PHYS 172: Modern Mechanics
Lecture 6 – Ball-Spring Model of Solids, Friction
Spring 2012
1
Read 4.1-4.8
Can we really predict the future?
BASIC IDEA
We give you the initial positions, velocities, and the interactions.
You predict everything! .... Really Everything?
PHILOSOPHICAL PROBLEMS
Is there free will?
Is there more than we can detect?
Emergence: some laws can only be discovered with 1023 particles.
PRACTICAL PROBLEMS
More than 1023 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
2
d
radial force (N)
Model of solid: chemical bonds
F ≈ linear
0
If atoms don’t move too far
away from equilibrium, force
looks like a spring force!
3
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
4
Initial conditions for circular motion
5
Length of a bond: diameter of copper atom
density ρ = 8.94 g/cm3:
molecular weight = 63.55 g/mole
NA molecules
1. Number of atoms in one cm3
8.94 g/cm3
atoms
atoms
N=
⋅ 6.022 × 1023
= 8.47 × 1022
63.55 g/mole
mole
cm3
2. Volume per one atom:
3
1
−23 cm
VCu =
= 1.18 ×10
22
3
8.47 ×10 atoms/cm
atom
3. Bond length:
dCu = 3 1.18 × 10−23 cm 3 = 2.27 × 10−8 cm=2.27 × 10−10 m=2.27 Å
6
Ball-Spring Model of a Wire
How is the stiffness of the wire related to the stiffness of one of the short
springs (bonds)?
7
Two Springs in Series
Spring constant k
Mass M
Each spring must supply an upward force equal to Mg, thus, each stretches 8by s
giving a total stretch of 2s, or an effective spring constant of k/2.
Two Springs in Parallel
Mass M
Each spring provides an upward force of Mg/2, so each stretches s/2,
giving an effective spring constant of 2k.
9
Stiffness of a Copper Wire
2-meter long Cu wire
8.77 x 109 bonds
in series
Each side = 1 mm
1.92 x 1013 chains in parallel
The stiffness of the wire is much greater than the effective
spring stiffness between atoms due to the much greater
number of chains in parallel than bonds in series.
10
Estimating interatomic “spring” stiffness
∆L
strain =
L
stress =
FT
A
tension
stress = Y ⋅ strain
Y - Young’s modulus
depends only on material
FT
∆L
=Y
A
L
Compare:
Fspring = k s s
Fspring
A
Fspring
A
A = ks
s
L
L
L s
= ks
A L
ks =
A
Y
L
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Effective interatomic spring stiffness
ks =
A
Y
L
d2
ks = Y
d
Interatomic spring stiffness
ks = Yd
12
Limits of applicability of Young’s modulus
stress = Y ⋅ strain
FT
∆L
=Y
A
L
Aluminum alloy
13
Brick on a table: compression
FN
Mg
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Friction
Exert a force so that the
brick moves to the right
at a constant speed.
What is the net force on
the brick?
18
Friction Doesn’t Always
Oppose Motion
Box dropped onto moving
conveyor belt. What happens?
How is it that a sprinter can accelerate?
19
Static Friction
• What happens when Fapplied < µkFN ?
• Block does not move due to static
friction
• In general:
µk < µs
21