1. A simple model of elasticity
Solids
A simple model of elasticity

2. Objectives
Describe the deformation of a solid in response to a tension or compression.

3. What’s the point?
How do solids react when deformed?

4. Structure of Solids Atoms and molecules connected by chemical bonds
Considerable force needed to deform
compression
tension

5. Structure of Solids
Atoms are always “attracting each other when they are a little distance apart, but repelling upon being squeezed into one another”
apart
force
toward
equil
equil
apart
distance

6. Structure of Solids
Atoms are always “attracting each other when they are a little distance apart, but repelling upon being squeezed into one another”
distance
force
equil
apart
toward

7. Force and Distance
distance
force
equil
apart
toward

8. Elasticity of Solids Small deformations are proportional to force
small stretch
larger stretch
Hooke’s Law: ut tensio, sic vis (as the pull, so the stretch)
Robert Hooke, 1635–1703

9. Hooke’s Law Graph slope < 0 Force exerted by the spring
forward
slope < 0
Force exerted by the spring
backward
backward
forward
Displacement from equilibrium position

10. Hooke’s Law Formula F = –kx F = force exerted by the spring
k = spring constant; units: N/m; k > 0
x = displacement from equilibrium position
negative sign: force opposes distortion
restoring force.

11. Poll Question
forward
backward
What direction of force is needed to hold the object (against the spring) at its plotted displacement?
Forward (right).
Backward (left).
No force (zero).
Can’t tell.
forward
backward
Spring’s Force
Displacement

12. Group Work
A spring stretches 4 cm when a load of 10 N is suspended from it. How much will the combined springs stretch if another identical spring also supports the load as in a and b?
0 N
10 N
10 N
0 N
Hint: what is the load on each spring?
Another hint: draw force diagrams for each load.

13. Work to Deform a Spring To pull a distance x from equilibrium kx2
slope = k x kx
force
displacement
area = W
Work =
F·x ; 1 2
F = kx
Work =
kx·x 1 2
kx2 1 2 =

14. Potential Energy of a Spring
The potential energy of a stretched or compressed spring is equal to the work needed to stretch or compress it from its rest length.
Just as the work is always positive, so is the potential energy. This follows automatically from the x2 term.
PE = 1/2 kx2
The PE is positive for both positive and negative x.

15. Group Poll Question
Two springs are gradually stretched to the same final tension. One spring is twice as stiff as the other: k2 = 2k1. Which spring has the most work done on it?
The stiffer spring (k = 2k1).
The softer spring (k = k1).
Equal for both.

16. Reading for Next Time Vibrations Big ideas:
Interplay between Hooke’s force law and Newton’s laws of motion
New vocabulary that will also apply to waves

17. A Word