1

x(t) = x_m cos(ωt + φ)T = 1fω = 2πT = 2πf

2

v(t) = dxdt = −ωx_m sin(ωt)a(t) = dvdt = −ω²x_m cos(ωt)a(t) = −ω²x(t)

3

F = ma = −mω²xF = −kx → k = mω²ω = kmT = 2πmk

4

U = (1/2)kx²K = (1/2)mv²U + K = (1/2)kx²_m(cos² + sin²) = (1/2)kx²_m

5

mg − kx₀ = 0md²xdt² = mg − k(x + x₀) = −kx

6

d²θdt² = −gL sin θ ≈ −gLθT = 2πLgU = mgL(1 − cos θ)U + K = (1/2)mA²Lg

7

τ = Iα = −mgh sin θ ≈ −mghθω = mghIT = 2πImgh

8

md²xdt² + bdxdt + kx = 0x(t) = x_me^−bt/(2m) cos(ω′t + φ)ω′ = km − b²4m²E ≈ (1/2)kx²_me^−bt/m

9

10

y(x, t) = y_m sin(kx − ωt + φ)k = 2πλω = 2πT = 2πfv = ωk = λT = λf

11

F = 2τ sin θ ≈ τΔlRF = Δmv²R = μΔlv²Rv = τμ

12

dKdt = 12μvω²y²_m cos²(kx − ωt)P_avg = 2dKdt_avg = 12μvω²y²_m

13

∂²y∂x² = 1v²∂²y∂t²y = f(kx − ωt),  v = ω/k

14

y + y′ = (2y_m sin kx) cos ωtx_node = nπkx_antinode = (n + 12)πk

15

λ_n = 2Ln (string, open tube)λ_n = 4L2n − 1 (closed tube)f = vλ,  v = τμ

16

s(t) = [2s_m cos ω′t] cos ωtω′ = ω₁ − ω₂2,  ω = ω₁ + ω₂2f_beat = f₁ − f₂

17

f′ = fv ± v_Dvf′ = fvv ∓ v_Sf′ = fv ± v_Dv ∓ v_S

18

sin θ = vtv_St = vv_SMach = v_S/v