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

2024
Views:

197

Pages:

5
Author:

Jabin F.

**# Conservation of Angular Momentum** If L (L = Iω) is invariant under rotation of the coordinate axes about a certain direction, then the angular momentum in that direction is conserved. If Lx remains invariant for rotation about x-axis then Lx is conserved. If Ly remains invariant for rotation about y-axis then Ly is conserved. If Lz remains invariant for rotation about z-axis then Lz is conserved. **Example:** Rotation of coordinates about y-axis. x' = x cos θ + z sin θ y' = y z' = -x sin θ + z cos θ Take dot product with x: x' . x = x cos θ + z sin θ Take dot product with z: z' . x = -x sin θ + z cos θ **x' = x cosθ + y sinθ** (1) **y' = -x sinθ + y cosθ** (2) **z' = z** (3) Squaring and adding eq (1) & (2) x'² + y'² = x² + y² This combination of x² + y² doesn't change under rotation about the z-axis. x' = x cosθ + y sinθ (1) y' = -x sinθ + y cosθ (2) z' = z (3) Squaring and adding eq (1) & (2) x'² + y'² = x² + y² Now: x'y' = (x cosθ + y sinθ)(-x sinθ + y cosθ) = -x² sinθ cosθ + xy cos²θ + xy sin²θ - y² sinθ cosθ = xy cos²θ + xy sin²θ - x² sinθ cosθ - y² sinθ cosθ = xy(cos²θ + sin²θ) - xy sinθ cosθ - xy sinθ cosθ x'y' = xy - xy cosθ sinθ x'² + y'² - x'y' = x² + y² - xy