Rotation y Revolution
Understanding Angular Motion
Learn Acerca de RotationDesde Earth spinning on su axis un/una wheels on un/una car, rotational motion es everywhere. Understanding como nosotros medir y describe spinning objects—usando angles, angular velocity, y related concepts—es fundamental un/una physics, engineering, y cotidiano life.
Angular Position y Displacement
Angular Position
- Measured desde un/una referencia direction
- Usually in radians o degrees
- Symbol: θ (theta)
Angular Displacement
- Change in angular position: Δθ
- Positive = counterclockwise (by convention)
- Full rotation = 360° = 2π radians = 1 revolution
Units
| Unidad | Per Full Circle | Use |
|---|---|---|
| Degrees (°) | 360 | General, navigation |
| Radians (rad) | 2π ≈ 6.283 | Physics, calculus |
| Revolutions (rev) | 1 | Engineering, motors |
| Gradians (gon) | 400 | Some surveying |
Angular Velocity
Angular velocity (ω) measures como rapido something rotates.
Definition
ω = Δθ / Δt (change in angle / change in time)
Units
- rad/s: Radians per segundo (SI unidad)
- °/s: Degrees per segundo
- rpm: Revolutions per minuto (comun in engineering)
- 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
Common Rotation Speeds
| Object | Velocidad | rad/s |
|---|---|---|
| Earth rotation | 1 rev/dia | 7.27 × 10⁻⁵ |
| Hour hand | 1 rev/12 hr | 1.45 × 10⁻⁴ |
| Minute hand | 1 rev/hr | 1.75 × 10⁻³ |
| Second hand | 1 rpm | 0.105 |
| Ceiling fan (low) | 50-100 rpm | 5-10 |
| Washing machine spin | 1000-1600 rpm | 105-168 |
| Car engine (idle) | 600-1000 rpm | 63-105 |
| Car engine (highway) | 2000-3000 rpm | 210-314 |
| Hard drive (7200) | 7200 rpm | 754 |
| Dental drill | 400,000 rpm | 41,888 |
Relating Linear y Angular Motion
For circular motion, linear quantities relate un/una angular ones:
Key Relationships
- Arc length: s = rθ (θ in radians)
- Linear velocity: v = rω
- Linear acceleration: un/una = rα (tangential)
- Centripetal acceleration: un/una_c = rω² = v²/r
Ejemplo: Car Wheel
Un/Una car wheel with 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 Aceleracion
Angular acceleration (α) measures como quickly rotation speed changes.
Definition
α = Δω / Δt (change in angular velocity / time)
Units
- rad/s² (radians per segundo squared)
- rpm/s (revolutions per minuto per segundo)
Rotational Equations of Motion
Analogous un/una linear motion (with constant α):
- ω = ω₀ + αt
- θ = ω₀t + ½αt²
- ω² = ω₀² + 2αθ
Moment of Inertia y Par motor
Moment of Inertia (I)
Resistencia un/una 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
Par motor (τ)
τ = Iα (rotational analog of F = ma)
τ = r × F (force times lever arm)
Earth's Rotation y Revolution
Rotation (Daily Spin)
- Period: 23 horas, 56 minutos (sidereal dia)
- Angular velocity: 7.27 × 10⁻⁵ rad/s
- Surface speed at equator: ~1,670 km/h (1,040 mph)
- Surface speed decreases toward poles
Revolution (Yearly Orbit)
- Period: 365.25 dias
- Angular velocity: 1.99 × 10⁻⁷ rad/s
- Orbital speed: ~107,000 km/h (67,000 mph)
- Earth travels ~940 million km per ano
Aplicaciones
Gyroscopes y Navigation
- Maintain orientation via angular momentum
- Used in aircraft, smartphones, spacecraft
- Rigidity in space, precession under torque
Motors y Machines
- Motor specs incluir rpm y torque
- Gearing changes speed/torque ratio
- RPM × Par motor = Potencia
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
Conclusion
Rotation y revolution describe angular motion—spinning alrededor de internal axes versus orbiting external points. Angular velocity (medido in rad/s, rpm, o degrees/segundo) quantifies como rapido things rotate, while angular acceleration describes speed changes. These concepts connect un/una linear motion through el/la radius: v = rω. Desde Earth's rotation un/una motor specifications un/una sports physics, understanding angular motion es fundamental un/una describing el/la spinning world alrededor de us.