Rotation and Revolution

変換元 Earth spinning on its axis to wheels on a car, rotational motion is everywhere. 理解する how we measure and describe spinning objects—using angles, angular velocity, and related concepts—is fundamental to physics, engineering, and everyday life.

Angular Position and Displacement

Angular Position

• Measured from a reference direction
• Usually in radians or degrees
• Symbol: θ (theta)

Angular Displacement

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

Units

Angular Velocity

Angular velocity (ω) measures how fast something rotates.

Definition

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

Units

• rad/s: ラジアン per second (SI単位)
• °/s: 度 per second
• rpm: 回転 per minute (common in engineering)
• Hz: Cycles per second (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

Relating Linear and 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

Example: Car Wheel

A 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 Acceleration

Angular acceleration (α) measures how quickly rotation speed changes.

Definition

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

Units

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

Rotational Equations of Motion

Analogous to linear motion (with constant α):

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

Moment of Inertia and 変換先rque

Moment of Inertia (I)

Resistance 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

変換先rque (τ)

τ = Iα (rotational analog of F = ma)

τ = r × F (force times lever arm)

Earth's Rotation and Revolution

Rotation (Daily Spin)

• Period: 23 hours, 56 minutes (sidereal day)
• 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 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

Applications

Gyroscopes and Navigation

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

Motors and Machines

• Motor specs include rpm and torque
• Gearing changes speed/torque ratio
• RPM × 変換先rque = Power

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

まとめ

Rotation and revolution describe angular motion—spinning around internal axes versus orbiting external points. Angular velocity (measured in rad/s, rpm, or degrees/second) quantifies how fast things rotate, while angular acceleration describes speed changes. These concepts connect to linear motion through the radius: v = rω. 変換元 Earth's rotation to motor specifications to sports physics, understanding angular motion is fundamental to describing the spinning world around us.

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