Uniform Circular Motion – Exam Cram

Key Ideas

• Uniform circular motion (UCM): motion in a circle of constant radius at constant speed
• Velocity magnitude constant; direction changes continuously → acceleration and net force directed toward center (centripetal)
• Change in direction without change in speed possible when force is ⟂ to velocity at every instant

Kinematics

• Tangential (linear) speed: $v = \frac{2\pi r}{T} = 2\pi r f$
• Period–frequency relation: $T = \frac{1}{f},\quad f = \frac{1}{T}$
• Instantaneous velocity vector tangent to path
• Centripetal (radial) acceleration: $a_c = \frac{v^2}{r}$ (points to center)

Dynamics (Forces)

• Required centripetal force: $F<em>c = m a</em>c = \frac{m v^2}{r}$ (direction: inward)
• "Centripetal force" is not a new force type; it is any real force component that supplies $F_c$ (tension, gravity, normal, friction…)
• No outward (centrifugal) force in an inertial frame; object’s inertia makes it move tangentially if $F_c$ vanishes

Common Sources of $F_c$

• Tension: ball on string, merry-go-round chains
• Gravity: planetary & lunar orbits (approx. circular)
• Friction: car on flat curve
• Normal force component: banked roadway or track

Flat Curves & Friction Limits

• Static friction provides $F<em>c$: $F</em>{fr} = \mu<em>s F</em>N \ge \frac{m v^2}{r}$
• If $\mu_s$ insufficient → tires skid; kinetic friction < static and points opposite motion, reducing control

Banked Curves (No Friction Required at Design Speed)

• Horizontal component of normal force supplies $F_c$
• Force balance yields: $\tan \theta = \frac{v^2}{r g}$
  • Design speed met ⇒ friction not needed
• Normal force magnitude: $F_N = \frac{m g}{\cos \theta}$ (greater than weight)

Quick Check Concepts

• Break the string → object moves tangentially (Newton I)
• Increase speed or decrease radius → required $F<em>c$ and $a</em>c$ increase ( $F<em>c \propto v^2,\; a</em>c \propto v^2$ )
• Period, speed, radius interrelated: knowing any two allows full kinematics