1. SCI 340 L25 Hooke's law A simple model of elasticity
Solids
A simple model of elasticity
§ 10.1

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
apart
force
toward
equil
equil
apart
distance

6. Structure of Solids
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
SCI 340 L25 Hooke's law
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. Effect of Gravity Less than you might expect:
Changes equilibrium position x = 0
Does not change k

14. Elastic Potential Energy
SCI 340 L25 Hooke's law
Elastic Potential Energy

15. Potential Energy The energy of relative position of two objects
SCI 340 L25 Hooke's law
Potential Energy
The energy of relative position of two objects
gravity
springs
electric charges
chemical bonds

16. Potential Energy Energy is stored doing work against a potential
SCI 340 L25 Hooke's law
Potential Energy
Energy is stored doing work against a potential
Potential energy increases when “the potential” does negative (< 0 ) work
lifting a weight
stretching a spring

17. 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 =

18. Potential Energy of a Spring
SCI 340 L25 Hooke's law
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.
U = 1/2 kx2
The U is positive for both positive and negative x.

19. Hooke’s Law Potential Source: Young and Freedman, Figure 7.14.
SCI 340 L25 Hooke's law
Hooke’s Law Potential
Source: Young and Freedman, Figure 7.14.

20. SCI 340 L25 Hooke's law
Group Poll Question
Two springs are gradually stretched to the same final length. One spring is twice as stiff as the other: k2 = 2k1. Which spring has the most work done on it?
The stiffer spring has twice as much work done on it. ½ kx^2with the same x and different k’s. Work will be directly proportional to k.
The stiffer spring (k = 2k1).
The softer spring (k = k1).
Equal for both.

21. SCI 340 L25 Hooke's law
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?
Tension isn’t explicitly in the formula. But the force acting on the spring is F = kx, so W = ½ kx^2 = ½ Fx. Both have the same F, but the softer spring is stretched to a greater distance. Specifically, x = F/k, so W = ½ F^2/k. So half the k is twice the work, and vice versa.
The stiffer spring (k = 2k1).
The softer spring (k = k1).
Equal for both.

22. Work from Potential Energy
SCI 340 L25 Hooke's law
Work from Potential Energy
When a potential does >0 work on a body
(For a spring, this means moving toward the equilibrium length):
The body’s potential energy decreases
The body’s kinetic energy increases

23. Conservative force
A Hooke’s law force conserves mechanical energy.

24. SCI 340 L25 Hooke's law
Example Problem 7.17
A spring of negligible mass has a force constant of k = 1600 N/m.
How far must the spring be stretched/compressed for 3.32 J of potential energy to be stored in it?
You place the spring vertically with one end on the floor. You then drop a 1.2-kg book from a height of 0.8 m above the top of the spring. How far will the spring be compressed?

25. Further example problems
K = Ktr + Krot
Worksheet 5 Question 4