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Chapter – 1 Vectors and Kinematics
Text Book: AN INTRODUCTION TO MECHANICS by Kleppner and Kolenkow
Dr. Virendra Kumar VermaMadanapalle Institute of Technology and Science
(MITS)
Scalars and VectorsA scalar quantity is a quantity that has only magnitude.
A vector quantity is a quantity that has both a magnitude and a direction.
Scalar quantitiesLength, Area, Volume,
Speed, Mass, Density
Temperature, PressureEnergy, Entropy
Work, Power
Vector quantitiesDisplacement, Direction,
Velocity, Acceleration,Momentum, Force,
Electric field, Magnetic field
Volume
Velocity
Vector notationVector notation was invented by a physicist, Willard Gibbs of Yale University.
By using vector notation, physical laws can often be written in compact and simple form.
For example, Newton’s second law In old notation,
In vector notation,zz
yy
xx
maF
maFmaF
amF
Equal vectors If two vectors have the same length and the same direction they are equal.
B C
The vectors B and C are equal. B = C
The length of a vector is called its magnitude. e.g. Magnitude of vector B = |B|
Unit vectors
If the length of a vector is one unit. e.g. The vector of unit length parallel to A is Â. A = |A|Â.
A unit vector is a vector that has a magnitude of exactly 1. Ex: The unit vectors point along axes in a right-handed coordinate system.
Algebra of vectors Multiplication of a Vector by a Scalar:
If we multiply a vector a by a scalar s, we get a new vector. Its magnitude is the product of the magnitude of a and the absolute value of s.
Algebra of vectors Addition of two Vectors:
Algebra of vectors Subtraction of two Vectors:
Scalar Product (“Dot” Product)
cosabba
The scalar product of the vectors and is defined as a b
)cos)(())(cos( bababa
)on of Projection)(( babba
)on of Projection)(( ababa
The above equation can be re written as
)cos)(( baba
OR
Scalar Product (“Dot” Product)
The commutative law applies to a scalar product, so we can write
abba
When two vectors are in unit-vector notation, we write their dot product as
zzyyxx
zyxzyx
bababa
kbjbibkajaiaba
)ˆˆˆ()ˆˆˆ(
then ,0 If ba
a = 0or b = 0or cos θ = 0 ( is perpendicular to )a b
Note: 2aaa
Example 1.1 Law of Cosines
BAC
)()( BABACC
cos2222
BABAC
cos2222 ABBAC
This result is generally expressed in terms of the angle φ
]cos)cos(cos[
Example 1.2 Work and the Dot product
The work W done by a force F on an object is the displacement d of the object times the component of F along the direction of d. If the force is applied at an angle θ to the displacement,
dFW
ordFW
)cos(
Vector Product (“Cross” Product) The vector product of and , written , produces a third
vector whose magnitude is
A
B
BA
C
sinBABAC
ABC
)( ABBA
Vector Product (“Cross” Product) When two vectors are in unit-vector notation, we write their cross product as
zyx
zyx
xyyxxzxzzyzy
zyxzyx
bbbaaakji
kabbajabbaiabba
kbjbibkajaiaba
ˆˆˆ
ˆ)(ˆ)(ˆ)(
)ˆˆˆ()ˆˆˆ(
Note: If and are parallel or antiparallel, .a b
0ba
Example 1.4 Area as a Vector
Consider the area of a quadrilateral formed by two vectors, andThe area of the parallelogram A is given by
A = base ×height = C D sinθ
If we think of A as a vector, we have
C
D
DC
DCA
Magnitude of A is the area of the parallelogram, and the vector product defines the convention for assigning a direction to the area.
Example 1.5 Vector Algebra
Example 1.6 Construction of a perpendicular Vector
Find a unit vector in the xy plane which is perpendicular to A = (3,5,1).
LetA = (3,5,-7)B = (2,7,1)Find A + B, A – B, |A|, |B|, A · B, and the cosine of the angle between A and B.
Base VectorsBase vectors are a set of orthogonal (perpendicular) unit vectors, one for each dimension.
Displacement and the Position Vector
kSjSiSS
kzzjyyixxS
zyxˆˆˆ
ˆ)(ˆ)(ˆ)( 121212
The values of the coordinates of the initial and final points depend on the coordinates system, S does not.
Displacement and the Position Vector
kzjyixr ˆˆˆ
Rrr
'
'')'()'(
12
12
rrRrRr
12 rrS
A true vector, such as a displacement S, is independent of coordinate system.
Velocity and Acceleration
Motion in One Dimension
12
12 )()(tt
txtxv
The average velocity ‘v’ of the point between two times, t1 and t2, is defined by
)( 1tx )( 2tx
The instantaneous velocity ‘v’ is the limit of the average velocity as the time interval approaches zero.
ttxttxv
t
)()(lim0 dt
dxv
ttvttva
t
)()(lim0
Velocity and Acceleration
In a similar fashion, the instantaneous acceleration is
dtdva
Motion in Several Dimensions
),( 111 yxr
The position of the particle At time t1, At time t2,
),( 222 yxr
The displacement of the particle between times t1 and t2 is),( 121212 yyxxrr
Motion in Several DimensionsWe can generalize our example by considering the position at some time t, and some later time t+Δt. The displacement of the particle between these times is
)()( trttrr
This above vector equation is equivalent to the two scalar equations
)()( txttxx
)()( tyttyy
Motion in Several Dimensions
dtrd
trv
t
0
lim
The velocity v of the particle as it moves along the path is defined to be
dtdy
tyv
dtdx
txv
ty
tx
0
0
lim
lim
dtdz
tzv
tz
0
lim
In 3D, The third component of velocity
Motion in Several Dimensions
kdtdzj
dtdyi
dtdx
dtrdv ˆˆˆ
kvjvivv zyxˆˆˆ
The magnitude of ‘v’ is
21
222 )( zyx vvvv Similarly, the acceleration a is defined by
dtrd
kdt
dvjdt
dvi
dtdv
dtvda zyx
2
ˆˆˆ
Motion in Several DimensionsLet the particle undergo a displacement Δr in time Δt. In the time Δt→0, Δr becomes tangent to the trajectory.
tv
tdtrdr
Δt→0, v is parallel to Δr
The instantaneous velocity ‘v’ of a particle is everywhere tangent to the trajectory.
Motion in Several Dimensions
dtrd
kdt
dvjdt
dvi
dtdv
dtvda zyx
2
ˆˆˆ
Similarly, the acceleration a is defined by
Example 1.7 Finding v from rThe position of a particle is given by
)ˆˆ( jeieAr tt
Where α is a constant. Find the velocity, and sketch the trajectory.
)ˆˆ( jeieAdtrdv
tt
tx eAv t
y eAv ==>
Solution:
The magnitude of ‘v’ is
21
22
21
22
)(
)(
tt
yx
eeA
vvv
To sketch the trajectory, apply the limiting cases.At t = 0, we get At t = ∞, we get
)ˆˆ()0(
)ˆˆ()0(
jiAv
jiAr
axis. thealong pointed vectors
ˆˆlimit In this
0
xvandriAevandiAer
eande
tt
tt
Example 1.7 Finding v from r
Note:
1. Check
2. Find the acceleration.
3. Check
4. Check
5. Check the direction of , and at t = 0 and t ∞.
)0()0( vr
)0()0( va
r v a)()( trta
Example 1.8 Uniform Circular Motion
)ˆsinˆ(cos jtitrr
Consider a particle is moving in the xy plane according to
constant
)]sin(cos[ 2/1222
r
titrr
The trajectory is a circle.
)ˆsinˆ(cos jtitrr
Example 1.8 Uniform Circular Motion
dtrdv
)ˆcosˆsin( jtitrv
0)sincoscossin(2
ttttrrv
constant. rv is perpendicular to .v r
Example 1.8 Uniform Circular Motion
dtvda
r
jtitra
2
2 )ˆsinˆcos(
The acceleration is directed radially inward, and is known as the centripetal acceleration.
Note:
1. Check
2. Check
va
ar
Kinematical equations
ttatttatttatvtvtvttvttvtvtv
)()2()()()()()2()()()(
10001
10001
Suppose - velocity at time t1 and - velocity at time t0.
Dividing the time interval (t1-t0) in n parts,
)( 1tv )( 0tv
nttt /)( 01
For n∞ (Δt0), and the sum becomes an integral:
1
0
)()()( 01
t
t
dttatvtv
Kinematical equations
The above result is the same as the formal integration of
1
0
)()()( 01
t
t
dttatvtv
] velocityintial)([,)()(
)()()(
)()(
)()(
0001
01
1
0
1
0
1
0
1
0
vtvdttavtv
dttatvtv
dttatvd
dttatvd
t
t
t
t
t
t
t
t
Kinematical equations
If acceleration a is constant and t0 = 0, we get
tavtv 0)(
200
000
21)(
)()(
tatvrtr
or
dttavrtrt
More about the derivative of a vectorConsider some vector A(t) which is function of time. The change in A during the interval from t to t + Δt is
)()( tAttAA
We define the time derivative of A by
ttAttA
dtAd
t
)()(lim0
is a new vector.
Depending on the behaviour of A:
The magnitude of can be large or small.
The direction of can be point in any direction.
dtAd
dtAd
dtAd
More about the derivative of a vectorCase 1:
. tomagnitude thechange
but unaltereddirection ||
AA
AA
Case 2:
unalteredy practicall magnitude theleavebut direction in change
AA
Case 3: In general
direction. and magnitudeboth in change willA
Case 3
More about the derivative of a vector
change.cannot magnitude its since rotate,must
, lar toperpendicu always is
A
AAΔ
change. magnitude its and same ofdirection
, toparallel always is
A
AAΔ
More about the derivative of a vector
. oflocity angular ve thecalled is /
,0limit theTaking
22
getwe.2/2/sin 1,For 2
sin2
Adtd
dtdA
dtAd
t
tA
tA
and
AA
AA
AA
More about the derivative of a vector
magnitude.in constant is if zero is / and
),0/( rotatenot does if zero is /
limit, the takingand by dividing
small,ly sufficient For
.
||
||
||
||
AdtAd
dtdAdtAd
dtdA
dtAd
dtdA
dtAd
t
AA
AA
AAA
More about the derivative of a vector
rv
rtdtdr
dtdr
dtrd
trA
or
)(
and Then . vector rotating thebe Let
constant. is of magnitude the
, lar toperpendicu is ifagain seecan we
2)(
Then .let relation, second In the
)(
)(
)(
2
A
A/dtAddtAdAA
dtd
BAdtBdAB
dtAdBA
dtd
dtBdAB
dtAdBA
dtd
dtAdcA
dtdcAc
dtd
Some formal identities
Motion in Plane Polar Coordinates
xyyxr
arctan
22
Motion in Plane Polar Coordinates
The lines of constant x and of constant y are straight and perpendicular to each other. The lines of constant θ and constant r are perpendicular.
Motion in Plane Polar Coordinates
.directions fixed have ˆ and ˆ whereas
position,vary with ˆ and ˆ ofdirection The
ji
r
cosˆsinˆˆsinˆcosˆˆji
jir
Velocity in Polar Coordinates
dtrdrrr
rrdtd
dtrdv
ˆˆ
)ˆ(
jyixjyixdtd
dtrdv ˆˆ)ˆˆ(
-scoordinatecartesian in that Recall
]ˆcosˆsinˆ[ˆ)cosˆsinˆ(
cosˆsinˆ
)(sinˆ)(cosˆˆsinˆcosˆˆ
thatknow We
ji
ji
jidtdj
dtdi
dtrd
jir
ˆˆ rrrv
Velocity in Polar Coordinates
ˆˆ rrrv
direction. )ˆ (i.e., l tangentiain the is termsecond Theoutward.radially directed
velocity theofcomponent theisright on the first term The
circle. theof arc on the liesmotion the.,.l. tangentiaisvelocity
ˆ and 0 ,constant.2direction. radial fixed ain motion the.,.
radial. isvelocity ˆ and 0 constant, 1
ei
rvrrei
rrv.
Motion in Plane Polar Coordinates
]ˆsinˆcosˆ[ˆ
)sinˆcosˆ(
sinˆcosˆ
)(cosˆ)(sinˆˆ
cosˆsinˆˆ thatknow We
rjir
ji
jidtdj
dtdi
dtd
ji
rdtddtrd
ˆˆ
ˆˆ
Acceleration in Polar Coordinates
ˆˆˆˆˆ
)ˆˆ(
dtdrrrr
dtdrrr
rrrdtd
dtvda
get we,/ˆ and/ˆ of value theSubstitute dtddtrd
rrrrrrra ˆˆˆˆˆ 2
ˆ)2(ˆ)( 2 rrrrra
termTangential | Radial
ˆˆˆˆ
vrvrrrv
r
termTangential | Radial
ˆˆˆ)2(ˆ)( 2
ararrrrra
r
Velocity and Acceleration in Polar Coordinates
References
1. Fundamentals of Physics by Halliday, Resnick and Walker
2. Berkeley Physics Course Volume-1
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