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

2024
Views:

63

Pages:

5
Author:

Jabin F.

#Coupled Pendulum: Here θ₁ = θ₂ = θ For small oscillations: θ₁ = θ₂ = θ̇ Similarly, x₁ = lθ₁ = lθ x₂ = lθ₂ = lθ T = 1/2 m(lθ̇)^2 + 1/2 m(lθ̇)^2 + 1/2 k(lθ - lθ)^2 = ml^2θ̇^2 + 1/2 kl^2θ^2 U = -mgl(1 - cosθ) - mgl(1 - cosθ) + 1/2 kl^2θ^2 = -2mgl(1 - cosθ) + 1/2 kl^2θ^2 V = T + U = ml^2θ̇^2 - 2mgl(1 - cosθ) + 1/2 kl^2θ^2 = ml^2θ̇^2 - 2mgl + 2mglcosθ + 1/2 kl^2θ^2 For normal mode: [ml^2 - (mg + kl)l^2]ω^2 = 0 [mg + kl] - ω^2l^2 = 0 ω^2 = (mg + kl) / l^2 ω = √((mg + kl) / l^2) (mg + kl - ω^2ml)^2 - (kl)^2 = 0 (mg + kl - ω^2ml)^2 = (kl)^2 mg + kl - ω^2ml = ±kl ω^2 = (mg + kl ± kl) / ml ω = √((mg + 2kl) / ml) or √(mg / ml) ω = √(g/l) ± √(k/m) ∴ dV/dx = e^(-x^4) - 3x = 0 ⇒ x = 0 d²V/dx² = e^(-x^4) - 12x^3 + 9x^4e^(-x^4) |_{x=0} = 9 k = d²V/dx²|_{x=0} = 9; m = 1 ⇒ ω = √(k/m) = √9 = 3 rad/s