Velocity in Cylindrical Coordinate System | Lecture Note - Edubirdie

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

2024
Views:

404

Pages:

5
Author:

Jabin F.

# Velocity in Cylindrical Coordinate System: v = dr/dt * êᵣ + rdθ/dt * êθ + dz/dt * êz v = ṙ * êᵣ + rθ̇ * êθ + ż * êz Radial component of velocity: vᵣ = ṙ Azimuthal component of velocity: vθ = rθ̇ Axial component of velocity: vz = ż Speed: v = √(ṙ² + r²θ̇² + ż²) Kinetic Energy: T = (1/2)mv² = (1/2)m(ṙ² + r²θ̇² + ż²) # Acceleration (a): Rate of change of velocity w.r.t. time. a = dv/dt = d²x/dt² (instantaneous) = (change in velocity) / (time taken) = (v₂ - v₁) / (t₂ - t₁) (average) Note: ê₁, ê₂, ê₃ are constant unit vectors. êᵣ, êθ, êz are variable unit vectors. In differentiation and integration, make attention on these. As if êᵣ, êθ, êz come in integration or differentiation, then we should first express these unit vectors in terms of î, ĵ, k̂. Only then we can differentiate or integrate. # Formula for Component of Acceleration from Lagrangian mechanics: a_i = (1/m) * (d/dt)(∂L/∂q̇_i) - (∂L/∂q_i) # Component of acceleration in Cartesian coordinate: a = d²r/dt² = d²/dt² (xî + yĵ + zk̂) = ẍî + ÿĵ + z̈k̂ x component of acceleration: ax = d²x/dt² = ẍ y component of acceleration: ay = d²y/dt² = ÿ z component of acceleration: az = d²z/dt² = z̈ # If vᵢ = constant then aᵢ = 0 and vice versa. If vₓ = constant then ax = 0 If vy = constant then ay = 0 If vz = constant then az = 0

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Velocity in Cylindrical Coordinate System

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