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

2024
Views:

217

Pages:

5
Author:

Jabin F.

# Velocity & Acceleration in Different Coordinate Systems In a general co-ordinate system, infinitesimal displacement is written as: d\vec{l} = h_1 dq_1 \hat{e}_1 + h_2 dq_2 \hat{e}_2 + h_3 dq_3 \hat{e}_3 where: q_1, q_2, q_3 - coordinates h_1, h_2, h_3 - scale factors \hat{e}_1, \hat{e}_2, \hat{e}_3 - unit vectors Coordinate System q_1 q_2 q_3 h_1 h_2 h_3 \hat{e}_1 \hat{e}_2 \hat{e}_3 Cartesian x y z 1 1 1 \hat{i} \hat{j} \hat{k} Spherical Polar r θ φ 1 r r sinθ \hat{r} \hat{θ} \hat{φ} Cylindrical r φ z 1 r 1 \hat{r} \hat{φ} \hat{z} Експортувати в Таблиці # Velocity in Cartesian Coordinate System: \vec{v} = \frac{d\vec{l}}{dt} = \frac{dx\hat{i} + dy\hat{j} + dz\hat{k}}{ Velocity v = ẋi + ẏj + żk x-component of velocity (v_x) = ẋ y-component of velocity (v_y) = ẏ z-component of velocity (v_z) = ż |v| = √(ẋ² + ẏ² + ż²) Kinetic Energy (Non-relativistic) Kinetic Energy: T = 1/2 mv² = 1/2 m(ẋ² + ẏ² + ż²) Note: Relativistic Kinetic Energy: KE = mc² - m₀c² # Velocity in Spherical Polar Coordinate System v = dl/dt = dr * dr + r * dθ * dθ + r * sinθ * dφ * dφ v = dr/dt * r + r * dθ/dt * θ + r * sinθ * dφ/dt * φ Radial Component of Velocity: v_r = ṙ Transverse Component of Velocity: v_θ = rθ̇ Azimuthal Component of Velocity: v_φ = r sinθ φ̇ |v| = √(ṙ² + r²θ̇² + r²sin²θφ̇²) Kinetic Energy T = 1/2 m|v|² T = 1/2 m(ṙ² + r²θ̇² + r²sin²θφ̇²)