Unformatted text preview: Springy Things
Restoring Force Oscillation and Resonance Model for Molecules UCSD: Physics 8; 2006 Springs: supplying restoring force When you pull on (stretch) a spring, it pulls back (top picture) When you push on (compress) a spring, it pushes back (bottom) Thus springs present a restoring force:
F = - x k x (meters) is the displacement k (N/m) is the "spring constant" or strength of the spring. A stiff spring has a big value the negative sign means opposite to the direction of displacement Spring 2006 2 UCSD: Physics 8; 2006 Example If the springs in your 1000 kg car compress by 10 cm when lowered off of jacks: the springs must be exerting mg = 10,000 Newtons of force to support the car F = - x = 10,000 N, x = - m k 0.1 so k = 100,000 N/m (stiff spring) this is the effecitve spring constant for all the springs on the car Now if you pile 400 kg into your car, how much will it sink? 4,000 = (100,000) x, so x = 4/100 = 0.04 m = 4 cm Could have taken short-cut: springs are linear, so 400 additional kg will depress car an additional 40% (400/1000) of its initial depression How much does a spring...extend? Find the Spring constants. Spring 2006 3 UCSD: Physics 8; 2006 Energy Storage in Spring Applied force is kx (reaction from spring is - x) k starts at zero when x = 0 Increases linearly with the distance spring is compressed Work is force times distance: W=Fd Let's say we want to move spring a total distance of x would naively think W = k x2 but force starts out small (not full k x right away) works out that W = k x2 Which spring is storing more energy? Spring 2006 4 Work "Integral" UCSD: Physics 8; 2006 Since work is force times distance, and the force ramps up as we compress the spring further...
Force from spring increases as it is compressed further Area is a work: a force (height) times a distance (width) Force distance (x) Force Total work done is area of triangle under force curve Force distance (x) distance (x) takes more work (area of rectangle) to compress a little bit more (width of rectangle) as force increases (height of rectangle) if full distance compressed is k x, then force is k x, and area under force "curve" is (base)(height) = ( x)k x = k x2 area under curve is called an integral: work is integral of force
Spring 2006 5 UCSD: Physics 8; 2006 The Potential Energy Function Since the potential energy varies with the square of displacement, we can plot this as a parabola Call the low point zero potential Think of it like the drawing of a trough between two hillsides A ball would roll back and forth exchanging gravitational potential for kinetic energy Likewise, a compressed (or stretched) spring and mass combination will oscillate exchanges kinetic energy for potential energy of spring Spring 2006 6 UCSD: Physics 8; 2006 Example of Oscillation Plot shows position (displacement) on the vertical axis and time on the horizontal axis Oscillation is clear Damping is present (amplitude decreases) envelope is decaying exponential function Spring 2006 7 UCSD: Physics 8; 2006 Frequency of Oscillation frequency of oscillation is number of complete oscillations (returning to starting position) per second measured in Hertz, or cycles per second The frequency is proportional to the square root of the spring constant divided by the mass: Larger mass means more sluggish (lower freq.) Larger (stiffer) spring constant means faster (higher freq.) How does the frequency change..
Spring 2006 8 UCSD: Physics 8; 2006 Natural Frequencies & Damping Many physical systems exhibit oscillation guitar strings, piano strings, violin strings, air in flute lampposts, trees, rulers hung off edge of table buildings, bridges, parking structures Simple systems have a natural frequency at which they like to oscillate damping: energy loss mechanisms (friction, radiation) a tree has a lot of damping from air resistance cars have "shocks" (shock absorbers) to absorb oscillation energy elastic is a word used to describe lossless (or nearly so) systems "bouncy" also gets at the right idea
Spring 2006 9 If you apply a periodic force to a system at or near its natural frequency, it may resonate depends on how closely the frequency matches damping limits resonance Resonance UCSD: Physics 8; 2006 Driving below the frequency, it deflects with the force Driving above the frequency, it doesn't do much at all Picture below shows amplitude of response oscillation when driving force changes frequency Spring 2006 10 UCSD: Physics 8; 2006 Vibrations at Natural Frequency When the driving force is at the natural frequency A small force can lead to huge motions You use this idea when move your legs at the frequency of a rope swing hanging from a tree Movie Helicopter natural frequency Movie: Tacoma bridge Spring 2006 11 UCSD: Physics 8; 2006 Resonance Examples Shattering wine glass if "pumped" at natural frequency, amplitude builds up until it shatters Tacoma Narrows Bridge eddies of wind shedding off top and bottom of bridge in alternating fashion "pumped" bridge at natural oscillation frequency big lesson for today's bridge builders: include damping Spring 2006 12 UCSD: Physics 8; 2006 Wiggling Molecules/Crystals Now imagine models of molecules built out of spring connections Result is very wiggly Thermal energy (heat content) manifests itself as constant wiggling of the atoms composing molecules and crystals (solids) Important for: microwave ovens colors of materials optical properties heat conduction Spring 2006 13 UCSD: Physics 8; 2006 A model for crystals/molecules We can think of molecules as masses connected by springs Even neutral atoms attract when they are close, but repel when they get too close The trough looks just like the spring potential so the "connection" is spring-like Repulsion: energy > zero Attraction: pulled together, when energy is < zero
Spring 2006 14 UCSD: Physics 8; 2006 How fast do atoms move? A 1 kg block of wood takes 1000 J to heat by 1 C just a restatement of specific heat = 1000 J/kg/C so from 0 to 300 K, it takes 300,000 J If we assign some kinetic energy to each mass (atom), it must all add up to 300,000 J The velocities are randomly oriented, but we can still say that mv2 = 300,000 J so v2 = 600,000 (m/s)2 characteristic v = 800 m/s (very fast!) This is in the right ballpark for the velocities of atoms buzzing about within materials at room temperature it's what we mean by heat Spring 2006 15 ...
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