxx
BAC
BAC
x
y
C
Cx = 3.50 m
Cy = 2.60 m
θ
The length of C is:
( ) ( )
m 36.4
m 60.2m 50.322
22
=
+=
+== yx CCC C
The direction of C is:
( ) °==
===
!6.367429.0tan
7429.0m 3.50
m 60.2tan
1"
"x
y
C
C
From the +x-axis
Example continued:
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Example: Margaret walks to the store using the followingpath: 0.500 miles west, 0.200 miles north, 0.300 miles east.What is her total displacement? Give the magnitude anddirection.
x
y
r3
r2
r1
Δr
Take north to be inthe +y direction andeast to be along +x.
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Example continued:
The displacement is Δr = rf − ri. The initial position is theorigin; what is rf?
The final position will be rf = r1 + r2 + r3. The componentsare rfx = −r1 + r3 = −0.2 miles and rfy = +r2 = +0.2 miles.
miles 283.022=!+!=!
yxrrr
°==!
!= 45 and 1tan ""
x
y
r
r
Using the figure, the magnitude anddirection of the displacement are
x
y
ΔrΔry
Δrx
θ
N of W.
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3-4 Unit VectorsUnit vectors are dimensionless vectors of unitlength.
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3-4 Unit VectorsMultiplying unit vectors by scalars: the multiplierchanges the length, and the sign indicates thedirection.
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3-5 Position, Displacement, Velocity,and Acceleration Vectors
Position vector points from the origin tothe location in question.
The displacement vectorpoints from the originalposition to the final position.
frr
r!r
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3-5 Position, Displacement, Velocity,and Acceleration Vectors
Average velocity vector:
(3-3)
So is in the same
direction as .avvr
r!r
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3-5 Position, Displacement, Velocity,and Acceleration Vectors
Instantaneous velocity vector is tangent to thepath:
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3-5 Position, Displacement, Velocity,and Acceleration Vectors
Average acceleration vector is in the directionof the change in velocity:
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3-5 Position, Displacement, Velocity,and Acceleration Vectors
Velocity vector is always in the direction ofmotion; acceleration vector can point anywhere:
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Example (text problem 3.42): At the beginning of a 3 hourplane trip you are traveling due north at 192 km/hour. At theend, you are traveling 240 km/hour at 45° west of north.
(a) Draw the initial and final velocity vectors.
x (east)
y (north)
vivf
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(b) Find Δv.
km/hr 3.2245cos
km/hr 170045sin
!=!°+=!="
!=!°!=!="
ifiyfyy
fixfxx
vvvvv
vvvv
The components are
km/hr 17122
=!+!=!yxvvv
( ) °==!="
"= #
5.71312.0tan1312.0tan1$%
x
y
v
v South ofwest
Example continued:
The magnitude and direction are:
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(c) What is aav during the trip?
t!
!=v
aav
2
av,
2
av,
km/hr 43.7hr 3
km/hr 3.22
km/hr 7.56hr 3
km/hr 170
!=!
="
"=
!=!
="
"=
t
va
t
va
y
y
x
x
The magnitude and direction are:
°==!==
=+=
" 5.7)1310.0(tan1310.0tan
km/hr 2.57
1
av,
av,
22
av,
2
av,av
##x
y
yx
a
a
aaa
South ofwest
Example continued:
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3-6 Relative MotionThe speed of the passenger with respect tothe ground depends on the relative directionsof the passenger’s and train’s speeds:
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3-6 Relative Motion
This also works in two dimensions:
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Example: You are traveling in a car (A) at 60 miles/hour easton a long straight road. The car (B) next to you is travelingat 65 miles/hour east. What is the speed of car B relative tocar A?
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east miles/hour 5
eastmiles/hr 60 east miles/hr 65BA
AGBGBA
AGBGBA
BAAGBG
=
!=
!=
"!"="
"+"="
v
vvv
rrr
rrrFrom the picture:
A
B
A
B
t=0 t>0
ΔrAG
ΔrBG
ΔrBA
Divide by Δt:
+x
Example continued:
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Example: You are traveling in a car (A) at 60 miles/houreast on a long straight road. The car (B) next to you istraveling at 65 miles/hour west. What is the speed of car Brelative to car A?
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A
B
A
B
t=0
ΔrAG
ΔrBG ΔrBA
t>0t>0
estmiles/hr w 125
eastmiles/hr 60estmiles/hr w 65
AGBGBA
AGBGBA
=
!=
!=
"!"="
vvv
rrr
+x
From the picture:
Divide by Δt:
Example continued:
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Summary of Chapter 3
• Scalar: number, with appropriate units
• Vector: quantity with magnitude and direction
• Vector components: Ax = A cos θ, By = B sin θ
• Magnitude: A = (Ax2 + Ay
2)1/2
• Direction: θ = tan-1 (Ay / Ax)
• Graphical vector addition: Place tail of secondat head of first; sum points from tail of first tohead of last
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• Component method: add components ofindividual vectors, then find magnitude anddirection• Unit vectors are dimensionless and of unitlength• Position vector points from origin to location• Displacement vector points from originalposition to final position• Velocity vector points in direction of motion• Acceleration vector points in direction ofchange of motion• Relative motion:
Summary of Chapter 3
13 12 23v = v + vr r r