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

2024
Views:

199

Pages:

5
Author:

Jabin F.

# Component of acceleration in spherical polar co-ordinates From Lagrangian mechanics: a_i = (1/m)[d/dt(∂T/∂qi̇) - ∂T/∂qi] Radial Component: a_r = (1/m)[d/dt(∂T/∂ṙ) - ∂T/∂r] T = 1/2 m(ṙ² + r²θ̇² + r²sin²θφ̇²) a_r = (1/m)[d/dt(mṙ) - mrθ̇² - mr sin²θφ̇²] a_r = r̈ - rθ̇² - r sin²θφ̇² Transverse Component: a_θ = (1/m)[d/dt(∂T/∂θ̇) - ∂T/∂θ] a_θ = (1/m)[d/dt(mr²θ̇) - mr²sinθcosθφ̇²] a_θ = (1/m)[2mṙθ̇ + mr²θ̈ - mr²sinθcosθφ̇²] a_θ = 2ṙθ̇ + rθ̈ - r sinθcosθφ̇² Azimuthal Component: a_φ = (1/m)[d/dt(∂T/∂φ̇) - ∂T/∂φ] a_φ = (1/m)[d/dt(mr²sin²θφ̇) - 0] a_φ = (1/m)[2mr sin²θφ̇ + mr²sinθcosθθ̇φ̇ + mr²sin²θφ̈] a_φ = 2ṙsinθφ̇ + rsinθcosθθ̇φ̇ + rsin²θφ̈ # Plane polar co-ordinate system (r, θ) or (ρ, φ) First two components of spherical polar (r, θ, φ) are cylindrical coordinates (r, θ). Used on a plane when there is rotational motion. Displacement: r = r̂ + rθ̂ Velocity: v̇ = ṙr̂ + rθ̇θ̂ v = √(ṙ² + r²θ̇²) K.E.: T = 1/2 m(ṙ² + r²θ̇²) Radial acc.: ar = r̈ - rθ̇² Transverse acc.: aθ = 2ṙθ̇ + rθ̈ Net acc.: a = ar r̂ + aθθ̂