University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

127

Pages:

5
Author:

Jabin F.

# Frequency of normal modes of Double Pendulum V = -mgl [2 cos θ₁ + cos θ₂] Apply approximation of angle: V = -mgl [3 - 1/2θ₁² - 1/2θ₂²] = -mgl [3 - 1/2(θ₁² + θ₂²)] T = 1/2 ml² (θ₁² + θ₂² + 2θ₁θ₂ cosα) V = [ 0 mgl/2 ] [ θ₁ ] [ mgl/2 0 ] [ θ₂ ] T = [ ml²/2 ml²cosα/2 ] [ θ₁² ] [ ml²cosα/2 ml²/2 ] [ θ₂² ] 144 For normal mode ω = ω₁f₁ = 0 [ mgl - ω₁²ml² mgl cosα ] [ θ₁ ] = 0 [ mgl cosα mgl - ω₁²ml² ] [ θ₂ ] ⇒ (mgl - ω₁²ml²)² - (mgl cosα)² = 0 ⇒ m²g²l² (1 - ω₁²/g)² - m²g²l² cos²α = 0 ⇒ (1 - ω₁²/g)² = cos²α ⇒ 1 - ω₁²/g = ± cosα ⇒ ω₁² = g(1 ± cosα) ⇒ ω₁ = √g(1 ± cosα) ω₁ = √g(1 + cosα) ω₁ = √g(1 - cosα) **Problem P-25** A particle of mass m is moving in a potential of the form: V(x,y,z)= 2 1 mω 2 (3x 2 +3y 2 +2z 2 +2xy) The oscillation frequencies of the three normal modes of the particle are given by: **Solution** V = 1/2 mω²(3x² + 3y² + 2z² + 2xy) T = 1/2 m(ẋ² + ẏ² + ż²) V = [ 3ω²/2 ω²/2 0 ] [ x ] [ ω²/2 3ω²/2 0 ] [ y ] [ 0 0 ω² ] [ z ] T = [ 1 0 0 ] [ ẋ² ] [ 0 1 0 ] [ ẏ² ] [ 0 0 1 ] [ ż² ] For normal mode: (T - ω²V) = 0 [ 3ω²/2 - ω² ω²/2 0 ] [ x ] = 0 [ ω²/2 3ω²/2 - ω² 0 ] [ y ] [ 0 0 ω² - ω² ] [ z ] ⇒ (3ω²/2 - ω²)(3ω²/2 - ω²) - (ω²/2)² = 0 ⇒ ω²(3ω² - 2ω²) - ω⁴/4 = 0 ⇒ 3ω⁴ - 2ω⁴ - ω⁴/4 = 0 ⇒ ω²(9 - 8 - 1) = 0 ⇒ ω = 0, ±√2ω₀, ±ω₀