Understanding Solid Behavior: Ball-Spring Model & Interatomic Forces, Summaries of Mechanics

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Spring 2012

PHYS 172: Modern Mechanics Lecture 6 –

Ball-Spring Model of Solids, Friction

Read 4.1-4.

Can we

really

predict the future?

BASIC IDEA We give you the initial positions, velocities, and the interactions.You predict everything! PHILOSOPHICAL PROBLEMS Is there free will?

.... Really Everything?

Is there free will? Is there more than we can detect?Emergence: some laws can only be discovered with 10

23

particles.

PRACTICAL PROBLEMS More than 10

23

particles in a glass of water.

Can't measure them all.

Sensitivity to initial conditions (chaos)Quantum mechanics:

Probabilities

determine outcomes

Quantum mechanics: Heisenberg uncertainty principle

A ball-spring model of a solid

Ball-spring model of a solid To model need to know:

• spring length s- spring stiffness- mass of an atom

Initial conditions for circular motion

Ball-Spring Model of a Wire

How is the stiffness of the wire related to the stiffness of one of the shortsprings (bonds)?

Two Springs in Series

Spring constant

k

Each spring must supply an upward force equal to Mg, thus, each stretches by

s

giving a total stretch of 2

s,

or an effective spring constant of

k

/2.

Mass M

8

Stiffness of a Copper Wire

2-meter long Cu wire

8.77 x 10

9

bonds

Each side = 1 mm

8.77 x 10

9

bonds

in series

1.92 x 10

13

chains in parallel

The stiffness of the wire is much greater than the effectivespring stiffness between atoms due to the much greaternumber of chains in parallel than bonds in series.

Estimating interatomic

“

spring

”

stiffness

strain

L L

stress

T F^ A =

tension

s tr

e

s s =

Y

⋅

s

tr

a

in

Y -

Young

’s modulus

depends only on material

T F

L

Y

Compare:

depends only on material

Y

A

L

spring

s

F

k

s

spring

s

F

s

A

k

L

A

L

spring

s

F

s

L

k

A

A

L

s

A

k

Y

L

Limits of applicability of Young

’s modulus

s tr

e

s s

=

Y

⋅

s

tr

a

in

T F

L

Y

A

∆ L

Aluminum alloy

Brick on a table: compression

Mg

N^

F

Friction Doesn

t Always

Oppose Motion

Box dropped onto movingconveyor belt. What happens?

How is it that a sprinter can accelerate?

Static Friction

• What happens when F

applied

μ

k

F

N

• Block does not move due to static

frictionfriction

• In general:

k

s