Circular motion is all about objects moving in curves. It's like when you swing a ball on a string or drive around a roundabout. The key is that something's always pulling the object toward the center.

This pulling force is called centripetal force. It changes the direction of motion but not the speed. The faster you go or the tighter the turn, the stronger this force needs to be. It's what keeps things spinning instead of flying off.

Uniform Circular Motion

Centripetal acceleration and velocity

• Centripetal acceleration ($a_c$) directed towards center of circular path
  • Always perpendicular to velocity vector
  • Changes direction of object, not speed
• Magnitude of $a_c$ depends on object's speed ($v$) and radius ($r$) of circular path
  • Formula: $a_c = \frac{v^2}{r}$
  • Higher speed increases $a_c$
  • Smaller radius increases $a_c$
• $a_c$ changes only direction of velocity, not magnitude
  • Velocity vector always tangent to circular path (ball on a string, car turning)
• Circular motion occurs when an object moves in a curved path and its velocity changes direction continuously

Forces in uniform circular motion

• Net force acting on object in uniform circular motion is centripetal force ($F_c$)
  • Directed towards center of circular path
  • Formula: $F_c = ma_c = m\frac{v^2}{r}$, where $m$ is object's mass
• Tension force can provide $F_c$
  • Ball attached to string rotating in horizontal circle (tetherball, yo-yo)
    • Tension in string provides $F_c$
• Friction force can provide $F_c$
  • Car making turn on flat road (racetrack, highway ramp)
    • Static friction between tires and road provides $F_c$
• Gravity can provide $F_c$
  • Satellite orbiting Earth (GPS, weather satellites)
    • Gravitational force between satellite and Earth provides $F_c$
• Inertia resists changes in motion, requiring a continuous centripetal force for circular motion

Applications of centripetal equations

• Identify force providing $a_c$
  • Tension, friction, gravity, or combination of forces
• Determine relevant variables
  • Mass of object ($m$)
  • Speed of object ($v$)
  • Radius of circular path ($r$)
• Use appropriate equations to solve for unknown variable
  • Centripetal acceleration: $a_c = \frac{v^2}{r}$
  • Centripetal force: $F_c = ma_c = m\frac{v^2}{r}$
• Example problem: 1000 kg car makes turn with 50 m radius at 15 m/s. Calculate $F_c$ and determine force responsible for circular motion
  1. Given: $m = 1000 \text{ kg}$, $v = 15 \text{ m/s}$, $r = 50 \text{ m}$
  2. Calculate $F_c$: $F_c = m\frac{v^2}{r} = 1000 \text{ kg} \cdot \frac{(15 \text{ m/s})^2}{50 \text{ m}} = 4500 \text{ N}$
  3. Force responsible for circular motion is static friction between tires and road (roller coaster loop, merry-go-round)

Special cases of circular motion

• Banked curves: Angled roads that use both friction and normal force components to provide centripetal force
• Conical pendulum: A weight suspended by a string that moves in a horizontal circle, with tension providing the centripetal force