Why study springs? Springs are_____________. This means that if you: a/ apply a ___________to them, and b/ they become ____________ (bent out of shape), and c/ then you remove the ___________, then they d/ __________________ . Many things have some elasticity, and so they behave like springs: wood _____________ plastic concrete humans water air the sun atoms quartz speakers water ______________________________ strings, air, drums… elastic force deformed force bounce back metal musical instruments Many elastic objects obey…

___________ Law: The compression or elongation x of an ideal spring from its ________________ position (x = 0) is ____________________________to the applied force Fs. Hooke's equilibrium directly proportional Fs = Fs = kx compression: stretching or elongation: x = 0 x = 0 x x Fs Fs stretch compression More F  more ____________ or __________________.

Hooke's Law is often written: Fs = -kx This is because it also describes the force that the _______________ exerts on an ___________ that is attached to it. The negative sign indicates that the direction of the spring force is always _____________ to the displacement of the object spring itself object opposite -x compressed spring: Fs ___ 0 > Fs x = 0 ______________ position, Fs = __ equilibrium undisturbed spring +x stretched spring: Fs ___ 0 < Fs

Ex. A weight of 8.7 N is attached to a spring that has a spring constant of 190 N/m. How much will the spring stretch? w/o weight w/ weight Given: 8.7 N Fs = 190 N/m x k = Unknown: 8.7 N x = ? Equation: Fs = kx 8.7 N = (190 N/m) x x = 4.6 x 10-2 m

Fs = kx Fs direct Ex: A force of 5.0 N causes the spring to stretch 0.015 m. How far will it stretch if the force is 10 N? 10 5 2 (0.015 m) = 0.030 m .015 ? x What quantity does the slope represent? slope = = Dy/Dx Compare to Fs = kx Solve for Fs/x = k Fs/x the spring constant, k. The slope represents _______________________________

What are the units of the spring constant, k? Solve… Fs = kx …for k: k = units of k: [k] = [ ]/[ ] = (derived) This can be also seen from the graph: Fs/x Fs x N/m Fs (N) k = the slope = Dy/Dx So k has the same units as Dy/Dx: N/m x (m)

stiffer spring  _________ slope  _________ k Ex. Comparing two springs that stretch different amounts. Fs spring B spring A Applying the same force F to both springs x xB xA Which spring stretches more? Which is stiffer? A B greater stiffer spring  _________ slope  _________ k larger

____________ PE - the energy stored in a spring when work is done on it to stretch or compress it Elastic PEs = (½)kx2 Ex. A spring with a spring constant of 370 N/m is stretched a distance 6.4 x 10-2 m. How much elastic PE will be stored in the spring? How much work was done to stretch the spring by this amount?

____________ PE - the energy stored in a spring when work is done on it to stretch or compress it Elastic PEs = (½)kx2 Ex. A spring with a spring constant of 370 N/m is stretched a distance 6.4 x 10-2 m. How much elastic PE will be stored in the spring? PEs = (½)kx2 = (0.5)( 370 N/m)(6.4 x 10-2 m)2 = 0.76 (N/m)(m2) = 0.76 Nm = 0.76 J How much work was done to stretch the spring by this amount? W = DPE = 0.76 J

PEs = (½)kx2 PES prop. to square What happens to PEs when you double x? The PEs quadruples. When you triple x? 9x more PEs. x Ex: The elastic PE stored in a spring is 0.70 J when it is stretched 0.010 cm. If the same spring is stretched 0.030 cm, how much PE will then be stored in it? x changes from 0.010 to 0.030  it triples  9x more PEs  9x (0.70 J) = 6.3 J

Ex: Plot F vs. x for an ideal spring F F = kx x What does the grey area represent? area = (½)bh = = It represents the ____________ on the spring, and the ______________________ in it. (½)xF (½)x(kx) (½)kx2 PEs work done energy stored W = DPEs

Ex: By looking at the area, you can see why the PE is proportional to the ___________ of the displacement x: square F F = kx x 1 x 2 x 3 x 4 3 A 1 2 1 x 3 x 2 x ____ triangle area = ____ so PE = ____ 1 ____ triangles area = ____ so PE = ____ 4 ____ triangles area = ____ so PE = ____ 9 A 4A 9A A 4A 9A

NO NO YES YES and NO One last warning:  If a spring is stretched too much, it ____________ permanently, and Hooke's Law (Fs = kx ) _________ ______________ Ex: Which of the graphs below shows a spring that obeys Hooke's Law? deforms is no longer valid. F x F x NO NO A/ C/ YES and NO F x F x NO YES B/ D/ YES