UNIVERSITY PHYSICS I CHAPTER 7 Simple Harmonic Oscimation §7.1hook' s force law 1. Ideal model-spring oscillator k 0000000m x A x=0 +A elasticity inertia Equilibrium position 2. Hook's force law Horizontal direction: =-k i 1
1 §7.1 Hook’s force law 1. Ideal model—spring oscillator Equilibrium position − A + A x Horizontal direction: 2. Hook’s force law F kxi ˆ s = − r elasticity inertia
87.2 Simple harmonic oscillation 1. Differential equation of a simple harmonic oscillation and its solution F=lxi =ma -kri=m-i dt d'x +kx=0 cco0000mi/ General Solution: x=0 x x(t=acos at+bsin at A=(a +b or x(=Acos(at+o) tang= 87.2 Simple harmonic oscillation 2. The quantities describing the oscillation Displacemnt at time t Phase x()=Acos(ot+φ) Amplitude Time m Angular frequency constant angle
2 §7.2 Simple harmonic oscillation 1. Differential equation of a simple harmonic oscillation and its solution x 0 d d 2 2 + x = m k t x i t x kxi m ˆ d d ˆ 2 2 Fx kxi max − = r r = − =ˆ General Solution: or ( ) cos( ) ( ) cos sin ω φ ω ω = + = + x t A t x t a t b t a b A a b = − = + tanφ ( ) 2 2 1 2 §7.2 Simple harmonic oscillation A m x 2. The quantities describing the oscillation
87.2 Simple harmonic oscillation Angular frequency, period and frequency dx k k a2x(t)+x()=0 dt x(r)=Acos(ar+小)∴.a2 =—or Angular frequency a is related to the spring constant and the mass. It is decided by the nature of the system. Unit is rad/s The period T: x(t)=Acos(at+o) x(t+T)=Acos[o(t+r)+oI 87.2 Simple harmonic oscillation 2丌 OT=2丌T Unit is s Frequency v: v=2= 0=2元V 2兀 Unit is 1/s or Hertz(Hz). ② Amplitude dx(t) x(t)=Acos(at+o) v(t)=dr -Asin(at+o) Fo=x(0=Acos t=0 vo=v(0)=-A@sin o A=xm=x5+ The range of the oscillation is 2A=2x
3 §7.2 Simple harmonic oscillation ① Angular frequency , period and frequency 0 d d 2 2 + x = m k t x x(t) = Acos(ωt +φ ) m k m k x t m k x t ∴ = = − + = ω ω ω or ( ) ( ) 0 2 2 Q Angular frequency ω is related to the spring constant and the mass. It is decided by the nature of the system. Unit is rad/s. The period T : ( ) cos[ ( ) ] ( ) cos( ) ω φ ω φ = + = + + = + x t T A t T x t A t §7.2 Simple harmonic oscillation ω π ω π 2 T = 2 T = Unit is s. Frequency ν : ω πν π ω ν 2 2 1 = = = T Unit is 1/s or Hertz (Hz). ②Amplitude x(t) = Acos(ωt +φ ) sin( ) d d ( ) ( ) = = −Aω ωt +φ t x t v t t =0 ω φ φ (0) sin (0) cos 0 0 v v A x x A = = − = = 2 2 2 0 0 ω v A x x = m = + The range of the oscillation is 2A=2xm
87.2 Simple harmonic oscillation Initial phase angle and phase t=0 o=(0)=Acos d 如=tanx(-" v=v(0)=- Asinφ o describe the initial state of the spring oscillator It is called initial phase or phase constant A and is related to the initial states or conditions of the system at+o is called the phase of the motion. It describes the states of the oscillation system 87.2 Simple harmonic oscillation x(t)=Acos(at+p) v()> dx(t)_-A@sin(at+o) dt at+o=T/3 x(t)=Ai v(t) oAi aH-m3x()2写训=Y3 3. The graphs ofx(o), v(o) and a(t) of simple harmonic oscillation
4 §7.2 Simple harmonic oscillation ③Initial phase angle and phase t =0 ω φ φ (0) sin (0) cos 0 0 v v A x x A = = − = = tan ( ) 0 1 0 ω φ x v = − − φ describe the initial state of the spring oscillator. It is called initial phase or phase constant. A and φ is related to the initial states or conditions of the system. ωt+φ is called the phase of the motion. It describes the states of the oscillation system. x(t) = Acos(ωt +φ ) sin( ) d d ( ) ( ) = = −Aω ωt +φ t x t v t §7.2 Simple harmonic oscillation ωt+φ=π/3 x t Ai v t Ai ˆ 2 3 ( ) ˆ 2 1 ( ) = = − ω r r ωt+φ= -π/3 x t Ai v t Ai ˆ 2 3 ( ) ˆ 2 1 ( ) = = ω r r 3. The graphs of x(t), v(t) and a(t) of simple harmonic oscillation
87.2 Simple harmonic oscillation x(t)=Acos(@t +o) +4 v(= dx(t)=-Aasin(at+P)E o d d'x(t (a) a()= +a4 dt -A@ cos(at +o) (b) +x.p=0a=-a x=0v=-V.a=0 x=-X.=0a=+a 87.2 Simple harmonic oscillation 4. How to determine if an oscillatory motion is simple harmonic oscillation? (criterion) Criterion 1: F total One force or the sum of several forces Criterion 2: k +“x=0 Criterion 3: x(t)=Acos(at+o)
5 §7.2 Simple harmonic oscillation x(t) = Acos(ωt +φ ) sin( ) d d ( ) ( ) = = −Aω ωt +φ t x t v t cos( ) d d ( ) ( ) 2 2 2 = − ω ω +φ = A t t x t a t m m m m m x x v a a x v v a x x v a a = − = = + = = − = = + = = − 0 0 0 0 +A +A +ωA −ωA A 2 +ω A 2 +ω φ=0 4. How to determine if an oscillatory motion is simple harmonic oscillation?(criterion) F kxi ˆ total = − r Criterion 1: One force or the sum of several forces Criterion 2: 0 d d 2 2 + x = m k t x Criterion 3: x(t) = Acos(ωt +φ ) §7.2 Simple harmonic oscillation