A simple model of elasticity Solids A simple model of elasticity

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

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

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

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

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

Force and Distance distance force equil apart toward

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

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

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.

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

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.

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 =

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.

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.

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

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