chapter 2( classical mechanics)

8/9/2019 Chapter 2( Classical Mechanics)

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PHYS3032

Classical Mechanics

http://teaching.phys.ust.hk/phys3032/


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Introduction

Review of Newton’s laws


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Classical Mechanics is the subject of studying whyan object (macroscopic) moves in the way it does.

Newtonian mechanics (Vectorial mechanics)

Analytical mechanics uses two scalars, the kinetic and potentialenergies, instead of vector forces, to analyze the motion

Lagrangian mechanics 1788

Hamiltonian mechanics 1833 (briefly covered in phys3032)

Isaac Newton1642-1727 Hamilton 1805-1865Irish mathematician

Lagrange, 1736-1813

French mathematician


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Why should you study Classical Mechanics ?

1. Because you love physics/math.

2. Because it is a required course.

3. Because it is one of the four fundamental subjects:Classical Mechanics,Electricity and Magnetism,

Quantum Mechanics,Statistical Mechanicsthat a physics major must have a good understanding of.

4. Hamiltonian and Lagrangian formulations connects toQuantum Mechanics, Statistical Mechanics, QuantumField Theory, General Relativity and Chaos theory.

5. Useful in engineering and technology


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How much mathematics do you need to know? Vector analysis (chapter 1):

+, -, dot product and cross product

Differentiation of a vector function .

Simple integration of a scalar function .

Gradient of scalar functions of several variables.(Mostly up to three variables – the x,y,z coordinatesof a particle).

Line (or path) integrals of a vector function (e.g.work done by a force). Complicated line integrals notrequired


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know how to calculate the “curl ” of a vectorfunction (need it on conservative forces)

used non-Cartesian coordinate systems:

Polar Coordinates, Cylindrical Coordinates,

Spherical Coordinates.

(we will review them for you)

Differential equations (only ordinary differentialequations ) (we will review them for you)

Other mathematical methods (e.g. Fourier series,matrice algebra) will be introduced when they areneeded.


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11 chapters of the textbook; some sub-topics

will be omitted

1. Fundamental (Math) Concepts: Vectors 2. Newtonian Mechanics: Rectilinear Motion of a Particle 3. Oscillations 4. General Motions of a Particle in Three Dimensions 6. Gravitation and Central Forces

5. Non-inertial Reference Systems (Rotating CoordinateSystems)

10. Lagrangian Mechanics

7. Dynamics of Systems of Particles 8. Mechanics of Rigid Bodies: Planar Motion 9. Motion of Rigid Bodies in Three Dimensions

11 Coupled Oscillators and Normal Modes


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What topics will be covered?

Newtonian Mechanics about a particle

Lagrangian MechanicsNewtonian Mechanics about systems of particles

Newtonian Mechanics about Rigid bodies

Statics, in balance,

Kinematics, how things move. no F, no m Dynamics, why things move. F = ma

0 F


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A Brief review of Newton’s Laws

0 0 F aI. (Law of inertia) Every objectcontinues in its state of rest or uniform

motion in a straight line (i.e. constantvelocity motion) unless a net force actson it to change that state.

II. The rate of change of momentum ofan object is directly proportional to theforce applied and takes place in thedirection of the force.

III. Every action has an equal andopposite reaction. 12 21 F F

( )d mv F dt

mF a

a special case ofthe 2nd law ?


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What if the 3rd Law fails?

If F12+F210, the object will

automatically accelerate disagree with the 2nd law ! We cannot survive in such

a dangerous world

F12

F21

The 2nd law F12+F21=0 the 3rd law


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Newton’s 1st and 2nd laws only hold in Inertial

Reference Frames

The observer on theground found

ma = Tsin , 2nd law holds.

An observer in the cartfounda = 0, Tsin > 0,2nd law breaks down !

The boxcar

has a constantacceleration

But what is inertial frame?


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Inertial Reference Frame An inertial reference frame is a reference frame in which

Newton’s first law is correct. (loop logic?)

One logic: 1

st

law defines a special type of reference frames,Then introduce 2nd law which only works in such frames. An equivalent logic: 1st law is a special case of F=ma, but

F=ma holds only in frames when F=0, a=0.

Do you think our Earth is such an inertial frame?(a) Yes, because we can use Newton’s first law here(b) No, but it is very close to an inertial frame

(c) No at all! The distant stars is the best approximation of an inertial

frame

Does 3rd law holds in non-inertia frames?


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Restore the 2nd

law

To restore the 2nd law,

the observer on thecart have to introduce afictitious force Ff=–ma

T

mg

ficm F Τ g F

sin 0 x F Τ ma cos 0 y F Τ mg

2nd law can be restored by including fictitious forces


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Fictitious Force

A fictitious force appears to act on an object in

the same way as a real force, but you cannotidentify the second object exert the action

A results from an accelerated frame of reference

Although fictitious forces are not real forces,they can have real effects: objects in the car doslide


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Real Forces

•Gravitational forceBetween any objects

•Electromagnetic force

•Between electriccharges, magnets•Nuclear force

Between subatomic

particles•Weak force

Arise in certainradioactive decay

processes

PHENOMELOGICALFORCES:

•Friction force•Contact force•Tension in rope

•Spring force F = -kx•Constraint force(a force that confinesan object to move along

a particular path orsurface)

FUNDAMENTAL FORCES:


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g

a

-a

effective g

fictitious force ma

gg

or

Accelerate a bottle of water,

Which figure is correct?


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Spinning Water

Spin a bucket of water

2

2 2

2

tan

2

m r dz

mg dr

dz rdr z g

r g

mg

z

r

m 2r

Fictitious

Effective g

parabolic water surface

(independent to the shapeof the container)


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Newton's Rotating Bucket (not required)

Newton designed it to demonstrate that truerotational motion cannot be defined as the relativerotation of the body with respect to theimmediately surrounding bodies.

-- Exist an absolute motionless space.

Mach: no absolute space. The curved surface is dueto the relative motion between water and the restof the universe. If earth, stars rotate around astationary water, water surface could becomecurved. Nobody can say how the experiment would

turn out if the bucket wall is massive with milesthick.

Fictitious force on water is real: exerted by therest of the universe due to the relative motion

between water and the rest of universe Mach (1838–1916)


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General Relativity (1915)

Two assumptions:

laws of physics have the same form

in all reference frames.

-- all frames have equivalent status

-- Mach's principle

Speed of light c = constant in all frames Predicted a massive rotating body drags space-time

round with it, e.g. a massive rotating hollow sphere

affects the object inside it. Confirmed in experiments around year 2000:

frame dragging effect of the rotating earth inducesa small precession of gyroscope in a satellite


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g

??? ???

Where am I?


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Einstein’s Postulation

Einstein claimed that the two situations were equivalent No local experiment can distinguish them

The principle of equivalence : a gravitational field isequivalent to an accelerated frame of reference ingravity-free space


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Einstein’s Prediction

A beam of light is bent downward in an acceleratedelevator

A beam of light should be bent by a gravitational field


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Testing General Relativity

General relativity predicts that a light ray passingnear the Sun should be deflected due to the gravityof the Sun

In 1919, Eddington took pictures of the starsin the region around the Sun during thesolar eclipse. His results confirmedEinstein’s prediction.


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Limits of Classical Mechanics


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Particle Dynamics in OneDimension


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1.The applied force is constant:Ex. The gravitational force acting on an object near the

surface of the earth.2.The applied force depends only on time:

Ex. The force acting on an electron due to

the electric field of an EM wave.3.The applied force depends only on velocity:

Ex. Air resistance acting on a moving object.

4.The applied force depends only on the positionof the particle:

Ex. The force exerted on an object attached to a spring.

Four common types of 1D problems

( )t F F

( ) xF F

F = constant

( )vF F


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Example

0

t dvm F F edt

0

t F F e

t F

dv e dt m

A block of mass m is initially at rest on a frictionless

surface at the origin. At time t=0, a decreasing force

is applied. Calculate x(t) and v(t)

From Newton’s second law,

t F ev C

m

,

F C

m (1 )t

F v e

m

(1 )t F dx vdt e dt m

( / 1/ )t F

x t em

Integrate on both sides

Using initial condition t = 0, v = 0

Integrating and using initial

condition t = 0, x = 0


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Solve the motion of an electron of mass m and charge -e,

initially at rest and interacting with electromagnetic wave

e- experiences an electric force

Integrating over time,

Integrating over time again,

0( ) sin( ).eE dv F t

a t

dt m m

0( ) sin( ). F t eE eE t

0 sin( ). E E t

0 0( ) cos( ) cos .eE eE

v t t m m

0 0 0

2 2( ) sin( ) ( cos ) sin .eE eE eE x t t t

m m m

Example

2nd law


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Case 3: F = F(v)

Integrating both sides,

This function can be inverted to

1( )

( )

dv F v m dt m dv

dt F v

1

( )t m dv C

F v

( )v v t

( ) ( ) x t v t dt Further integrate

In air, roughly F -v2

In viscous fluid like water, F -v


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Further integration

Since at t = 0, v = v0

, C1 = lnv0

Thus the complete solution is .

If x ( t = 0) = 0, one readily determines that

A cart moving on a horizontal frictionless track through amedium that produces a linear resistive (drag) force -kmv.

Example

0( ) (1 )kt v

x t e

k


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Z(t=0)=h


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Problem A bullet is shot upward. The air drag v2 . Solve x(v).

2

( ) dv

F mg v ma m dt 2nd law:

Ask for x(v), so try to transform to an equationabout x and v: times dx on both sides:

2( ) dv

mg Av dx m dx mvdvdt

0

2

22 0

0 2ln( ) ln

2 2

v

v

mvdv

dx mg Av

mg Avm m x x mg Av

A mg Av


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Case 4: F = F( x

) 2nd order, harder to solve

Rewrite as a 1st order eq. Using dx = v dt to eliminate dt and transform the

equation about dv and dx:

( )

( )

dv F x dx m dx mvdvdt

F x dx mvdv

2

2

( )( )

d x t F x m

dt

( )

dv

F x m dt

0

2 2

0

1 1( )

2 2

x

xW F x dx mv mv Work = K.E.

Kinetic energy often denoted as K, K.E. or T.


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Next, we define the potential energy of a particle whenit is at a position x by

where x0 is a fixed reference position that can be chosento be any point you like. The potential energy is zero atthe reference point x

0

.

With the above definitions, the work done by a forceon a particle is equal to the negative of the change of the

potential energy of the particle .

( ) ( ) s

x

x

V x F x dx

(2)


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the conservation of mechanical energy.

rewritten as

0

2 2

0 0

1 1( ) ( ) ( )

2 2

x

xmv mv F x dx V x V x

2 2

0 0

1 1( ) ( ) ( ) ( ) constant2 2mv x V x mv x V x E

the total mechanical energy of a particle moving in onedimension under the action of a position-dependentforce is always conserved .


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By making use of conservation of mechanical energy,the solution for any arbitrary position dependentforce F(x) can be obtained by doing an integral,

The plus or minus sign is determined by the direction ofthe initial velocity.


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Constant gravitational force

Gravitational attractive force

Coulomb force

Spring force

2( ) ( ) s s

x x

x x

GMm GMm

V x F x dx dx x x

2

0

1( ) ( )

2 s s

x x

x x

V x F x dx kxdx kx

1 2 1 2

20 0

1 1( ) ( )

4 4 s s

x x

x x

Q Q Q QV x F x dx dx

x x

0( ) ( ) s s

x x

x xV x F x dx mgdx mgx

Some typical 1D potential energies


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0

x

xDetermine the motion x(t) of ablock attached on a spring witha restoring forcef(x) = -kx

Integrate and

obtain t = t(x)

where

The potential energy for spring force is 21

( )2

V x kx

( )

1/2 1

2 1/2( 0) ( ) cos ( / )2 1[ ( )]

2

x t

x t

dx m

t x A C k E kxm

1/22( ) E

Ak

By defining and

rewrite t(x) to x(t) as

where the constant 0 is determined from the initial position

1/2( )k

m

0 C

0( ) cos( ) x t A t

0 0( 0) cos x x t A

Example

F li t d F( )’ th


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( )( )

dV x F x

dx

P.E. diagramFor complicated F(x)’s, the

integration required to find x(t)

may be very difficult or

impossible to carry out. In such

cases, we can still qualitatively

get a good picture of thepossible motions by using

energy conservation

consideration with the

aid of a P.E. diagram

force = - slope


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Some notations ofderivative

( )

Leibniz 's notation :

Lagrange's notation:

( )( )

' ' Newton 's notat

(ion :

)

y f xdy df x d

f xdx dx dx

y f x y

2 2 2

2 2 2

( )( )

'' ''( )

d y d f x d x

dx dx dx

y f x y

1st derivative 2nd derivative

chapter 2( classical mechanics)

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