Rotational Motion & Types of Motion

Definitions and Overall Classification of Motion

Physics traditionally separates motion into two macro–categories:
- Translational Motion: Every point of an object follows the same path; the body changes position as a whole.
- Rotational Motion: The body spins about an internal or external axis; points on the object trace circles about that axis.
Circular motion can appear in either category:
- As a special path of translational motion (whole object goes around another center – e.g.
  Earth around Sun).
- As the natural path of individual particles during rotation (every particle except those on the axis moves in a circle).

Path possibilities:
- Straight-line (Linear) Path
- Simplest case; constant direction.
- Example sketch: A \rightarrow B.
- Circular Path
- Body’s center of mass traces a circle around an external point.
- Illustration: Earth – Sun system.
Qualitative features of circular translational motion:
- Two speed patterns
1. Uniform Circular Translation
  - Magnitude of velocity v is constant (only direction changes).
  - Example numbers written in transcript: 20\,\text{m/s} \rightarrow 20\,\text{m/s} \rightarrow 20\,\text{m/s} every quarter-turn.
2. Non-Uniform Circular Translation
  - v changes with time (accelerates or decelerates).
  - Sketch shows 20\,\text{m/s} then 40\,\text{m/s} at symmetric locations, signaling speed variation.
Conceptual emphasis:
- Translational axis (any straight line fixed in the body) moves along with the object.
- Useful for analyzing projectiles, satellites, or vehicles on curved tracks.

Defining idea: motion of a body that spins about an axis.
Key properties
- All points move in circles centered on the axis.
- The axis itself can be fixed in space, moving, real, or imaginary (see later headings).
Notation reminder from syllabus: angular displacement \theta (rad), angular velocity \omega (rad/s), angular acceleration \alpha (rad/s^2).

Because every point in a rotating rigid body travels in a circle, circular motion is a special case of rotational kinematics.
Distinguish from the earlier “circular translational” case by examining whether the center of the circle is inside the object (rotation) or outside (translation).

A second layer of classification (applies to both rotating objects and objects translating around an external center):
1. Horizontal Circle
- Plane of motion is horizontal; height remains constant (single level).
- Transcript note: “one height (same).”
- Examples: spinning record, tether ball going around at constant elevation, amusement-park swing in steady state.
1. Vertical Circle
- Plane of motion is vertical; height (gravitational potential) varies continuously.
- Transcript annotation: “different height; top ↑, bottom ↓.”
- Examples: roller-coaster loop, pendulum, bucket of water swung in a vertical loop.
Physical implications:
- In vertical circles, weight plays a role: at the top tension/normal force is reduced; at the bottom it is increased.
- Horizontal circles mostly involve centripetal forces without weight-component changes.

Axis = straight line about which the body turns; can be:
- Fixed in Space (laboratory frame)
- Ferris wheel axle: vertical line perpendicular through the center; does not translate.
- Ceiling fan: axis aligned along the central rod; points on the rod itself have zero linear velocity.
- Moving With the Object (body axis)
- Rolling ball: axis goes through ball, but both ball and axis translate across floor.
Orientation references in transcript sketches:
- “Center Perpendicular” for Ferris wheel = axis exits board toward viewer.
- “Side Parallel” indicates axis lying parallel to the board/screen.

Real (Physical) Axis
- Material rod, spindle, or axle you can touch.
- Examples: Ferris-wheel support beam, motor shaft.
Imaginary Axis
- Geometric line internal to the body, not a separate physical piece.
- Examples:
- Planetary spin axis inside Earth.
- Any diagonal line through a tossed-bat about which it tumbles.
Both obey the same kinematic relations; distinction matters for engineering practicalities (bearings, torque transmission).

Ferris wheel example ties classroom definitions to amusement-park engineering.
Planet rotation vs. revolution (spin vs. orbital path) connect to astronomy and gravitational discussion in later parts of Chapter 1.
Understanding vertical circles preludes to energy conservation, tension calculations, and looping aircraft maneuvers.

Uniform-speed sketches: constant v = 20\,\text{m/s}.
Non-uniform example: 20\,\text{m/s} then 40\,\text{m/s} representing acceleration.
Symbolic angular metrics were not yet provided on these pages, but recall relations:
v = r\,\omega, a_c = \frac{v^2}{r} = r\,\omega^2, \Delta\theta = \omega\,\Delta t + \tfrac12 \alpha\,\Delta t^2.

Fixed vs. moving axes influence ease of calculation (inertial vs. non-inertial frames).
Imaginary axes remind us that physics often uses idealized constructs to simplify problems, underscoring the conceptual nature of “motion.”
Recognizing the dual use of the term “circular motion” (translation vs. rotation) prevents common student misconceptions and supports clearer problem-solving.