SHM SHM. T is the period or time it takes to complete 1 cycle. T = = 2π. f is the frequency or the number of cycles completed per unit time.

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1 SHM A ω = k d x x = Acos ( ω +) dx v = = ω Asin( ω + ) vax = ± ωa dv a = = ω Acos + k + x Apliude ( ω ) = 0 a ax = ± ω A SHM x = Acos is he period or ie i akes o coplee cycle. ω = π ( ω +) π = = π ω k f is he frequency or he nuber of cycles copleed per uni ie. k ω = π π Unis for f are Hz (Herz)

2 SHM - Energy K +U = consan K = v = ω A sin + K = ka sin ( ) ω + U = kx = ka cos K + U = consan = ka ( ω ) ( ω +) SHM A Connecion o Unifor Circular Moion θ = ω + ( ω ) y = Rsinθ = Rsin + ( ω ) x = Rcosθ = Rcos + A = R

3 he Siple Pendulu d θ g + θ = 0 l ( ω ) θ = θ cos + ax θ ax ω = g l Apliude is he period or ie i akes o coplee cycle. ω = π π = = π ω l g f is he frequency or he nuber of cycles copleed per uni ie. g ω = π l π Unis for f are Hz (Herz) he Siple Pendulu d θ g + θ = 0 l ( ω ) θ = θ cos + ax ( ω ) θ = ωθ sin + ax θ = ± ax ωθ ax ( ω ) θ = ω θ cos + ax ax θ = ±ω θ ax Energy of a siple pendulu: 3

Mgr ω = π I π Unis for f are Hz (Herz) he orsion Pendulu d θ γ + θ = 0 I ( ω ) θ = θ cos + ax θ ax ω = γ I Apliude is he

4 he Physical Pendulu d θ Mgr + θ = 0 I ( ω ) θ = θ cos + ax θ ax ω = Mgr I Apliude is he period or ie i akes o coplee cycle. ω = π π = = π ω I Mgr f is he frequency or he nuber of cycles copleed per uni ie. Mgr ω = π I π Unis for f are Hz (Herz) he orsion Pendulu d θ γ + θ = 0 I ( ω ) θ = θ cos + ax θ ax ω = γ I Apliude is he period or ie i akes o coplee cycle. ω = π π I = = π ω γ f is he frequency or he nuber of cycles copleed per uni ie. γ ω = π I π Unis for f are Hz (Herz) 4

DHM A L = = b α ω = π = ω d x dx + b + kx = 0 L x = Ae cos k b 4 Apliude Mean Lifeie Period Frequency ( ω + ) SHM DHM When, his is criical daping, and he value of b for which his occurs is b c : When

5 DHM A L = = b α ω = π = ω d x dx + b + kx = 0 L x = Ae cos k b 4 Apliude Mean Lifeie Period Frequency ( ω + ) SHM DHM When, his is criical daping, and he value of b for which his occurs is b c : When b > b c, overdaped When b < b c, underdaped Forced Vibraion Resonance d x + b dx x = Asin + kx = F cos ( ω +) 0 ( ω ) ω ω an( ) =, ωo ωb o = k A = F o ( ω ω ) + b ω o b A = ax a ωax = ω0 = F o cos( ω ) 5

Waves Waves ransfer energy fro one poin o anoher.

any of he paricles of he ediu being displaced

Insead here are oscillaions around fixed posiions

ypes of Waves ransverse Shear Copression Pressure

6 Waves Waves ransfer energy fro one poin o anoher. For echanical waves he disurbance propagaes wihou any of he paricles of he ediu being displaced peranenly. here is no associaed ass ranspor. Insead here are oscillaions around fixed posiions (SHM?). Requires an elasic resoring force. ypes of Waves ransverse Shear Copression Pressure Surface Waves on waer Wave diensions D ransverse D Surface 3D Sound 6

v is he wave velociy or he speed a which a waves ravels.

7 Basic Definiions Apliude is he axiu peak or rough value. λ is he wavelengh or disance fro peak o peak. is he period or ie i akes o for one wavelengh o pass a given poin. f is he frequency or he nuber of waves ha pass a given poin per uni ie. v is he wave velociy or he speed a which a waves ravels. λ v = λ f = v = Velociy, Energy, Power, and Inensiy F μ F is he ension, μ is he ass/uni lengh. v = B ρ B is he bulk odulus, ρ is he densiy. v = E ρ Power Inensiy = Area E is he elasic odulus, ρ is he densiy. Energy = kdm = π f D Energy = π μvf D Power = π μvf D M M M Energy π f D Power = = ie M 7

8 he Wave Equaion D D x ( x, ) D( x, ) = v ( x, ) = D ( k x ± ω + ) M sin - wave oves in +x + wave oves in -x D M ω πf π k = = = v λf λ ω Max Apliude Wave nuber he following syses exhibi siple haronic oion. Wha is he period of each oion? (a) A oy ighrope walker, wih geoery as shown, whose body has ass uch less han ha of he barbell weighs, each of ass 50 g. he oy sways fro side o side in haronic oion. (b) A ass aached o wo parallel springs, each of spring consan k. Ignore graviy. (c) A spring wih spring consan k and a ass aached o each end. (d) A ass hanging verically fro a spring, of spring consan k, under he influence of graviy. 8

Thus the force is proportional but opposite to the displacement away from equilibrium.

Thus the force is proportional but opposite to the displacement away from equilibrium. Chaper 3 : Siple Haronic Moion Hooe s law saes ha he force (F) eered by an ideal spring is proporional o is elongaion l F= l where is he spring consan. Consider a ass hanging on a he spring. In equilibriu