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

2024
Views:

388

Pages:

5
Author:

Jabin F.

# Action (S): S = ∫ₜ₁ₜ₂ L dt is the integration of Lagrangian w.r.t. time. First convert L into a function of time by expressing q_i, q̇_i in terms of time. Then calculate action. P-67 A particle of mass m = 0.1 kg is initially at rest at the origin. It starts moving with a uniform acceleration a = 10 m/s² at t = 0. The action S of the particle, in units of J⋅s, at t = 1 s is: P-67 A particle of mass m = 0.1 kg is initially at rest at the origin. It starts moving with a uniform acceleration a = 10 m/s² at t = 0. The action S of the particle, in units of J⋅s, at t = 1 s is: Solution: A particle is at rest at the origin. Total energy = KE + PE = 0. As constant acceleration: x = ut + (1/2)at² = (1/2)(10)t² = 5t² KE = (1/2)mv² = (1/2)(0.1)(10t)² = 5t² PE = -Sx = -5t² L = 5t² - (-5t²) = 10t² S = ∫₀¹ 10t² dt = (10/3)t³ |₀¹ = (10/3)(1)³ = 10/3 ≈ 3.33 J⋅s P-68 A projectile of mass 1 kg is launched at an angle of 30° from the horizontal direction at t = 0 and takes time T before hitting the ground. If its initial speed is 10 m/s, the value of the action integral for the entire flight in the units of J⋅s is: Solution: u = 10 m/s θ = 30° g = 10 m/s² x = utcosθ - (1/2)gt² y = utsinθ - (1/2)gt² T = (2usinθ)/g = (2 * 10 * sin(30°))/10 = 1 s KE = (1/2)mv² = (1/2)(1)(10)²(t² - t + 1) = 50(t² - t + 1) PE = mgy = 10 * 5(t - t²) = 50(t - t²) L = KE - PE = 50(t² - t + 1) - 50(t - t²) = 100t² - 100t + 50 Action = ∫₀¹ L dt = ∫₀¹ (100t² - 100t + 50) dt = (100/3)t³ - (50)t² + (50)t |₀¹ = (100/3) - 50 + 50 = 100/3 ≈ 33.33 J⋅s P-104 Consider the following transformation of the phase space coordinates: (q, p) → (Q, P), where Q = cos(bp), P = (1/a)sin(ap). For what values of a and b will the transformation be canonical? Solution: If this transformation is canonical, then it follows [Q, P] = 1. [Q, P] = (∂Q/∂q) * (∂P/∂p) - (∂Q/∂p) * (∂P/∂q) = (-b)sin(bp) * (1/a)cos(ap) - 0 * 0 = -(b/a)sin(bp)cos(ap) For this to be satisfied: ab = 1 → a = 1/b