Dynamics

Motion of bodies and the action of forces, including velocity, acceleration, forces, and torque.

Key Equations

Newton's Laws

F = ma

ΣF = 0 (equilibrium)

F₁₂ = -F₂₁

Kinematics

v = v₀ + at

s = v₀t + ½at²

v² = v₀² + 2as

Applications

• Machine Design

• Vehicle Dynamics

• Robotics

• Aerospace Systems

• Manufacturing Equipment

• Structural Analysis

Key Concepts

• Force and Motion

• Energy and Work

• Momentum

• Rotational Motion

• Oscillations

• Impact and Collision

Linear Dynamics

Kinematics Equations

Constant Acceleration:

v = v₀ + at

s = v₀t + ½at²

v² = v₀² + 2as

s = ½(v₀ + v)t

Variable Acceleration:

a = dv/dt = d²s/dt²

v = ds/dt

s = ∫v dt

Newton's Laws

First Law (Inertia):

An object at rest stays at rest, and an object in motion stays in motion at constant velocity, unless acted upon by a net external force.

Second Law:

F = ma = m(dv/dt)

Third Law:

For every action, there is an equal and opposite reaction.

Rotational Dynamics

Angular Kinematics

Angular Motion:

ω = ω₀ + αt

θ = ω₀t + ½αt²

ω² = ω₀² + 2αθ

θ = ½(ω₀ + ω)t

Linear-Angular Relations:

v = rω

a = rα

s = rθ

Torque and Moment

Torque:

τ = r × F = rF sin θ

τ = Iα (rotational analog of F = ma)

Στ = 0 (rotational equilibrium)

Moment of Inertia:

I = Σmᵢrᵢ² (discrete masses)

I = ∫r²dm (continuous mass)

I = mk² (radius of gyration)

Energy and Work

Work-Energy Theorem

Linear Motion:

W = F·s = Fs cos θ

W = ΔKE = ½mv² - ½mv₀²

KE = ½mv²

PE = mgh (gravitational)

Rotational Motion:

W = τθ

KE = ½Iω²

W = ΔKE = ½Iω² - ½Iω₀²

Power

Linear Power:

P = dW/dt = F·v

P = Fv cos θ

Average Power = W/t

Rotational Power:

P = τω

P = dW/dt = τ(dθ/dt)

Momentum and Impulse

Linear Momentum

Momentum:

p = mv

F = dp/dt

Impulse = FΔt = Δp

Conservation:

Σp = constant (no external forces)

m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

Angular Momentum

Angular Momentum:

L = Iω = r × p

τ = dL/dt

Angular Impulse = τΔt = ΔL

Conservation:

ΣL = constant (no external torques)

I₁ω₁ + I₂ω₂ = I₁ω₁' + I₂ω₂'

Oscillations and Vibrations

Simple Harmonic Motion

SHM Equations:

x = A cos(ωt + φ)

v = -Aω sin(ωt + φ)

a = -Aω² cos(ωt + φ)

F = -kx

Parameters:

ω = √(k/m)

T = 2π/ω = 2π√(m/k)

f = 1/T = ω/(2π)

Damped Oscillations

Damped Motion:

mẍ + cẋ + kx = 0

ζ = c/(2√(km)) (damping ratio)

ωd = ωn√(1 - ζ²)

Types:

ζ < 1: Underdamped

ζ = 1: Critically damped

ζ > 1: Overdamped

Common Moment of Inertia Values

Point Mass:

I = mr²

Solid Cylinder:

I = ½mr²

Hollow Cylinder:

I = mr²

Solid Sphere:

I = ⅖mr²

Hollow Sphere:

I = ⅔mr²

Thin Rod (center):

I = 1/12 mL²

Thin Rod (end):

I = ⅓mL²

Rectangular Plate:

I = 1/12 m(a² + b²)

Disk (center):

I = ½mr²

Impact and Collision

Collision Types

Elastic Collision:

Momentum conserved: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

KE conserved: ½m₁v₁² + ½m₂v₂² = ½m₁v₁'² + ½m₂v₂'²

e = 1 (coefficient of restitution)

Inelastic Collision:

Momentum conserved

KE not conserved

0 ≤ e < 1

Coefficient of Restitution

Definition:

e = (v₂' - v₁')/(v₁ - v₂)

e = relative velocity after / relative velocity before

Values:

e = 1: Perfectly elastic

e = 0: Perfectly inelastic

0 < e < 1: Partially elastic