Rotational and Circular Motion

Rotational Motion Overview

• Definition: Rotational motion is the turning or spinning motion of an object about a fixed axis.

• Axis of Rotation: Line around which rotation occurs; remains fixed while the object rotates.

  • Examples: pivot points, hinges.

• Motion: Each point in a rotating object moves in a circle around the axis of rotation, maintaining a consistent angular displacement.

Rotational Kinematics

• Study of Motion: Focuses on motion along circular paths without regard to forces or torques.

1. Angular Position (θ)

• Definition: The angle through which an object is displaced from a reference direction.

• Mathematical Representation:

  • When an object moves a distance S along a circle of radius r, the angular position is the angle θ.

  • Similar to identifying position in translational motion with distance x

• Example: Angle θ is marked between the position vector r and a fixed reference line (usually the +x-axis).

2. Angular Displacement (Δθ)

• Definition: The change in angular position with respect to a chosen reference direction.

• Positive/Negative Convention:

  • Positive Δθ = anti-clockwise motion.

  • Negative Δθ = clockwise motion.

• Units: radians (rad), degrees, and revolutions.

• Conversion: 1 full rotation = 360° = 2π radians.

  • 1 rad = 57.39°.

3. Angular Velocity (ω)

• Definition: The rate of change of angular displacement.

• Mathematical Formula:

  • For small angular displacement Δθ in time Δt:

  • ω = Δθ / Δt

• Units: rad/s, deg/s, rev/s.

4. Angular Acceleration (α)

• Definition: The time rate of change of angular velocity.

• Formula:

  • α = Δω / Δt.

• Units: rad/s², deg/s².

• Direction: Defined using the right-hand rule for angular velocity; positive when same direction as ω.

Relationships between Linear and Angular Quantities

• Linear vs. Angular Displacement:

  • S = r * Δθ (S = arc length, r = radius).

• Linear vs. Angular Velocity:

  • v = r * ω (tangential velocity).

• Linear vs. Angular Acceleration:

  • a = r * α (tangential and radial components).

Centripetal Acceleration and Force

• Definition: Centripetal acceleration is directed towards the center of the circular path.

  • a_c = v² / r

• Centripetal Force (F_c): The net force causing centripetal acceleration.

  • Formula: F_c = mv² / r

• Application Examples:

  • Tension for circular motion (e.g., swinging a ball on a string).

  • Gravitational forces for celestial bodies.

Moment of Inertia (I)

• Definition: A body's resistance to angular acceleration.

• Formula: I = Σ(m * r²) for discrete masses.

• Comparison to Mass: Larger moment of inertia means greater difficulty in accelerating, similar to mass in linear motion.

Angular Momentum (L)

• Definition: The product of moment of inertia and angular velocity.

• Formula: L = I * ω.

• Conservation Principle: In the absence of external torque, angular momentum remains constant.

Torque (τ)

• Definition: The effectiveness of a force to cause rotation.

• Formula: τ = r * F * sin(θ).

• Relation to Angular Acceleration: τ = I * α.

Weightlessness in Satellites

• Explanation: Astronauts experience weightlessness due to free fall in orbital motion, not the absence of gravity.

• Health Issues: Long-term exposure can weaken muscles and bones; adaptations in living conditions are necessary.

Artificial Gravity

• Creation Method: Achieved by rotating space stations around their axis to provide centripetal force mimicking gravity.

• Considerations: Required centripetal acceleration should equate to gravitational acceleration (g).

Summary of Key Concepts

• Angular Motion: Involves angular position, velocity, acceleration, and displacement.

• Centripetal Concepts: Include acceleration and forces guiding circular motion.

• Torque and Inertia: Fundamental for rotational dynamics.

• Angular Momentum: Conserved unless acted upon by torque.