physics 319 classical mechanics - jefferson...

Undergraduate Classical Mechanics Spring 2017

Physics 319

Classical Mechanics

G. A. Krafft

Old Dominion University

Jefferson Lab

Lecture 2


Scalar Vector or Dot Product

• Takes two vectors as inputs and yields a number (scalar)

• Linear in both inputs and symmetrical

• For normal (perpendicular) vectors vanishes

• Maximum when vectors parallel (co-linear)

• Get the vector norm by

b

a

cos

x x y y z z

a b ab b a

a b a b a b a b

2 2 2 2 2

x y zv v v v v v v


Vector Cross Product

• Takes two vectors as inputs and yields a normal vector

• Linear in both inputs and anti-symmetrical

• For co-linear vectors vanishes

• Maximum when vectors perpendicular

• Norm of the vector product is the area of parallelogram

spanned by the input vectors

• Important when rotations involved

b

a

3 3 3

1 1 1

ˆ

ˆ ˆ ˆ

det

ijk i j k

i j k

x y z

x y z

a b a b e

x y z

a a a

b b b

a b

sina b ab


3D Velocity and Acceleration

• Consider a particle orbit

• The velocity is

• By product rule from calculus

• Acceleration is

3

1

ˆi i

i

r t x t e t

0limt

r t t r tdrv t

dt t

3 3 3

1 1 1

ˆˆ ˆi i

i i i i

i i i

dx t de tdx t e t e t x t

dt dt dt

2 23 3 3

2 21 1 1

ˆ ˆˆ 2

i i i i

i i

i i i

d x t dx t de t d e tdva t e t x t

dt dt dt dt dt


Simplest Case

• In frames where the unit vectors do not depend on time

• For constant forces have

• Directions solved separately with the freshman physics result.

Simple expressions and equations only in inertial frame.

2

2

23

21

ˆ ˆ0

ˆ

i i

i

i

i

de t d e t

dt dt

d x ta t e

dt

2

2

2

0 0

ˆ

1

2

i i i

ii i i

d x t F e F

dt m m

Fx t x v t t

m


Incline Plane Problem


Inertial frame

• Frame of reference where the frame unit vectors do not

depend on time (there is no rotation) and where there are

no external forces and accelerations. Operational

definition: frame where observer feels no inertial forces.

• By the previous result: only if the frame is at rest or

uniformly translating at a constant velocity

• Newton’s first law: in the absence of forces a particle

moves at a constant velocity. A frame of reference tied to

the particle motion is an inertial frame.

• Newton’s second law: a particle of mass m acted on by a

vector force accelerates with the vector acceleration

F ma


What Newton Really Said

• He defined something that we call momentum today

and said the total vector force was the time derivative of

the momentum

• This formula actually works in relativistic situations if

include proper relativistic mass in the momentum

• In classical mechanics the two formulations are obviously

equivalent because

p mv

dpF

dt

dpma

dt


Newton’s Third Law

• Essentially what we now call conservation of total

momentum: if object 1 exerts a force F21 on object 2, then

object 2 exerts an equal and opposite force on object 1:

F12 = ‒F21

,

,

,0

0 0

aab ext a

b a

tot a

a

tot aab ext a

a a b a a

ext a tot

a

tottot

dpF F

dt

p p

dp dpF F

dt dt

F F

dpF

dt


Polar coordinates

• First experience with a rotating coordinate system. Unit

vectors can now have time derivatives.

• You show

2 2

1

cos

sin tan

ˆ ˆcos sinˆ ˆ ˆcos sin

x r r x y

yy r

x

r t t x r t t yrr t t x t y

r r t

ˆ ˆ ˆsin cost t x t y


Derivatives and Second Derivatives

• Derivatives

• Second Derivatives

ˆˆˆ ˆsin cos

ˆˆ ˆ ˆcos sin

dr tt x t y t

dt

d tt x t y r t

dt

2

2

2

2

2

2

ˆˆ ˆ ˆ ˆ

ˆˆˆ ˆ ˆ

d r tt t t r t

dt

d tr t r t r t t

dt


Acceleration

• Acceleration calculated from functions r(t) and θ(t)

• Resolved into cylindrical components

22

2

2 2ˆ ˆˆ ˆ ˆ2

d r t dr tda t r t r t r t t r t t r t

dt dt dt

22

2

2

2

2

2

ˆ

ˆ2

ˆ

r

z

F r t Fd rr

dt m m

F t Fdr r

dt m m

Fd z F z

dt m m


In this Rotating Coordinate System

• The form of the equations of motion is no longer the same.

It has extra terms

– Centrifugal “force” pushing away from the rotation axis

– So-called Coriolis acceleration (more general

expression later)

• When Fθ vanishes

This is the conserved angular momentum in the z

direction. Important for the planetary motion problem

(Kepler’s second law)

2mr

2mr

2 0d

mrdt


Skateboard Problem

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