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Mechanics: Scalars and Vectors, Lecture notes of Classical Mechanics

Scalars, Vectors, Rules of vector algebra

Typology: Lecture notes

2018/2019

Uploaded on 08/07/2019

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Scalars and Vectors
Scalars are physical quantities specified completely by magnitude
i.e. a number possibly followed by a unit of measurement.
Examples :
Distance to brightest star Sirius 8.6 ly
Average rainfall in Cherrapunji 11777 mm
Hottest spot on earth (Lut Desert, Iran) 70oC
Heaviest weight lifted (Clean & jerk by Rezazadeh, Iran) 263 kg
Maximum speed in animal kingdom (Peregrine falcon) 389 km/h
Physics examples work done, flux, mass, electrical resistance etc.
Properties :
IRemain unchanged under coordinate transformation.
IIndependent of coordinate system used.
IOrdinary rules of arithmetic apply addition & subtraction,
multiplication & division.
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Scalars and Vectors

Scalars are physical quantities specified completely by magnitude

i.e. a number possibly followed by a unit of measurement.

Examples :

Distance to brightest star Sirius – 8.6 ly

Average rainfall in Cherrapunji – 11777 mm

Hottest spot on earth (Lut Desert, Iran) – 70

o C

Heaviest weight lifted (Clean & jerk by Rezazadeh, Iran) – 263 kg

Maximum speed in animal kingdom (Peregrine falcon) – 389 km/h

Physics examples – work done, flux, mass, electrical resistance etc.

Properties :

I (^) Remain unchanged under coordinate transformation.

I Independent of coordinate system used.

I (^) Ordinary rules of arithmetic apply – addition & subtraction,

multiplication & division.

Scalars and Vectors

Vectors are physical quantities specified completely by magnitude

and direction w.r.t. some reference.

a

^

|A|

A −A

|A|

−a

^

Examples :

Displacement, Area, Velocity, Momentum, Force, Torque, Electric

field, Magnetic field, Runge-Lenz vector etc.

Properties :

I (^) May change direction under coordinate transformation

(rotation).

I (^) Denoted by

A or A whose magnitude is |

A| or A and direction

given by

A or more often by unit vector ˆa =

A/|

A|.

I (^) Represented by a directed line i.e. an arrow in direction of ˆa.

Vectors – additional properties

Result of vector addition (for dot product wait till next slide):

C =

A +

B and C =

C ·

C

C

2 = A

2

  • B

2

  • 2AB cos θ and sin φ =

B

C

sin θ

Sometimes (or more familiar), angle φ expressed in terms

A and

B

tan φ =

B sin θ

A + B cos θ

=

A

B

A A

B B

C C

(A+B)+C

A+(B+C)

C=A+B

A+B

B+C

=

C=A+B

B

A

  1. Commutative:

A +

B =

B +

A

  1. Associative: (

A +

B) +

C =

A + (

B +

C )

A few more rules of vector algebra...

Vector multiplication –

  1. Scalar or Dot product:

A ·

A = |

A||

B| cos θ

1.1 Results in a scalar quantity.

1.2 Gives projection of one vector on another.

~ A ·

~ B = 0 and

~ A 6 = 0 6 =

~ B ⇒ θ = π/2 meaning

~ A ⊥

~ B.

1.4 Commutative:

~ A ·

~ B =

~ B ·

~ A

1.5 Distributive:

~ A · (

~ B +

~ C ) =

~ A ·

~ B +

~ A ·

~ C

1.6 Examples: Mechanical work =

~ F · ~s, Magnetic flux =

~ B · ~a.

  1. Vector or Cross product:

A ×

B = ˆn|

A||

B| sin θ

2.1 Results in a vector quantity.

2.2 Gives area of a parallelogram of sides A and B.

~ A ×

~ B = 0 and

~ A 6 = 0 6 =

~ B ⇒ θ = nπ meaning

~ A ‖

~ B.

2.4 Not commutative:

~ A ×

~ B = −

~ B ×

~ A

2.5 Distributive:

~ A × (

~ B +

~ C ) =

~ A ×

~ B +

~ A ×

~ C

2.6 Examples: Angular momentum = ~r ×~p, Lorentz force = ~v ×

~ B.

Vector applications: Example 2.

Velocity & acceleration in uniform circular motion

r

r’

v

v’

v’ (^) v

R

v

r

P

P’

An object in uniform circular motion, at two instants of time, is at

P and P

′ of position vectors ~r and ~r

′ ( |~r | = |~r

′ | = R) having

velocities ~v and ~v

′ (|~v | = |~v

′ | = v ).

Triangle law of vector addition : ~v

′ − ~v = ∆~v & ~r

′ − ~r = ∆~r

∆~v ⊥ ∆~r and if ∆t is lapsed from P to P

′ →

average acceleration ~a av

= ∆~v /∆t i.e. |~a av

| = |∆~v |/∆t.

Similar triangles : |∆~v |/v = |∆~r |/R →

|~aav| =

v

R

|∆~r |

∆t

⇒ |~a| =

v

R

lim

∆t→ 0

|∆~r |

∆t

=

v

2

R

Vectors : components and bases

Practical necessities –

  • knowing direction in conventional sense
  • figuring resultant of complicated vector system

Bring in coordinate system – Cartesian system for instance

Components of vector

A – projections of

A along positive

directions of the coordinate axes (x, y , z) → (A x

, A

y

, A

z

I

A = (A

x

, A

y

, A

z

), A =

A

2

x

+ A

2

y

+ A

2

z

I A

x

= A cos α, A y

= A cos β, A z

= cos γ, where α, β, γ are

direction cosines.

I

A =

B ⇒ A

x

= B

x

, A

y

= B

y

, A

z

= B

z

I

A ·

B = A

x

B

x

+ A

y

B

y

+ A

z

B

z

m=x,y ,z

A

m

B

m

I

A ×

B = wait a while!!

Something more about vectors...

Vectors as row / column matrices – Consider two vectors

A = (Ax , Ay , Az ) and

B = (Bx , By , Bz ) and their scalar product

can be written as,

~ A =

Ax

Ay

Az

 ,

~ B =

Bx

By

Bz

~ A ·

~ B = (Ax Ay Az )

Bx

By

Bz

 (^) = A x Bx +^ Ay By +^ Az Bz

Can you tell how unit vectors would be in column matrix notation?

Rotation of vector – Rotating a vector in a fixed basis is called

active transformation. An example is rotation of vector about

z-axis – active transformation through angle θ (counterclockwise),

A

x =^ Ax cos^ θ^ −^ Ay sin^ θ

A

y =^ Ax sin^ θ^ +^ Ay cos^ θ

A

z

= Az

A

x

A

y

A

z

cos θ − sin θ 0

sin θ cos θ 0

0 0 1

Ax

Ay

Az

 .

Vector applications: Examples

An object, starting from origin (0,0), made the following

displacements ~s 1

= (4, 4), ~s 2

= (6, −2), ~s 3

= (9, 8). Find how far

the object is from the origin and at what direction (w.r.t. x-axis)?

Vector applications: Examples

An object, starting from origin (0,0), made the following

displacements ~s 1

= (4, 4), ~s 2

= (6, −2), ~s 3

= (9, 8). Find how far

the object is from the origin and at what direction (w.r.t. x-axis)?

Net displacement

~s = ~s 1

  • ~s 2

  • ~s 3

= ˆi (4 + 6 + 9) + ˆj (4 − 2 + 8) = ˆi19 + ˆj 10

Distance from origin |~s| =

2

  • 10

2 = 21.5 units.

Vector applications: Examples

An object, starting from origin (0,0), made the following

displacements ~s 1

= (4, 4), ~s 2

= (6, −2), ~s 3

= (9, 8). Find how far

the object is from the origin and at what direction (w.r.t. x-axis)?

Net displacement

~s = ~s 1

  • ~s 2

  • ~s 3

= ˆi (4 + 6 + 9) + ˆj (4 − 2 + 8) = ˆi19 + ˆj 10

Distance from origin |~s| =

2

  • 10

2 = 21.5 units.

Direction i.e. at an angle w.r.t. x-axis

cos θ = (~s ·

i)/(|~s||

i|) = 0. 885 ⇒ θ = 27. 75

o

Vector applications: Examples

An object, starting from origin (0,0), made the following

displacements ~s 1

= (4, 4), ~s 2

= (6, −2), ~s 3

= (9, 8). Find how far

the object is from the origin and at what direction (w.r.t. x-axis)?

Net displacement

~s = ~s 1

  • ~s 2

  • ~s 3

= ˆi (4 + 6 + 9) + ˆj (4 − 2 + 8) = ˆi19 + ˆj 10

Distance from origin |~s| =

2

  • 10

2 = 21.5 units.

Direction i.e. at an angle w.r.t. x-axis

cos θ = (~s ·

i)/(|~s||

i|) = 0. 885 ⇒ θ = 27. 75

o

Two particles A and B move in opposite directions along a circle,

of radius R, with constant angular speed ω. At t = 0, they are

both at point ~r = R

j. Find the velocity of A w.r.t. B.

Relative separation

d = 2R sin(ωt)ˆi ⇒ ~v A/B

d = 2Rω cos(ωt)ˆi

Vector applications: Examples

An object, starting from origin (0,0), made the following

displacements ~s 1

= (4, 4), ~s 2

= (6, −2), ~s 3

= (9, 8). Find how far

the object is from the origin and at what direction (w.r.t. x-axis)?

Net displacement

~s = ~s 1

  • ~s 2

  • ~s 3

= ˆi (4 + 6 + 9) + ˆj (4 − 2 + 8) = ˆi19 + ˆj 10

Distance from origin |~s| =

2

  • 10

2 = 21.5 units.

Direction i.e. at an angle w.r.t. x-axis

cos θ = (~s ·

i)/(|~s||

i|) = 0. 885 ⇒ θ = 27. 75

o

Two particles A and B move in opposite directions along a circle,

of radius R, with constant angular speed ω. At t = 0, they are

both at point ~r = R

j. Find the velocity of A w.r.t. B.

Relative separation

d = 2R sin(ωt)ˆi ⇒ ~v A/B

d = 2Rω cos(ωt)ˆi

Relative velocity ~v A

− ~v B

= ~v A/B

Polar coordinate : position, velocity, acceleration

ˆr , ϕˆ different from point to point – change with ϕ but not with r

dˆr

= ˆϕ,

d ϕˆ

= −ˆr ⇒

dˆr

dt

= ˙ϕ ϕ,ˆ

d ϕˆ

dt

= − ϕ˙ ˆr

Position vector

~r = x

i + y

j = r ˆr

Velocity vector

~v =

~r = x˙

i + ˙y

j = r˙ ˆr + r

ˆr = r˙ ˆr + r ϕ˙ ϕˆ

Here, v r

= ˙r is radial velocity and v ϕ

= r ϕ˙ is angular velocity.

Acceleration vector

~a =

~v =

~r = ¨x

i + ¨y

j =

¨r − r ϕ˙

2

ˆr + (r ϕ¨ + 2˙r ϕ˙) ˆϕ

Can you identify the centripetal / centrifugal & coriolis terms?

Coordinate systems : Cylindrical

Cylindrical coordinate : (ˆρ, ϕ,ˆ

k)

Useful when problems have cylindrical symmetry! e.g. motion on

cylinder etc. (quite useful in electromagnetism)

x = ρ cos ϕ

y = ρ sin ϕ

z = z

ρ ˆ = cos ϕ

i + sin ϕ

j

ϕ ˆ = − sin ϕ

i + cos ϕ

j

k =

k

where, ρ ≥ 0 , 0 ≤ ϕ ≤ 2 π, |z| ≥ 0.

Position vector

~r = x

i + y

j + z

k = ρρˆ + z

k

Velocity vector

~v = ρ˙ ρˆ + ρ ϕ˙ ϕˆ + ˙z

k

Acceleration vector

~a =

ρ¨ − ρ ϕ˙

2

ρˆ + (ρ ϕ¨ + 2 ˙ρ ϕ˙) ˆϕ + ¨z

k