Rotation e Revolution

De Earth spinning on its axis to wheels on a car, rotational motion é everywhere. Understanding como we measure e describe spinning objects—usando angles, angular velocity, e related concepts—é fundamental to física, engenharia, e cotidiano life.

Angular Position e Displacement

Angular Position

• Measured from a reference direction
• Usually in radians ou degrees
• Símbolo: θ (theta)

Angular Displacement

• Change in angular position: Δθ
• Positive = counterclockwise (by convention)
• Full rotation = 360° = 2π radians = 1 revolution

Unidades

Angular Velocidade

Angular velocity (ω) measures como rápido something rotates.

Definition

ω = Δθ / Δt (change in angle / change in time)

Unidades

• rad/s: Radians per segundo (SI unidade)
• °/s: Degrees per segundo
• rpm: Revolutions per minute (comum in engenharia)
• Hz: Cycles per segundo (1 Hz = 1 rev/s)

Conversions

• 1 rpm = 2π/60 rad/s ≈ 0.1047 rad/s
• 1 rad/s ≈ 9.55 rpm
• 1 Hz = 60 rpm = 2π rad/s

Comuns Rotation Speeds

Relating Linear e Angular Motion

For circular motion, linear quantities relate to angular ones:

Key Relationships

• Arc length: s = rθ (θ in radians)
• Linear velocity: v = rω
• Linear acceleration: a = rα (tangential)
• Centripetal acceleration: a_c = rω² = v²/r

Exemplo: Car Wheel

A car wheel com radius 0.3m rotating at 100 rad/s:

• Linear speed: v = 0.3 × 100 = 30 m/s (67 mph)
• In rpm: 100 rad/s ÷ 0.1047 ≈ 955 rpm

Angular Aceleração

Angular acceleration (α) measures como rapidamente rotation speed changes.

Definition

α = Δω / Δt (change in angular velocity / time)

Unidades

• rad/s² (radians per segundo squared)
• rpm/s (revolutions per minute per segundo)

Rotational Equations of Motion

Analogous to linear motion (com constant α):

• ω = ω₀ + αt
• θ = ω₀t + ½αt²
• ω² = ω₀² + 2αθ

Momento of Inércia e Torque

Momento of Inércia (I)

Resistência to rotational acceleration—rotational analog of mass.

• Solid cylinder: I = ½MR²
• Hollow cylinder: I = MR²
• Solid sphere: I = ⅖MR²
• Thin rod (center): I = 1/12 ML²

Torque (τ)

τ = Iα (rotational analog of F = ma)

τ = r × F (force times lever arm)

Earth's Rotation e Revolution

Rotation (Diário Spin)

• Period: 23 hours, 56 minutes (sidereal day)
• Angular velocity: 7.27 × 10⁻⁵ rad/s
• Superfície speed at equator: ~1,670 km/h (1,040 mph)
• Superfície speed decreases toward poles

Revolution (Yearly Orbit)

• Period: 365.25 days
• Angular velocity: 1.99 × 10⁻⁷ rad/s
• Orbital speed: ~107,000 km/h (67,000 mph)
• Earth travels ~940 million km per year

Aplicações

Gyroscopes e Navigation

• Maintain orientation via angular momentum
• Usado in aircraft, smartphones, spacecraft
• Rigidity in space, precession under torque

Motors e Machines

• Motor specs include rpm e torque
• Gearing changes speed/torque ratio
• RPM × Torque = Potência

Sports

• Figure skaters: spin faster by pulling arms in (conservation of angular momentum)
• Golf/tennis: club/racket angular velocity determines ball speed
• Curve balls: spin creates pressure differences

Conclusão

Rotation e revolution describe angular motion—spinning around internal axes versus orbiting external points. Angular velocity (measured in rad/s, rpm, ou degrees/segundo) quantifies como rápido things rotate, while angular acceleration describes speed changes. Estes concepts connect to linear motion through o radius: v = rω. De Earth's rotation to motor specifications to sports física, understanding angular motion é fundamental to describing o spinning mundo around us.

Artigos Relacionados

• Degrees vs Radians: Quando to Usar Cada Ângulo Unidade
• História of Ângulo Medição: De Ancient Astronomy to Modern Engenharia