C2 – Vectors C3 – Interactions transfer...
Transcript of C2 – Vectors C3 – Interactions transfer...
General Physics GP7-Vectors (Ch 4) 1
• C2 – Vectors• C3 – Interactions transfer momentum
General Physics GP7-Vectors (Ch 4) 2
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General Physics GP7-Vectors (Ch 4) 3
Vectors and Scalars
• scalar quantity• completely specified by a single value
with an appropriate unit and has nodirection.
• vector quantity• completely described by a number and
appropriate units plus a direction.
General Physics GP7-Vectors (Ch 4) 4
Vectors and Scalars
• scalar quantity• completely specified by a single value with an
appropriate unit and has no direction.• Use an italic letter:
• vector quantity• completely described by a number and
appropriate units plus a direction.• Use an arrow:
A
A
r
General Physics GP7-Vectors (Ch 4) 5
Vector Example
• A particle travels from Ato B along the pathshown by the dottedred line• This is the distance
traveled and is a scalar
• The displacement isthe solid line from A toB• The displacement is
independent of the pathtaken between the twopoints
• Displacement is a vector
General Physics GP7-Vectors (Ch 4) 6
Equality of Two Vectors
• Two vectors are equal ifthey have the samemagnitude and the samedirection
• if A = B and theypoint along parallel lines
• All of the vectors shown inthe diagram at right areequal
BA
rr=
General Physics GP7-Vectors (Ch 4) 7
Adding Vectors Graphically
• Draw the vectors “tip-to-tail”
• The resultant is drawnfrom the origin ofto the end of the lastvector
• Measure the length ofand its angle• Use the scale factor to
convert length to actualmagnitude
A
r
R
r
General Physics GP7-Vectors (Ch 4) 8
Adding Multiple Vectors
• tip-to-tail for allvectors
• The resultant, ,is still drawn fromthe origin of the firstvector to the end ofthe last vector
R
r
General Physics GP7-Vectors (Ch 4) 9
Adding Vectors, Rules
• When two vectorsare added, the sumis independent ofthe order of theaddition.• This is the
commutative lawof addition
ABBA
rrrr+=+
General Physics GP7-Vectors (Ch 4) 10
Adding Vectors, Rules cont.
• When adding three ormore vectors, their sumis independent of theway in which theindividual vectors aregrouped• This is called the
Associative Propertyof Addition
( ) ( )CBACBA
rrrrrr++=++
General Physics GP7-Vectors (Ch 4) 11
Adding Vectors, Rules final
• When adding vectors, all of the vectorsmust have the same units
• All of the vectors must be of the sametype of quantity• For example, you cannot add a
displacement to a velocity
General Physics GP7-Vectors (Ch 4) 12
Negative of a Vector
• The negative of a vector is defined as thevector that, when added to the originalvector, gives a resultant of zero• Represented as –
• The negative of the vector will have thesame magnitude, but point in the oppositedirection
A
r
( ) 0=!+ AA
rr
General Physics GP7-Vectors (Ch 4) 13
Subtracting Vectors
• Special case ofvector addition
• If , then use
• Continue withstandard vectoraddition procedure
( )BA
BArr
rr
!+
!
General Physics GP7-Vectors (Ch 4) 14
Vector representation
• A vector has both magnitude (numbervalue) and direction.
zpypxppzyxˆˆˆ ++=
r
Show magnitude or value Show direction
General Physics GP7-Vectors (Ch 4) 15
Column Vector
• How can we describe this vector using ausing column vector?
zpypxppzyxˆˆˆ ++=
rtypical vector notation
column vector notation
!!!
"
#
$$$
%
&
=
z
y
x
p
p
p
pr
General Physics GP7-Vectors (Ch 4) 16
Components of a Vector
• A component is a part• We will use rectangular
components• These are the projections of the
vector along the x- and y-axes
• are thecomponent vectors of• They are vectors and follow all
the rules for vectors
• Ax and Ay are scalars, and willbe referred to as thecomponents of
yx AArr
and
A
r
A
r
General Physics GP7-Vectors (Ch 4) 17
Magnitude and Direction
• A vector may also be represented by it’smagnitude and direction.
• Magnitude
• The magnitude of the vector hasphysical units
• The magnitude of a vector is always apositive number
( ) 222
zyxppppmagp ++==
r
General Physics GP7-Vectors (Ch 4) 18
Multiplying or Dividing a Vectorby a Scalar
• The result of the multiplication or division is avector
• The magnitude of the vector is multiplied ordivided by the scalar
• If the scalar is positive, the direction of theresult is the same as of the original vector
• If the scalar is negative, the direction of theresult is opposite that of the original vector
General Physics GP7-Vectors (Ch 4) 19
Think-Pair-Share
• A boat is crossing a river that flowsfrom North to South at a rate of 3 m/s.The boat starts at the east end of theriver and heads directly west with aspeed of 4 m/s.(a) What is the boats total velocity?(b) If the river is 1000 m in the E-Wdirection, how long does it take theboat to cross?
General Physics GP7-Vectors (Ch 4) 20
Components of a Vector, 2
• The x-component of a vector is the projectionalong the x-axis
• The y-component of a vector is the projectionalong the y-axis
• Then,
!
r A = Ax
ˆ x + Ayˆ y
General Physics GP7-Vectors (Ch 4) 21
Components of a Vector, 3
• The previous equations arevalid only if θ is measuredwith respect to the x-axis
• The components are the legsof the right triangle whosehypotenuse is A
General Physics GP7-Vectors (Ch 4) 22
Components of a Vector, final
• The components canbe positive ornegative and willhave the same unitsas the originalvector
• The signs of thecomponents willdepend on the angle
General Physics GP7-Vectors (Ch 4) 23
Think-Pair-Share
• A displacement vector in the x-y plane is15 m long and directed at an angle of 30degrees above the positive x axis.Determine(a) the x-component(b) the y-component
rr
General Physics GP7-Vectors (Ch 4) 24
Think-Pair-Share
• A displacement vector in the x-y plane is10 m long and directed at an angle of 190degrees counterclockwise from the positivex axis. Determine(a) the x-component(b) the y-component
rr
General Physics GP7-Vectors (Ch 4) 25
Think-Pair-Share
• A fly lands on one wall of a room. Thelower left-hand corner of the wall isselected as the origin of the two-dimensional Cartesian coordinatesystem. If the fly is located at the pointhaving coordinates (2.00, 1.00) m, (a)how far is it from the corner of theroom? (b) What is its location in polarcoordinates?
General Physics GP7-Vectors (Ch 4) 26
Think-Pair-Share
• A person walks 25.00 north of east for3.10 km. How far would she have towalk due north and due east to arrive atthe same location?
General Physics GP7-Vectors (Ch 4) 27
Think-Pair-Share
• Jane leaves her houseand walks 5.0 blockseast and then proceedsnorth until she is 7.8blocks from home at anangle of 50 degreesNorth of east. Howmany blocks north didshe travel?
General Physics GP7-Vectors (Ch 4) 28
Example
• The Cartesian coordinates of apoint in the xy plane are (x,y)= (-3.50, -2.50) m, as shownin the figure. Find the polarcoordinates of this point.
• Solution:
and,
General Physics GP7-Vectors (Ch 4) 29
Unit Vectors
• A unit vector is a dimensionless vector witha magnitude of exactly 1.
• Unit vectors are used to specify a direction(x,y, and z) and have no other physicalsignificance
• The symbols for x, y, z are
• They form a set of mutually perpendicularvectors
zyx ˆ and ˆ,ˆ
yx
z
General Physics GP7-Vectors (Ch 4) 30
Unit Vectors in Vector Notation
• The complete vectorcan be expressed as
yAA
xAA
xy
xx
ˆ
ˆ
=
=r
r
zAyAxAA zyxxˆˆˆ ++=
r
x
y
General Physics GP7-Vectors (Ch 4) 31
Adding Vectors Using UnitVectors
• Using• Then
• and so Rx = Ax + Bx
• and Ry = Ay + By
( ) ( )
( ) ( )
yRxRR
yBAxBAR
yBxByAxAR
BAR
yx
yyxx
yxyx
ˆˆ
ˆˆ
ˆˆˆˆ
+=
+++=
+++=
+=
r
r
r
rrr
!
r R =
Ax
Ay
"
# $
%
& ' +
Bx
By
"
# $
%
& ' =
Ax + Bx
Ay + By
"
# $
%
& '
General Physics GP7-Vectors (Ch 4) 32
Adding Vectors with Unit Vectors
General Physics GP7-Vectors (Ch 4) 33
Adding Vectors Using UnitVectors – Three Directions
• Using
• Rx = Ax + Bx , Ry = Ay + By and Rz = Az + Bz
( ) ( )
( ) ( ) ( )
!!!
"
#
$$$
%
&
+
+
+
=
!!!
"
#
$$$
%
&
=
!!!
"
#
$$$
%
&
=
++=
+++++=
+++++=
+=
zz
yy
xx
z
y
x
z
y
x
zyx
zzyyxx
zyxzyx
BA
BA
BA
R
B
B
B
B
A
A
A
A
zRyRxRR
zBAyBAxBAR
zByBxBzAyAxAR
BAR
rrr
r
r
r
rrr
ˆˆˆ
ˆˆˆ
ˆˆˆˆˆˆ
General Physics GP7-Vectors (Ch 4) 34
Think-Pair-Share
• (a) In unit-vector notation, what is the sumof , where
a = (4.0 m) i + (3.0 m) j
b = (-13.0 m) i + (7.0 m) j
• What are the (b) magnitude and (c) directionof (relative to )?
yx
yx
ba
rr+
ba
rr+
x
General Physics GP7-Vectors (Ch 4) 35
Think-Pair-Share
• Repeat the problem using columnvector notation.
General Physics GP7-Vectors (Ch 4) 36
Coordinate Systems
• Used to describe the position of a pointin space
• Coordinate system consists of• a fixed reference point called the origin• specific axes with scales and labels• instructions on how to label a point relative
to the origin and the axes
General Physics GP7-Vectors (Ch 4) 37
Cartesian Coordinate System
• Also calledrectangularcoordinate system
• x- and y- axesintersect at theorigin
• Points are labeled(x,y)
General Physics GP7-Vectors (Ch 4) 38
Polar Coordinate System
• Origin and referenceline are noted
• Point is distance rfrom the origin in thedirection of angle θ,counter-clockwisefrom reference line
• Points are labeled (r,θ)
General Physics GP7-Vectors (Ch 4) 39
Polar to Cartesian Coordinates
• Based onforming a righttriangle from rand θ
• x = r cos θ• y = r sin θ• SOH-CAH-TOA
H
O
A
General Physics GP7-Vectors (Ch 4) 40
Cartesian to Polar Coordinates
• r is the hypotenuse andθ an angle
• θ must be counter-clockwise frompositive x axis forthese equations to bevalid