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

2024
Views:

279

Pages:

5
Author:

Jabin F.

Wedge and Block System → all surfaces are smooth. tan θ = h / (x₁ - x₂) y₁ = y₂ = (x₁ - x₂)tan θ yg = 0 (for triangular block) d.o.f = 3 (holonomic & scleronomic) there are two generalized coordinates. T = (1/2)m(ẋ₁² + ẋ₂²) + (1/2)Mẋ₁² → let us choose x & S as generalized coordinates. A block of mass m has velocity ṡ down the inclined plane of a wedge of mass M with inclination angle θ. Wedge is also moving with velocity ẋ. Now net velocity of m: vnet = √(ẋ² + ṡ² + 2ṡẋcosθ) by cosine law T = (1/2)m₁ẋ₁² + (1/2)m₂ẋ₂² + (1/2)mṡ² + (1/2)ṡ² + ṡẋcosθ T = (1/2)(m₁ + m₂)ẋ² + (1/2)(m + m)ṡ² + ṡẋcosθ V = 0 = mg - mg sin θ X-EOM: ẋ is present but not x → x is cyclic coordinate T = (1/2)(m₁ + m₂)ẋ² + (1/2)(m + m)ṡ² + ṡẋcosθ S-EOM: ḋ = (d/dt)(∂L/∂ṡ) - ∂L/∂s = 0 → mṡ + mẋcosθ - mgsinθ = 0 ṡ + ẋcosθ - gsinθ = 0 → ṡ = gsinθ - ẋcosθ put this value in T, we get (m + m)ẋ + mcosθ(gsinθ - ẋcosθ) = 0 (m + m - mcos²θ)ẋ + mgsinθcosθ = 0 ẋ = -mgsinθcosθ / (m + m - mcos²θ) Here -ve sign indicates that wedge will move in -x direction. P-102 Generating function for an identity transformation is: Solution: Identity Transformation: qᵢ' = qᵢ, pᵢ' = pᵢ for F₁(qᵢ, pᵢ) → ∂F₁/∂qᵢ = pᵢ → ∂F₁ = pᵢdqᵢ → F₁ = qᵢpᵢ ∂F₁/∂pᵢ = 0 → ∂F₁ = 0 → F₁ = 0 for F₂(pᵢ, qᵢ) → ∂F₂/∂pᵢ = -qᵢ → ∂F₂ = -qᵢdpᵢ → F₂ = -qᵢpᵢ ∂F₂/∂qᵢ = -pᵢ → ∂F₂ = -pᵢdqᵢ → F₂ = -qᵢpᵢ F₂ = -qᵢpᵢ