Overview
This lecture explains how circular and rotational motion are described using tangential and angular variables, how to convert between these descriptions, and highlights key relationships and formulas.
Types of Motion
- Circular motion is when an object moves along a circular path centered some distance away.
- Rotational motion is when an object rotates about its own center or axis.
- Examples: A car driving in a circle exhibits circular motion; a record spinning shows rotational motion.
Description of Motion
- The description of motion refers to how we measure and analyze motion (choice of axis, units, directions).
- The real motion is independent of the description we choose.
Tangential vs. Angular Descriptions
- Tangential description measures motion along the circumference in units of length (meters).
- Angular description measures motion as angles swept out, in radians or degrees.
Applying Descriptions
- Circular motion usually uses tangential variables: tangential displacement (Δs), velocity (v), and acceleration (a).
- Circular motion can also use angular descriptions: angular displacement (Δθ), angular velocity (ω), and angular acceleration (α).
- Rotational motion typically uses angular descriptions.
- A point on a rotating object (ex: a fly on a record) is in circular motion and can be described tangentially.
Key Relationships & Formulas
- All points on a rotating object have the same angular velocity, but different tangential velocities depending on radius.
- Arc length (Δs) = radius (r) × angular displacement (Δθ) [in radians]: Δs = rΔθ
- Tangential velocity (v) = radius × angular velocity: v = rω
- Tangential acceleration (a_tan) = radius × angular acceleration: a_tan = rα
- When converting, always use radians for angular variables.
Examples
- Two objects with the same angular velocity but different radii have different tangential velocities.
- If objects move at the same tangential velocity but on different radii, the one with the smaller radius has higher angular velocity.
Key Terms & Definitions
- Circular motion — movement along a circular path with a center outside the object.
- Rotational motion — spinning about the object’s own axis.
- Tangential (linear) variables — describe motion along the path (Δs, v, a).
- Angular variables — describe motion in terms of angles (Δθ, ω, α).
- Radius (r) — distance from center to point on the circle.
- Radian — angle subtended by an arc equal to the circle’s radius.
Action Items / Next Steps
- Practice converting between tangential and angular variables using the provided formulas.
- Review definitions of motion types and variable descriptions.
- Complete assigned homework problems on circular and rotational motion.