Uniform Circular Motion
Angular kinematics (θ, ω), centripetal acceleration (a = v²/r = ω²r), centripetal force, banked roads, conical pendulum, and vertical circular motion.
Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.
Learning Objectives
- Relate linear velocity to angular velocity (v = rω) and centripetal acceleration (a = v²/r).
- Analyze banking angle conditions on curved roads (tan θ = v² / rg) and critical speed in a vertical loop.
Core Theory & Concepts
An object moving along a circular path of radius r at constant speed v experiences continuous inward directional change. This centripetal acceleration requires a real net inward centripetal force to sustain rotation.
Centripetal Acceleration & Force
Acceleration is directed towards the circle center: a_c = v² / r = ω² r. Centripetal force: F_c = m v² / r = m ω² r.
Banking of Curved Roads
To prevent vehicles skidding without relying on tire friction, roads are banked at angle θ: tan θ = v² / (rg).
Common Misconceptions
Treating "centrifugal force" as a real physical force acting on an object in an inertial frame.
Syllabus Topic Outline
- 1Angular displacement θ and angular speed ω = dθ/dt
- 2Centripetal acceleration derivation a_c = v²/r = ω²r
- 3Banked roadway design for friction-free curving: tan θ = v²/rg
- 4Motion in a vertical circle: critical apex speed v_min = √(gr)