1. Summer 2011 PHYS 172: Modern Mechanics Lecture 6 – Ball-Spring Model of Solids, Friction Read 4.1-4.8

2. Model of solid: chemical bonds If atoms don’t move too far away from equilibrium, force looks like a spring force! 0 radial force (N) d

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 N A molecules 1. Number of atoms in one cm 3 density  = 8.94 g/cm 3 : molecular weight = 63.55 g/mole 2. Volume per one atom: 3. Bond length: angstroms

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 Each spring must supply an upward force equal to Mg, thus, each stretches by s giving a total stretch of 2s, or an effective spring constant of k/2. Mass M Spring constant k

8. 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 Each side = 1 mm 2-meter long Cu wire 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 effective spring stiffness between atoms due to the much greater number of chains in parallel than bonds in series.

10. Compare: Estimating interatomic “spring” stiffness tension Y - Young’s modulus depends only on material

11. Effective interatomic spring stiffness Interatomic spring stiffness

12. Limits of applicability of Young’s modulus Aluminum alloy Demo

13. Brick on a table: compression

14. Friction Exert a force so that the brick moves to the right at a constant speed. What is the net force on the brick?

15. Friction Doesn’t Always Oppose Motion Box dropped onto moving conveyor belt. What happens? How is it that a sprinter can accelerate?

16. Sliding Friction When one object slides on another, the component of force exerted by one object on the other has a component parallel (or antiparallel) to the motion: –f friction ~  k F N  k is the coefficient of kinetic friction F N is the “normal force” – the perpendicular component of the force that is squeezing the two objects into each other

17. Static Friction What happens when F applied <  k F N ? Block does not move due to static friction In general: –  k <  s