Dynamics Final

Position along x-axis in constant velocity motion (no acceleration in x-direction)

X = X₀ + v₀ₓt

Definitions of velocity and acceleration as derivatives of displacement and velocity respectively.

v = ds/dt; a = dv/dt

Finds displacement in uniformly accelerated motion.

s = s₀ + v₀t + (1/2)at²

Velocity-displacement relationship without involving time.

v² = v₀² + 2a(s - s₀)

Relates acceleration and velocity with displacement; useful for deriving motion relationships.

a ds = v dv

Breaks total acceleration into tangential (speed change) and normal (direction change) components.

a = aₜ t̂ + aₙ n̂

Calculates centripetal (normal) acceleration when moving along a curved path of radius ρ.

aₙ = v²/ρ

Acceleration in cylindrical (polar) coordinates; useful in rotating systems or polar motion.

ρ = [1 + (dy/dx)²]³ᐟ² / |d²y/dx²|

Kinetic energy of a moving object.

EK = (1/2)mv²

Gravitational potential energy.

EP = mgh

Power: rate of doing work. Also, 1 horsepower = 550 ft·lb/s.

P = W/t = Fv

Use: Work-energy principle: initial kinetic energy + work = final kinetic energy.

T₁ + ∑U₁₋₂ = T₂

Impulse-momentum principle in vector form.

14. mv₁ + ∫F dt = mv₂

Use: Breaks impulse-momentum equation into individual axes for 3D motion.

Component form (x, y, z):

Angular velocity from constant angular acceleration.

ω = ω₀ + αt

Angular displacement in uniformly accelerated rotation.

θ = θ₀ + ω₀t + (1/2)αt²

Angular version of v² = v₀² + 2aΔs.

ω² = ω₀² + 2α(θ - θ₀)

Tangential velocity from angular velocity.

v = ωr

Angular velocity change with time when angular acceleration is known.

dω = αdt

Torque equals moment of inertia times angular acceleration.

M = Iα

Defines moment of inertia in terms of mass and radius of gyration.

I = mk²

aₜ = αr; aₙ = ω²r

Tangential and normal acceleration in rotational systems

ρ (rho)

Radius of curvature

aₙ

Normal (centripetal) acceleration

aₜ

Tangential acceleration

t̂, n̂

Unit vectors (tangential, normal)

r

Radial distance

ṙ (r dot)

Radial velocity

r double dot

radial accel

theta dot

rate of rotation

theta double dot

angular accel

radial and tangential unit vectors

ûᵣ, ûθ

Linear speed from rotation

v = ωr

Tangential acceleration in rotation

aₜ = αr

Tangential acceleration in rotation

aₙ = ω²r